lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma AddAction.automorphize_smul_left [AddGroup α] [AddAction α β]  (f : β → M)
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/SubMulAction.lean:lemma AddSubgroupClass.zsmulMemClass {S M : Type*} [SubNegMonoid M] [SetLike S M]
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/SubMulAction.lean:lemma AddSubmonoidClass.nsmulMemClass {S M : Type*} [AddMonoid M] [SetLike S M]	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma AlexandrovDiscrete.sup {t₁ t₂ : TopologicalSpace α} (_ : @AlexandrovDiscrete α t₁)	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma AlgHom.fieldRange_le_normalClosure  (f : K →ₐ[F] L) : f.fieldRange ≤ normalClosure F K L :=
lake-packages/mathlib/Mathlib/CategoryTheory/Noetherian.lean:lemma ArtinianObject.subobject_lt_wellFounded (X : C) [ArtinianObject X] :   
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma BiTotal.rel_exists (h : BiTotal R)
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma BiTotal.rel_forall (h : BiTotal R) :   
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma BijOn.iterate {f : α → α} {s : Set α} (h : BijOn f s s) : ∀ n, BijOn f^[n] s s	 
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma BijOn.perm_inv (hf : BijOn f s s) : BijOn ↑(f⁻¹) s s := hf.symm f.invOn 
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma BijOn.perm_pow : BijOn f s s → ∀ n : ℕ, BijOn (f ^ n) s s := by 
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma BijOn.perm_zpow (hf : BijOn f s s) : ∀ n : ℤ, BijOn (f ^ n) s s 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma BijOn.symm {g : β → α} (h : InvOn f g t s) (hf : BijOn f s t) : BijOn g t s :=	
lake-packages/mathlib/Mathlib/Topology/Bornology/Basic.lean:lemma Bornology.ext (t t' : Bornology α)	   
lake-packages/mathlib/Mathlib/Topology/Bornology/Basic.lean:lemma Bornology.ext_iff (t t' : Bornology α) :	   
lake-packages/mathlib/Mathlib/Topology/Bornology/Basic.lean:lemma Bornology.le_cofinite (α : Type*) [Bornology α] : cobounded α ≤ cofinite :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma C_addPolynomial :	    
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean:lemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R} [CanonicallyOrderedCommSemiring R]	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma Cauchy.mono_uniformSpace {u v : UniformSpace β} {F : Filter β} (huv : u ≤ v)	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Cauchy.pi [Nonempty ι] {l : ∀ i, Filter (α i)} (hl : ∀ i, Cauchy (l i)) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/ConeCategory.lean:lemma Cocone.toCostructuredArrow_comp_proj {F : J ⥤ C} (c : Cocone F) :	    
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/ConeCategory.lean:lemma Cocone.toCostructuredArrow_comp_toOver_comp_forget {F : J ⥤ C} (c : Cocone F) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean:lemma Cofork.IsColimit.π_desc' {s : Cofork f g} (hs : IsColimit s) {W : C} (k : Y ⟶ W)	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean:lemma CokernelCofork.IsColimit.isIso_π {X Y : C} {f : X ⟶ Y} (c : CokernelCofork f)	
lake-packages/mathlib/Mathlib/Algebra/Group/Defs.lean:lemma CommSemigroup.IsLeftCancelMul.toIsCancelMul (G : Type u) [CommSemigroup G]   
lake-packages/mathlib/Mathlib/Algebra/Group/Defs.lean:lemma CommSemigroup.IsLeftCancelMul.toIsRightCancelMul (G : Type u) [CommSemigroup G]	
lake-packages/mathlib/Mathlib/Algebra/Group/Defs.lean:lemma CommSemigroup.IsRightCancelMul.toIsCancelMul (G : Type u) [CommSemigroup G]  
lake-packages/mathlib/Mathlib/Algebra/Group/Defs.lean:lemma CommSemigroup.IsRightCancelMul.toIsLeftCancelMul (G : Type u) [CommSemigroup G]	
lake-packages/mathlib/Mathlib/CategoryTheory/Monad/Basic.lean:lemma ComonadHom.ext' {T₁ T₂ : Comonad C} (f g : T₁ ⟶ T₂) (h : f.app = g.app) : f = g :=	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/Compact.lean:lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α]  
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/Compact.lean:lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α]	
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:lemma CompletePartialOrder.scottContinuous {f : α → β} :   
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma Complex.coe_realPart (z : ℂ) : (ℜ z : ℂ) = z.re := calc	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma Complex.coe_selfAdjointEquiv (z : selfAdjoint ℂ) :  
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma Complex.rank_rat_complex : Module.rank ℚ ℂ = continuum := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/ConeCategory.lean:lemma Cone.toStructuredArrow_comp_proj {F : J ⥤ C} (c : Cone F) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/ConeCategory.lean:lemma Cone.toStructuredArrow_comp_toUnder_comp_forget {F : J ⥤ C} (c : Cone F) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma Connected.adj_union {H K : G.Subgraph}  
lake-packages/mathlib/Mathlib/Topology/DiscreteSubset.lean:lemma Continuous.discrete_of_tendsto_cofinite_cocompact [T1Space X] [WeaklyLocallyCompactSpace Y]	
lake-packages/mathlib/Mathlib/Topology/Basic.lean:lemma ContinuousOn.comp_fract {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma CoordinateRing.C_addPolynomial :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma CoordinateRing.C_addPolynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/Data/Countable/Defs.lean:lemma Countable.exists_injective_nat (α : Sort u) [Countable α] : ∃ f : α → ℕ, Injective f :=	
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma Covby.of_le_of_lt (hac : a ⋖ c) (hab : a ≤ b) (hbc : b < c) : b ⋖ c :=	
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma Covby.of_lt_of_le (hac : a ⋖ c) (hab : a < b) (hbc : b ≤ c) : a ⋖ b :=	
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma DifferentiableAt.lineDeriv_eq_fderiv (hf : DifferentiableAt 𝕜 f x) :	
lake-packages/mathlib/Mathlib/Order/Directed.lean:lemma DirectedOn.is_bot_of_is_min {s : Set α} (hd : DirectedOn (· ≥ ·) s)	
lake-packages/mathlib/Mathlib/Order/Directed.lean:lemma DirectedOn.is_top_of_is_max {s : Set α} (hd : DirectedOn (· ≤ ·) s)	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Discrete.lean:lemma Discrete.monoidal_tensorUnit_as : (𝟙_ (Discrete M)).as = 1 := rfl		
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Star/Multiplier.lean:lemma DoubleCentralizer.ext (𝕜 : Type u) (A : Type v) [NontriviallyNormedField 𝕜]	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma EffectiveEpi.fac {X Y W : C} (f : Y ⟶ X) [EffectiveEpi f]	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma EffectiveEpi.uniq {X Y W : C} (f : Y ⟶ X) [EffectiveEpi f]	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma EffectiveEpiFamily.fac {B W : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B))	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma EffectiveEpiFamily.hom_ext {B W : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B))	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma EffectiveEpiFamily.uniq {B W : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B))	
lake-packages/mathlib/Mathlib/Data/Quot.lean:lemma Equivalence.quot_mk_eq_iff {α : Type _} {r : α → α → Prop} (h : Equivalence r) (x y : α) :	
lake-packages/mathlib/Mathlib/Init/Logic.lean:lemma Equivalence.reflexive {r : β → β → Prop} (h : Equivalence r) : Reflexive r := h.refl	
lake-packages/mathlib/Mathlib/Init/Logic.lean:lemma Equivalence.symmetric {r : β → β → Prop} (h : Equivalence r) : Symmetric r := λ _ _ => h.symm	
lake-packages/mathlib/Mathlib/Init/Logic.lean:lemma Equivalence.transitive {r : β → β → Prop}(h : Equivalence r) : Transitive r :=  
lake-packages/mathlib/Mathlib/Algebra/Homology/Exact.lean:lemma Exact.epi_kernel_lift (h : Exact f g) : Epi (kernel.lift g f h.w) := by		       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma Exact.hasHomology (h : S.Exact) : S.HasHomology := 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma Exact.hasZeroObject (h : S.Exact) : HasZeroObject C :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma Exact.op (h : S.Exact) : S.op.Exact := by		   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma Exact.unop {S : ShortComplex Cᵒᵖ} (h : S.Exact) : S.unop.Exact := by	
lake-packages/mathlib/Mathlib/Topology/ExtremallyDisconnected.lean:lemma ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen [ExtremallyDisconnected A]	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma FactorsThrough.apply_extend {δ} {g : α → γ} (hf : FactorsThrough g f)   
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) : 
lake-packages/mathlib/Mathlib/NumberTheory/FLT/Basic.lean:lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :	
lake-packages/mathlib/Mathlib/Order/Filter/Bases.lean:lemma FilterBasis.ofSets_sets (s : Set (Set α)) :	      
lake-packages/mathlib/Mathlib/Data/Set/Finite.lean:lemma Finite.exists_minimal_wrt' [PartialOrder β] (f : α → β) (s : Set α) (h : (f '' s).Finite)	
lake-packages/mathlib/Mathlib/Data/Set/Finite.lean:lemma Finite.of_forall_not_lt_lt (h : ∀ ⦃x y z : α⦄, x < y → y < z → False) : Finite α := by	
lake-packages/mathlib/Mathlib/LinearAlgebra/FiniteDimensional.lean:lemma FiniteDimensional.exists_mul_eq_one (F : Type*) {K : Type*} [Field F] [Ring K] [IsDomain K]	
lake-packages/mathlib/Mathlib/Analysis/InnerProductSpace/TwoDim.lean:lemma FiniteDimensional.finiteDimensional_of_fact_finrank_eq_two {K V : Type*} [DivisionRing K]	
lake-packages/mathlib/Mathlib/Analysis/Convex/Combination.lean:lemma Finset.centerMass_id_mem_convexHull_of_nonpos (t : Finset E) {w : E → R}	
lake-packages/mathlib/Mathlib/Analysis/Convex/Combination.lean:lemma Finset.centerMass_mem_convexHull_of_nonpos (t : Finset ι) (hw₀ : ∀ i ∈ t, w i ≤ 0)	
lake-packages/mathlib/Mathlib/Analysis/Convex/Combination.lean:lemma Finset.centerMass_of_sum_add_sum_eq_zero {s t : Finset ι}	 
lake-packages/mathlib/Mathlib/Analysis/Convex/Combination.lean:lemma Finset.centerMass_smul_left {c : R'} [Module R' R] [Module R' E] [SMulCommClass R' R R]	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Finset.insert_eq_cons {α : Type*} [DecidableEq α] (a : α) (s : Finset α) (h : a ∉ s) :	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Finset.range_succ' {n nn n' : ℕ} (pn : NormNum.IsNat n nn) (pn' : nn = Nat.succ n') :	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Finset.range_zero' {n : ℕ} (pn : NormNum.IsNat n 0) :  
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Finset.univ_eq_elems {α : Type*} [Fintype α] (elems : Finset α) 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean:lemma FirstObj.ext (z₁ z₂ : FirstObj P R) (h : ∀ (Y : C) (f : Y ⟶ X)		     
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean:lemma Fork.IsLimit.lift_ι' {s : Fork f g} (hs : IsLimit s) {W : C} (k : W ⟶ X) (h : k ≫ f = k ≫ g) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Elements.lean:lemma Functor.Elements.ext {F : C ⥤ Type w} (x y : F.Elements) (h₁ : x.fst = y.fst)   
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma Functor.commShiftIso_eq_ofInduced (a : A) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory.lean:lemma Functor.mapHomotopyCategory_map (F : V ⥤ W) [F.Additive] {c : ComplexShape ι}		 
lake-packages/mathlib/Mathlib/CategoryTheory/Functor/Basic.lean:lemma Functor.map_comp_assoc {C : Type u₁} [Category C] {D : Type u₂}	    
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma FunctorsInverting.ext {W : MorphismProperty C} {F₁ F₂ : FunctorsInverting W D}  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma FunctorsInverting.hom_ext {W : MorphismProperty C} {F₁ F₂ : FunctorsInverting W D}	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Basic.lean:lemma Gleason (X : CompHaus.{u}) :   
lake-packages/mathlib/Mathlib/GroupTheory/Nilpotent.lean:lemma Group.IsNilpotent.nilpotent (G : Type*) [Group G] [IsNilpotent G] :      
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasFDerivAt.hasLineDerivAt (hf : HasFDerivAt f L x) (v : E) :  
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasFDerivWithinAt.hasLineDerivWithinAt (hf : HasFDerivWithinAt f L s x) (v : E) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma HasHomology.mk' (h : S.HomologyData) : HasHomology S :=  
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasLineDerivAt.hasLineDerivWithinAt (hf : HasLineDerivAt 𝕜 f f' x v) (s : Set E) :  
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasLineDerivWithinAt.congr_of_eventuallyEq (hf : HasLineDerivWithinAt 𝕜 f f' s x v)	
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasLineDerivWithinAt.lineDifferentiableWithinAt (hf : HasLineDerivWithinAt 𝕜 f f' s x v) :	
lake-packages/mathlib/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean:lemma HasLineDerivWithinAt.mono (hf : HasLineDerivWithinAt 𝕜 f f' s x v) (hst : t ⊆ s) :  
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma HasPullbackOpenEmbedding : HasPullback f i := 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean:lemma Hom.comp_eq_comp {X Y Z : WalkingMulticospan fst snd}	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean:lemma Hom.comp_eq_comp {X Y Z : WalkingMultispan fst snd}  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/Scheme.lean:lemma Hom.continuous {X Y : Scheme} (f : X ⟶ Y) : Continuous f.1.base := f.1.base.2	
lake-packages/mathlib/Mathlib/CategoryTheory/Monad/Algebra.lean:lemma Hom.ext' (X Y : Algebra T) (f g : X ⟶ Y) (h : f.f = g.f) : f = g := Hom.ext _ _ h	
lake-packages/mathlib/Mathlib/CategoryTheory/Monad/Algebra.lean:lemma Hom.ext' (X Y : Coalgebra G) (f g : X ⟶ Y) (h : f.f = g.f) : f = g := Hom.ext _ _ h	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean:lemma Hom.id_eq_id (X : WalkingMulticospan fst snd) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean:lemma Hom.id_eq_id (X : WalkingMultispan fst snd) : Hom.id X = 𝟙 X := rfl	
lake-packages/mathlib/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean:lemma Hom.toIsometry_injective (V W : QuadraticModuleCat.{v} R) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma HomologyData.exact_iff (h : S.HomologyData) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma HomologyData.exact_iff' (h : S.HomologyData) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma HomologyData.exact_iff_i_p_zero (h : S.HomologyData) :	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma HomologyData.leftRightHomologyComparison'_eq (h : S.HomologyData) :		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma Icc_subset_uIcc : Icc a b ⊆ [[a, b]] := Icc_subset_Icc inf_le_left le_sup_right	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma Icc_subset_uIcc' : Icc b a ⊆ [[a, b]] := Icc_subset_Icc inf_le_right le_sup_left	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/Basic.lean:lemma Ici_eq_singleton_iff_isTop {x : α} : (Ici x = {x}) ↔ IsTop x := by 
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma IndepFun_iff {β γ} [MeasurableSpace Ω] [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma IndepFun_iff_Indep [MeasurableSpace Ω] [mβ : MeasurableSpace β]	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma IndepSet_iff [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω) :	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma IndepSet_iff_Indep [MeasurableSpace Ω] (s t : Set Ω) (μ : Measure Ω) :	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma IndepSets_iff [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω) :	   
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma Indep_iff (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :     
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma Indep_iff_IndepSets (m₁ m₂ : MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω ) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma Inducing.alexandrovDiscrete {f : β → α} (h : Inducing f) : AlexandrovDiscrete β where	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma InfClosed.codirectedOn (hs : InfClosed s) : DirectedOn (· ≥ ·) s := 
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma InfClosed.finsetInf'_mem (hs : InfClosed s) (ht : t.Nonempty) :  
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma InfClosed.finsetInf_mem [OrderTop α] (hs : InfClosed s) (ht : t.Nonempty) :	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma InfClosed.inter (hs : InfClosed s) (ht : InfClosed t) : InfClosed (s ∩ t) :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma InjOn.image_inter {s t u : Set α} (hf : u.InjOn f) (hs : s ⊆ u) (ht : t ⊆ u) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma InjOn.iterate {f : α → α} {s : Set α} (h : InjOn f s) (hf : MapsTo f s s) :	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma Injective.FactorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=		
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma Injective.apply_extend {δ} (hf : Injective f) (F : γ → δ) (g : α → γ) (e' : β → γ) (b : β) :	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma Injective.dite (p : α → Prop) [DecidablePred p]	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma InvOn.comp (hf : InvOn f' f s t) (hg : InvOn g' g t p) (fst : MapsTo f s t)	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma Ioc_subset_uIoc : Ioc a b ⊆ Ι a b := Ioc_subset_Ioc (min_le_left _ _) (le_max_right _ _)	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma Ioc_subset_uIoc' : Ioc a b ⊆ Ι b a := Ioc_subset_Ioc (min_le_right _ _) (le_max_left _ _)	
lake-packages/mathlib/Mathlib/Topology/DiscreteSubset.lean:lemma IsClosed.tendsto_coe_cofinite_iff [T1Space X] [WeaklyLocallyCompactSpace X]	
lake-packages/mathlib/Mathlib/Topology/DiscreteSubset.lean:lemma IsClosed.tendsto_coe_cofinite_of_discreteTopology	       
lake-packages/mathlib/Mathlib/Topology/Constructions.lean:lemma IsClosedMap.restrictPreimage {f : α → β} (hcl : IsClosedMap f) (T : Set β) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean:lemma IsColimit.inl_desc {t : PushoutCocone f g} (ht : IsColimit t) {W : C} (h : Y ⟶ W) (k : Z ⟶ W)	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean:lemma IsColimit.inr_desc {t : PushoutCocone f g} (ht : IsColimit t) {W : C} (h : Y ⟶ W) (k : Z ⟶ W)	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma IsCycle.ne_bot : ∀ {p : G.Walk u u}, p.IsCycle → G ≠ ⊥	 
lake-packages/mathlib/Mathlib/Probability/Kernel/Basic.lean:lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma IsInvertedBy.iff_comp {C₁ C₂ C₃ : Type*} [Category C₁] [Category C₂] [Category C₃]	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma IsInvertedBy.iff_map_subset_isomorphisms (W : MorphismProperty C) (F : C ⥤ D) :	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma IsInvertedBy.isoClosure_iff (W : MorphismProperty C) (F : C ⥤ D) :	    
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma IsInvertedBy.map_iff {C₁ C₂ C₃ : Type*} [Category C₁] [Category C₂] [Category C₃]	
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma IsLUB_of_ScottContinuous {c : Chain α} {f : α → β} (hf : ScottContinuous f) : 
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma IsLUB_range_ωSup (c : Chain α) : IsLUB (Set.range c) (ωSup c) := by			  
lake-packages/mathlib/Mathlib/Algebra/GroupWithZero/Defs.lean:lemma IsLeftCancelMulZero.to_isCancelMulZero [IsLeftCancelMulZero M₀] :	
lake-packages/mathlib/Mathlib/Algebra/GroupWithZero/Defs.lean:lemma IsLeftCancelMulZero.to_isRightCancelMulZero [IsLeftCancelMulZero M₀] :	
lake-packages/mathlib/Mathlib/Algebra/Ring/Basic.lean:lemma IsLeftCancelMulZero.to_noZeroDivisors [Ring α] [IsLeftCancelMulZero α] :	
lake-packages/mathlib/Mathlib/Algebra/Regular/Pow.lean:lemma IsLeftRegular.prod (h : ∀ i ∈ s, IsLeftRegular (f i)) :		
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean:lemma IsLimit.lift_fst {t : PullbackCone f g} (ht : IsLimit t) {W : C} (h : W ⟶ X) (k : W ⟶ Y)	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean:lemma IsLimit.lift_snd {t : PullbackCone f g} (ht : IsLimit t) {W : C} (h : W ⟶ X) (k : W ⟶ Y)	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean:lemma IsLocalizedEquivalence.mk' [CatCommSq Φ.functor L₁ L₂ G] [IsEquivalence G] :	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean:lemma IsLocalizedEquivalence.of_equivalence [IsEquivalence Φ.functor]	      
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean:lemma IsLocalizedEquivalence.of_isLocalization_of_isLocalization	      
lake-packages/mathlib/Mathlib/Algebra/Module/LocalizedModule.lean:lemma IsLocalizedModule.eq_iff_exists [IsLocalizedModule S f] {x₁ x₂} :	
lake-packages/mathlib/Mathlib/Algebra/Module/LocalizedModule.lean:lemma IsLocalizedModule.surj [IsLocalizedModule S f] (y : M') : ∃ x : M × S, x.2 • y = f x.1 :=	
lake-packages/mathlib/Mathlib/Probability/Kernel/Basic.lean:lemma IsMarkovKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]	
lake-packages/mathlib/Mathlib/Geometry/Manifold/PartitionOfUnity.lean:lemma IsOpen.exists_msmooth_support_eq_aux {s : Set H} (hs : IsOpen s) :    
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma IsOpen.exterior_eq (h : IsOpen s) : exterior s = s :=    
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma IsOpen.exterior_subset (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t :=  
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma IsOpen.exterior_subset_iff (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : ¬ p.Nil) : (p.tail hp').IsPath := by	
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma IsPreconnected.induction₂ {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop)	
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma IsPreconnected.induction₂' {s : Set α} (hs : IsPreconnected s) (P : α → α → Prop)	
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.continuous (h : IsProperMap f) : Continuous f := h.toContinuous	
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f := by	    
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.isCompact_preimage (h : IsProperMap f) {K : Set Y} (hK : IsCompact K) :	
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.pi_map {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]      
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.prod_map {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :    
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma IsProperMap.ultrafilter_le_nhds_of_tendsto (h : IsProperMap f) ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄	
lake-packages/mathlib/Mathlib/Algebra/Regular/Pow.lean:lemma IsRegular.prod (h : ∀ i ∈ s, IsRegular (f i)) :  
lake-packages/mathlib/Mathlib/Algebra/GroupWithZero/Defs.lean:lemma IsRightCancelMulZero.to_isCancelMulZero [IsRightCancelMulZero M₀] :	
lake-packages/mathlib/Mathlib/Algebra/GroupWithZero/Defs.lean:lemma IsRightCancelMulZero.to_isLeftCancelMulZero [IsRightCancelMulZero M₀] :	
lake-packages/mathlib/Mathlib/Algebra/Ring/Basic.lean:lemma IsRightCancelMulZero.to_noZeroDivisors [Ring α] [IsRightCancelMulZero α] :	
lake-packages/mathlib/Mathlib/Algebra/Regular/Pow.lean:lemma IsRightRegular.prod (h : ∀ i ∈ s, IsRightRegular (f i)) :		
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma IsSelfAdjoint.coe_realPart {x : A} (hx : IsSelfAdjoint x) :	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma IsSelfAdjoint.imaginaryPart {x : A} (hx : IsSelfAdjoint x) :	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma IsSeparable.prod {β : Type*} [TopologicalSpace β]	 
lake-packages/mathlib/Mathlib/LinearAlgebra/BilinearForm/TensorProduct.lean:lemma IsSymm.baseChange {B₂ : BilinForm R M₂} (hB₂ : B₂.IsSymm) : (B₂.baseChange A).IsSymm :=	
lake-packages/mathlib/Mathlib/LinearAlgebra/BilinearForm/TensorProduct.lean:lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}	
lake-packages/mathlib/Mathlib/Analysis/Asymptotics/Theta.lean:lemma IsTheta.add_isLittleO {f₁ f₂ : α → E'}  
lake-packages/mathlib/Mathlib/Analysis/Asymptotics/Theta.lean:lemma IsTheta.isBigO (h : f =Θ[l] g) : f =O[l] g := h.1	   
lake-packages/mathlib/Mathlib/Analysis/Asymptotics/Theta.lean:lemma IsTheta.isBigO_symm (h : f =Θ[l] g) : g =O[l] f := h.2	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma IsTopologicalBasis.eq_of_forall_subset_iff {t : Set α} (hB : IsTopologicalBasis B)	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma IsTopologicalBasis.subset_of_forall_subset {t : Set α} (hB : IsTopologicalBasis B)	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean:lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :	    
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean:lemma IsTree.existsUnique_path (hG : G.IsTree) : ∀ v w, ∃! p : G.Walk v w, p.IsPath :=		
lake-packages/mathlib/Mathlib/Data/ZMod/Units.lean:lemma IsUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) :  IsUnit ((a : ZMod m) : ZMod n) :=	
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma IsZeckendorfRep.sum_fib_lt : ∀ {n l}, IsZeckendorfRep l → (∀ a ∈ (l ++ [0]).head?, a < n) →	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean:lemma KernelFork.IsLimit.isIso_ι {X Y : C} {f : X ⟶ Y} (c : KernelFork f) 
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma LT.lt.exists_disjoint_Iio_Ioi (h : a < b) :   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma LeftHomologyData.exact_iff [S.HasHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.homologyIso_hom_comp_leftHomologyIso_inv	       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.homologyIso_leftHomologyData [S.HasHomology] : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.homologyπ_comp_homologyIso_hom (h : S.LeftHomologyData) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.leftHomologyIso_hom_comp_homologyIso_inv	       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.leftHomologyIso_hom_naturality 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.leftHomologyIso_inv_naturality 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma LeftHomologyData.liftCycles_comp_cyclesIso_hom :		  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma LeftHomologyData.lift_K_comp_cyclesIso_inv :   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma LeftHomologyData.π_comp_homologyIso_inv (h : S.LeftHomologyData) [S.HasHomology] :       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma LeftHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData}	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma LeftTotal.rel_exists (h : LeftTotal R) : 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma LeftUnique.flip (h : LeftUnique r) : RightUnique (flip r) :=	
lake-packages/mathlib/Mathlib/LinearAlgebra/LinearIndependent.lean:lemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])	    
lake-packages/mathlib/Mathlib/LinearAlgebra/LinearIndependent.lean:lemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])	
lake-packages/mathlib/Mathlib/LinearAlgebra/LinearIndependent.lean:lemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])	
lake-packages/mathlib/Mathlib/LinearAlgebra/LinearIndependent.lean:lemma LinearIndependent.linear_combination_pair_of_det_ne_zero {R M : Type*} [CommRing R]	
lake-packages/mathlib/Mathlib/LinearAlgebra/LinearIndependent.lean:lemma LinearIndependent.pair_iff {x y : M} : 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/ToLin.lean:lemma LinearMap.toMatrix_pow (f : M₁ →ₗ[R] M₁) (k : ℕ) :  
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/lpSpace.lean:lemma LipschitzWith.uniformly_bounded [PseudoMetricSpace α] (g : α → ι → ℝ) {K : ℝ≥0}	 
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma List.range_succ_eq_map' {n nn n' : ℕ} (pn : NormNum.IsNat n nn) (pn' : nn = Nat.succ n') :	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma List.range_zero' {n : ℕ} (pn : NormNum.IsNat n 0) :     
lake-packages/mathlib/Mathlib/MeasureTheory/Function/LpSpace.lean:lemma Lp.coeFn_const : Lp.const p μ c =ᵐ[μ] Function.const α c :=   
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma MapsTo.comp_left (g : β → γ) (hf : MapsTo f s t) : MapsTo (g ∘ f) s (g '' t) :=	 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma MapsTo.comp_right {s : Set β} {t : Set γ} (hg : MapsTo g s t) (f : α → β) :  
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma MapsTo.perm_pow : MapsTo f s s → ∀ n : ℕ, MapsTo (f ^ n) s s := by      
lake-packages/mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean:lemma MeasurableSet.singleton [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) :   
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma MeasureTheory.Measure.eq_of_cdf (μ ν : Measure ℝ) [IsProbabilityMeasure μ]   
lake-packages/mathlib/Mathlib/MeasureTheory/Function/LpSpace.lean:lemma Memℒp.toLp_const : Memℒp.toLp _ (memℒp_const c) = Lp.const p μ c := rfl	
lake-packages/mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean:lemma Modification.comp_app' {F G : OplaxFunctor B C} {α β γ : F ⟶ G} (m : α ⟶ β) (n : β ⟶ γ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean:lemma Modification.id_app' {F G : OplaxFunctor B C} (α : F ⟶ G) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Monad/Basic.lean:lemma MonadHom.ext' {T₁ T₂ : Monad C} (f g : T₁ ⟶ T₂) (h : f.app = g.app) : f = g :=	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformGroup.lean:lemma MonoidHom.tendsto_coe_cofinite_of_discrete [T2Space G] {H : Type*} [Group H] {f : H →* G}	  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma MorphismProperty.top_eq : (⊤ : MorphismProperty C) = fun _ _ _ => True := rfl   
lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma MulAction.automorphize_smul_left [Group α] [MulAction α β] (f : β → M)	       
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Multiset.insert_eq_cons {α : Type*} [DecidableEq α] (a : α) (s : Multiset α) :	   
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Multiset.range_succ' {n nn n' : ℕ} (pn : NormNum.IsNat n nn) (pn' : nn = Nat.succ n') :	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:lemma Multiset.range_zero' {n : ℕ} (pn : NormNum.IsNat n 0) :  
lake-packages/mathlib/Mathlib/Analysis/SpecificLimits/Basic.lean:lemma Nat.tendsto_div_const_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (λ x ↦ x / n) atTop atTop := by	
lake-packages/mathlib/Mathlib/Algebra/Ring/Basic.lean:lemma NoZeroDivisors.to_isDomain [Ring α] [h : Nontrivial α] [NoZeroDivisors α] :	
lake-packages/mathlib/Mathlib/CategoryTheory/Noetherian.lean:lemma NoetherianObject.subobject_gt_wellFounded (X : C) [NoetherianObject X] :	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma Nonempty.exists_eq_singleton_or_nontrivial : s.Nonempty → (∃ a, s = {a}) ∨ s.Nontrivial :=	
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/Projections.lean:lemma P_succ (q : ℕ) : (P (q+1) : K[X] ⟶ K[X]) = P q ≫ (𝟙 _ + Hσ q) := rfl    
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/Projections.lean:lemma P_zero : (P 0 : K[X] ⟶ K[X]) = 𝟙 _ := rfl				      
lake-packages/mathlib/Mathlib/FieldTheory/Perfect.lean:lemma PerfectRing.ofSurjective (R : Type _) (p : ℕ) [CommRing R] [Fact p.Prime] [CharP R p]		
lake-packages/mathlib/Mathlib/FieldTheory/Perfect.lean:lemma PerfectRing.toPerfectField (K : Type _) (p : ℕ)	
lake-packages/mathlib/Mathlib/Topology/Constructions.lean:lemma Pi.continuous_restrict (S : Set ι) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.hom_ext {f : β → C} [HasProduct f] {X : C} (g₁ g₂ : X ⟶ ∏ f)			
lake-packages/mathlib/Mathlib/Topology/Constructions.lean:lemma Pi.induced_precomp [TopologicalSpace β] {ι' : Type*} (φ : ι' → ι) :	
lake-packages/mathlib/Mathlib/Topology/Constructions.lean:lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) : 
lake-packages/mathlib/Mathlib/Topology/Constructions.lean:lemma Pi.induced_restrict (S : Set ι) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_comp_map {f : α → C} {g h : β → C} [HasProduct f] [HasProduct g] [HasProduct h]	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_comp_map' {f : α → C} {g : β → C} {h : γ → C} [HasProduct f] [HasProduct g]	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_comp_π {f : α → C} {g : β → C} [HasProduct f] [HasProduct g] (p : β → α)	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_eq {f : α → C} {g : β → C} [HasProduct f] [HasProduct g] {p p' : β → α}	   
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_id {f g : α → C} [HasProduct f] [HasProduct g] (p : ∀ b, f b ⟶ g b) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map'_id_id {f : α → C} [HasProduct f] : Pi.map' id (fun a => 𝟙 (f a)) = 𝟙 (∏ f) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map_comp_map {f g h : α → C} [HasProduct f] [HasProduct g] [HasProduct h]  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map_comp_map' {f g : α → C} {h : β → C} [HasProduct f] [HasProduct g] [HasProduct h]	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Pi.map_id {f : α → C} [HasProduct f] : Pi.map (fun a => 𝟙 (f a)) = 𝟙 (∏ f) := by	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Pi.uniformContinuous_restrict (S : Set ι) :   
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Pi.uniformSpace_comap_precomp (φ : ι' → ι) :  
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Pi.uniformSpace_comap_precomp' (φ : ι' → ι) : 
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Pi.uniformSpace_comap_restrict (S : Set ι) :  
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma Pi.uniformSpace_eq :			    
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/PosDef.lean:lemma PosDef.eigenvalues_pos [DecidableEq n] [DecidableEq 𝕜] {A : Matrix n n 𝕜}	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/PosDef.lean:lemma PosSemidef.eigenvalues_nonneg [DecidableEq n] [DecidableEq 𝕜] {A : Matrix n n 𝕜}	
lake-packages/mathlib/Mathlib/Tactic/ToAdditive.lean:lemma Pow_lemma ...   
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma PreconnectedSpace.induction₂ [PreconnectedSpace α] (P : α → α → Prop)	
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma PreconnectedSpace.induction₂' [PreconnectedSpace α] (P : α → α → Prop)	
lake-packages/mathlib/Mathlib/Logic/Lemmas.lean:lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=	
lake-packages/mathlib/Mathlib/Logic/Lemmas.lean:lemma Prop.forall {f : Prop → Prop} : (∀ p, f p) ↔ f True ∧ f False :=	
lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma QuotientAddGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean:lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]	
lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma QuotientGroup.automorphize_smul_left (f : G → M) (g : G ⧸ Γ → R) :  
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean:lemma QuotientGroup.integral_eq_integral_automorphize {E : Type*} [NormedAddCommGroup E]	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean:lemma QuotientGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma Real.rank_rat_real : Module.rank ℚ ℝ = continuum := by	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma RespectsIso.isoClosure_eq {P : MorphismProperty C} (hP : P.RespectsIso) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma RightHomologyData.exact_iff [S.HasHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.homologyIso_hom_comp_rightHomologyIso_inv      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.homologyIso_hom_comp_ι			       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.homologyIso_inv_comp_homologyι		       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.homologyIso_rightHomologyData [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma RightHomologyData.opcyclesIso_hom_comp_descQ :	  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma RightHomologyData.opcyclesIso_inv_comp_descOpcycles :	       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.rightHomologyIso_hom_comp_homologyIso_inv      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.rightHomologyIso_hom_naturality	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma RightHomologyData.rightHomologyIso_inv_naturality	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma RightHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma RightTotal.rel_forall (h : RightTotal R) :	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma RingEquiv_coe_eq {X Y : Type _} [CommRing X] [CommRing Y] (e : X ≃+* Y) :	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma RingEquiv_coe_eq {X Y : Type _} [CommSemiring X] [CommSemiring Y] (e : X ≃+* Y) :	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma RingEquiv_coe_eq {X Y : Type _} [Ring X] [Ring Y] (e : X ≃+* Y) :	 
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma RingEquiv_coe_eq {X Y : Type _} [Semiring X] [Semiring Y] (e : X ≃+* Y) :	
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma ScottContinuous.continuous' {f : α → β} (hf : ScottContinuous f) : Continuous' f := by	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma SecondCountableTopology.mk' {b : Set (Set α)} (hc : b.Countable) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean:lemma SecondObj.ext (z₁ z₂ : SecondObj P S) (h : ∀ (Y Z : C) (g : Z ⟶ Y) (f : Y ⟶ X) 
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma Set.Finite.isCompact_exterior (hs : s.Finite) : IsCompact (exterior s) := by	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma Set.Ici_ciSup [Nonempty ι] {f : ι → α} (hf : BddAbove (range f)) :	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma Set.Iic_ciInf [Nonempty ι] {f : ι → α} (hf : BddBelow (range f)) :	
lake-packages/mathlib/Mathlib/Data/Set/Finite.lean:lemma Set.finite_diff_iUnion_Ioo (s : Set α) : (s \ ⋃ (x ∈ s) (y ∈ s), Ioo x y).Finite :=   
lake-packages/mathlib/Mathlib/Data/Set/Finite.lean:lemma Set.finite_diff_iUnion_Ioo' (s : Set α) : (s \ ⋃ x : s × s, Ioo x.1 x.2).Finite := by 
lake-packages/mathlib/Mathlib/Data/Set/Finite.lean:lemma Set.finite_of_forall_not_lt_lt (h : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ s, x < y → y < z → False) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma ShiftMkCore.shiftFunctorAdd_eq (h : ShiftMkCore C A) (a b : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma ShiftMkCore.shiftFunctorZero_eq (h : ShiftMkCore C A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma ShiftMkCore.shiftFunctor_eq (h : ShiftMkCore C A) (a : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma Sieve.generateFamily_eq {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean:lemma Sieve.generateSingleton_eq {X Y : C} (f : Y ⟶ X) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.hom_ext {f : β → C} [HasCoproduct f] {X : C} (g₁ g₂ : ∐ f ⟶ X)		
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map'_comp_map {f : α → C} {g h : β → C} [HasCoproduct f] [HasCoproduct g]	      
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map'_comp_map' {f : α → C} {g : β → C} {h : γ → C} [HasCoproduct f] [HasCoproduct g]	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map'_eq {f : α → C} {g : β → C} [HasCoproduct f] [HasCoproduct g]	   
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map'_id {f g : α → C} [HasCoproduct f] [HasCoproduct g] (p : ∀ b, f b ⟶ g b) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map'_id_id {f : α → C} [HasCoproduct f] :	      
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map_comp_map {f g h : α → C} [HasCoproduct f] [HasCoproduct g] [HasCoproduct h] 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map_comp_map' {f g : α → C} {h : β → C} [HasCoproduct f] [HasCoproduct g]	      
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.map_id {f : α → C} [HasCoproduct f] : Sigma.map (fun a => 𝟙 (f a)) = 𝟙 (∐ f) := by	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma Sigma.openEmbedding_ι {α : Type} [Fintype α] (Z : α → Stonean.{u}) (a : α) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Products.lean:lemma Sigma.ι_comp_map' {f : α → C} {g : β → C} [HasCoproduct f] [HasCoproduct g]	 
lake-packages/mathlib/Mathlib/Algebra/Star/Pointwise.lean:lemma StarMemClass.star_coe_eq {S α : Type*} [InvolutiveStar α] [SetLike S α]	 
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma StrictAnti.not_bddAbove_range [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β]	
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma StrictAnti.not_bddBelow_range [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β]	
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma StrictMono.not_bddAbove_range [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β]	
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma StrictMono.not_bddBelow_range [NoMinOrder α] [PredOrder β] [IsPredArchimedean β]	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma Subgraph.Connected.induce_verts {H : G.Subgraph} (h : H.Connected) :	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformGroup.lean:lemma Subgroup.tendsto_coe_cofinite_of_discrete [T2Space G] (H : Subgroup G) [DiscreteTopology H] :	
lake-packages/mathlib/Mathlib/RingTheory/Jacobson.lean:lemma Subring.mem_closure_image_of {S T : Type*} [CommRing S] [CommRing T] (g : S →+* T)	
lake-packages/mathlib/Mathlib/Order/Filter/Subsingleton.lean:lemma Subsingleton.isCountablyGenerated (hl : l.Subsingleton) : IsCountablyGenerated l := by	
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma SuccOrder.forall_ne_bot_iff   
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma SupClosed.directedOn (hs : SupClosed s) : DirectedOn (· ≤ ·) s :=	  
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma SupClosed.finsetSup'_mem (hs : SupClosed s) (ht : t.Nonempty) :	  
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma SupClosed.finsetSup_mem [OrderBot α] (hs : SupClosed s) (ht : t.Nonempty) :	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma SupClosed.inter (hs : SupClosed s) (ht : SupClosed t) : SupClosed (s ∩ t) :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma SurjOn.comp_left (hf : SurjOn f s t) (g : β → γ) : SurjOn (g ∘ f) s (g '' t) := by	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma SurjOn.comp_right {s : Set β} {t : Set γ} (hf : Surjective f) (hg : SurjOn g s t) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma SurjOn.iterate {f : α → α} {s : Set α} (h : SurjOn f s s) : ∀ n, SurjOn f^[n] s s	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma SurjOn.perm_pow : SurjOn f s s → ∀ n : ℕ, SurjOn (f ^ n) s s := by      
lake-packages/mathlib/Mathlib/Topology/Algebra/Module/Basic.lean:lemma TopologicalSpace.IsSeparable.span {R M : Type*} [AddCommMonoid M] [Semiring R] [Module R M]	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Tor.lean:lemma Tor'_map_app' (n : ℕ) {X Y : C} (f : X ⟶ Y) (Z : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Tor.lean:lemma Tor'_obj_map (n : ℕ) {X Y : C} (Z : C) (f : X ⟶ Y) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Basic.lean:lemma Triangle.eqToHom_hom₁ {A B : Triangle C} (h : A = B) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Basic.lean:lemma Triangle.eqToHom_hom₂ {A B : Triangle C} (h : A = B) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Basic.lean:lemma Triangle.eqToHom_hom₃ {A B : Triangle C} (h : A = B) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Basic.lean:lemma Triangle.hom_ext {A B : Triangle C} (f g : A ⟶ B)	   
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialSet.lean:lemma Truncated.hom_ext {X Y : Truncated n} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=	
lake-packages/mathlib/Mathlib/Data/UInt.lean:lemma UInt16.val_eq_of_lt {a : Nat} : a < UInt16.size -> (ofNat a).val = a := Nat.mod_eq_of_lt	
lake-packages/mathlib/Mathlib/Data/UInt.lean:lemma UInt32.val_eq_of_lt {a : Nat} : a < UInt32.size -> (ofNat a).val = a := Nat.mod_eq_of_lt	
lake-packages/mathlib/Mathlib/Data/UInt.lean:lemma UInt64.val_eq_of_lt {a : Nat} : a < UInt64.size -> (ofNat a).val = a := Nat.mod_eq_of_lt	
lake-packages/mathlib/Mathlib/Data/UInt.lean:lemma UInt8.val_eq_of_lt {a : Nat} : a < UInt8.size -> (ofNat a).val = a := Nat.mod_eq_of_lt	
lake-packages/mathlib/Mathlib/Data/UInt.lean:lemma USize.val_eq_of_lt {a : Nat} : a < USize.size -> (ofNat a).val = a := Nat.mod_eq_of_lt	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformFun.div_apply [Group β] : (f / g : α →ᵤ β) x = f x / g x := Pi.div_apply f g x 
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformFun.inv_apply [Group β] : (f : α →ᵤ β)⁻¹ x = (f x)⁻¹ := Pi.inv_apply f x	    
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformFun.mul_apply [Monoid β] : (f * g : α →ᵤ β) x = f x * g x := Pi.mul_apply f g x	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformFun.one_apply [Monoid β] : (1 : α →ᵤ β) x = 1 := Pi.one_apply x	    
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformOnFun.div_apply [Group β] : (f / g : α →ᵤ[𝔖] β) x = f x / g x := Pi.div_apply f g x	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformOnFun.inv_apply [Group β] : (f : α →ᵤ[𝔖] β)⁻¹ x = (f x)⁻¹ := Pi.inv_apply f x  
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformOnFun.mul_apply [Monoid β] : (f * g : α →ᵤ[𝔖] β) x = f x * g x := Pi.mul_apply f g x	
lake-packages/mathlib/Mathlib/Topology/Algebra/UniformConvergence.lean:lemma UniformOnFun.one_apply [Monoid β] : (1 : α →ᵤ[𝔖] β) x = 1 := Pi.one_apply x   
lake-packages/mathlib/Mathlib/Data/ByteArray.lean:lemma Up.WF (ub) : WellFounded (Up ub) := 
lake-packages/mathlib/Mathlib/Data/ByteArray.lean:lemma Up.next {ub i} (h : i < ub) : Up ub (i+1) i := ⟨Nat.lt_succ_self _, h⟩	    
lake-packages/mathlib/Mathlib/RingTheory/Valuation/ValuationRing.lean:lemma ValuationRing.cond {A : Type u} [CommRing A] [IsDomain A] [ValuationRing A] (a b : A) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma Walk.connected_induce_support {u v : V} (p : G.Walk u v) :	 
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma Wcovby.of_le_of_le (hac : a ⩿ c) (hab : a ≤ b) (hbc : b ≤ c) : b ⩿ c :=	
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma Wcovby.of_le_of_le' (hac : a ⩿ c) (hab : a ≤ b) (hbc : b ≤ c) : a ⩿ b :=	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma WellFounded.prod_lex {ra : α → α → Prop} {rb : β → β → Prop} (ha : WellFounded ra)       
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma XClass_ne_zero [Nontrivial R] : XClass W x ≠ 0 :=	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_hom_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₁ p₂ p₃ : ι}	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_hom_comp_XIsoOfEq_inv (K : HomologicalComplex V c) {p₁ p₂ p₃ : ι}	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_hom_comp_d (K : HomologicalComplex V c) {p₁ p₂ : ι} (h : p₁ = p₂) (p₃ : ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_hom_naturality {K L : HomologicalComplex V c} (φ : K ⟶ L) {n n' : ι} (h : n = n') :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_inv_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₁ p₂ p₃ : ι}	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_inv_comp_XIsoOfEq_inv (K : HomologicalComplex V c) {p₁ p₂ p₃ : ι}	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_inv_comp_d (K : HomologicalComplex V c) {p₂ p₁ : ι} (h : p₂ = p₁) (p₃ : ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_inv_naturality {K L : HomologicalComplex V c} (φ : K ⟶ L) {n n' : ι} (h : n = n') :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma XIsoOfEq_rfl (K : HomologicalComplex V c) (p : ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma XIsoOfEq_shift (K : CochainComplex C ℤ) (n : ℤ) {p q : ℤ} (hpq : p = q) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal'_eq :	  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal_add_eq :		    
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal_eq₁ : XYIdeal W x₁ (C y₁) = XYIdeal W x₁ (linePolynomial x₁ y₁ L) := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal_eq₂ (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal_mul_XYIdeal (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma XYIdeal_neg_mul : XYIdeal W x₁ (C <| W.negY x₁ y₁) * XYIdeal W x₁ (C y₁) = XIdeal W x₁ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma YClass_ne_zero [Nontrivial R] : YClass W y ≠ 0 :=	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma Y_eq_of_X_eq (hx : x₁ = x₂) : y₁ = y₂ ∨ y₁ = W.negY x₂ y₂ := by		     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma Y_eq_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.negY x₂ y₂) : y₁ = y₂ :=		     
lake-packages/mathlib/Mathlib/Algebra/Module/LinearMap.lean:lemma _root_.Function.Injective.injective_linearMapComp_left (hf : Injective f) :  
lake-packages/mathlib/Mathlib/Algebra/Module/LinearMap.lean:lemma _root_.Function.Surjective.injective_linearMapComp_right (hg : Surjective g) :	   
lake-packages/mathlib/Mathlib/Algebra/Lie/Normalizer.lean:lemma _root_.LieIdeal.idealizer_eq_normalizer (I : LieIdeal R L) :  
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Injective.lean:lemma _root_.LinearMap.toSpanSingleton_ker :	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean:lemma _root_.MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map	
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean:lemma _root_.MeasureTheory.NullMeasurableSet.prod {s : Set α} {t : Set β}	
lake-packages/mathlib/Mathlib/Data/Int/Order/Basic.lean:lemma _root_.Nat.cast_natAbs {α : Type*} [AddGroupWithOne α] (n : ℤ) : (n.natAbs : α) = |n| :=	
lake-packages/mathlib/Mathlib/Algebra/Algebra/Unitization.lean:lemma _root_.NonUnitalAlgHom.toAlgHom_zero :  
lake-packages/mathlib/Mathlib/NumberTheory/WellApproximable.lean:lemma _root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}	
lake-packages/mathlib/Mathlib/Data/Set/Semiring.lean:lemma _root_.Set.up_image [MulOneClass α] [MulOneClass β] (f : α →* β) (s : Set α) : 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma _root_.SimpleGraph.Walk.toSubgraph_connected {u v : V} (p : G.Walk u v) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma `comp_assoc` often requires a careful selection of degrees with good definitional	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean:lemma abelianImageToKernel_comp_kernel_ι :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean:lemma abelianImageToKernel_comp_kernel_ι_comp_cokernel_π :	
lake-packages/mathlib/Mathlib/Algebra/Order/AbsoluteValue.lean:lemma abv_add (x y) : abv (x + y) ≤ abv x + abv y := abv_add' x y	
lake-packages/mathlib/Mathlib/Algebra/Order/AbsoluteValue.lean:lemma abv_eq_zero {x} : abv x = 0 ↔ x = 0 := abv_eq_zero'	
lake-packages/mathlib/Mathlib/Algebra/Order/AbsoluteValue.lean:lemma abv_mul (x y) : abv (x * y) = abv x * abv y := abv_mul' x y	
lake-packages/mathlib/Mathlib/Algebra/Order/AbsoluteValue.lean:lemma abv_nonneg (x) : 0 ≤ abv x := abv_nonneg' x	
lake-packages/mathlib/Mathlib/Algebra/Quandle.lean:lemma act_act_self_eq (x y : S) : (x ◃ y) ◃ x = x ◃ y := by	
lake-packages/mathlib/Mathlib/Algebra/Quandle.lean:lemma act_idem (x : S) : (x ◃ x) = x := by rw [←act_one x, ←Shelf.self_distrib, act_one, act_one]	
lake-packages/mathlib/Mathlib/Algebra/Quandle.lean:lemma act_self_act_eq (x y : S) : x ◃ (x ◃ y) = x ◃ y := by			
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma addPolynomial_eq :	     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma addPolynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma add_app (f g : P ⟶ Q) (X : Cᵒᵖ) : (f + g).app X = f.app X + g.app X := rfl	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma add_assoc (P Q R : W.Point) : P + Q + R = P + (Q + R) :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma add_comm (P Q : W.Point) : P + Q = Q + P :=	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma add_def (P Q : W.Point) : P.add Q = P + Q := 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma add_eq_zero (P Q : W.Point) : P + Q = 0 ↔ P = -Q := by	 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma add_hom_app_obj (a b : A) (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma add_inv_app_obj (a b : A) (X : C) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma add_left_neg (P : W.Point) : -P + P = 0 := by 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma add_v {n : ℤ} (z₁ z₂ : Cochain F G n) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma add_zero_inv_app (h : ShiftMkCore C A) (n : A) (X : C) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Preadditive/Mat.lean:lemma additiveObjIsoBiproduct_hom_π (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :	
lake-packages/mathlib/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean:lemma adjoin_induction' {s : Set A} {p : adjoin R s → Prop} (a : adjoin R s)   
lake-packages/mathlib/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean:lemma ae_eq_zero_of_forall_set_integral_isClosed_eq_zero {μ : Measure β} {f : β → E}  
lake-packages/mathlib/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean:lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero   
lake-packages/mathlib/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean:lemma ae_eq_zero_of_forall_set_integral_isCompact_eq_zero'  
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/AEMeasurable.lean:lemma aemeasurable_indicator_const_iff {s} [MeasurableSingletonClass β] (b : β) [NeZero b] :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma alexandrovDiscrete_coinduced {β : Type*} {f : α → β} :   
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma alexandrovDiscrete_iSup {t : ι → TopologicalSpace α} (_ : ∀ i, @AlexandrovDiscrete α (t i)) :	
lake-packages/mathlib/Mathlib/Algebra/Algebra/Unitization.lean:lemma algHom_ext'' {F : Type*} [AlgHomClass F R (Unitization R A) C] {φ ψ : F}	
lake-packages/mathlib/Mathlib/Tactic/ComputeDegree.lean:lemma and returns two lists: the left-over goals of all the applications, followed by the	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma antitone_iff_forall_covby [PartialOrder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma antitone_iff_forall_wcovby [Preorder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean:lemma antitone_rpow_of_base_le_one {b : ℝ} (hb₀ : 0 < b) (hb₁ : b ≤ 1) : 
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma apply {ι : Type*} {p : ι → Type*} {g : α → ∀ x, p x}	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma asIsoHomologyι_inv_comp_homologyι (hg : S.g = 0) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma asIsoHomologyπ_inv_comp_homologyπ (hf : S.f = 0) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Quandle.lean:lemma assoc (x y z : S) : (x ◃ y) ◃ z = x ◃ y ◃ z := by	 
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma assoc : ∀ {X Y Z W : Σi, C i} (f : X ⟶ Y) (g : Y ⟶ Z) (h : Z ⟶ W), (f ≫ g) ≫ h = f ≫ g ≫ h	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma assoc_inv_app (h : ShiftMkCore C A) (m₁ m₂ m₃ : A) (X : C) :	
lake-packages/mathlib/Mathlib/Tactic/CategoryTheory/Coherence.lean:lemma assoc_liftHom {W X Y Z : C} [LiftObj W] [LiftObj X] [LiftObj Y]	
lake-packages/mathlib/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean:lemma associated_sq : associated (R₁ := R₁) sq = LinearMap.toBilin (LinearMap.mul R₁ R₁) :=    
lake-packages/mathlib/Mathlib/Order/Filter/Ultrafilter.lean:lemma atBot_eq_pure_of_isBot [LinearOrder α] {x : α} (hx : IsBot x) :	
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma atTop_eq_generate_of_forall_exists_le [LinearOrder α] {s : Set α} (hs : ∀ x, ∃ y ∈ s, x ≤ y) :	
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma atTop_eq_generate_of_not_bddAbove [LinearOrder α] {s : Set α} (hs : ¬ BddAbove s) :   
lake-packages/mathlib/Mathlib/Order/Filter/Ultrafilter.lean:lemma atTop_eq_pure_of_isTop [LinearOrder α] {x : α} (hx : IsTop x) :	
lake-packages/mathlib/test/norm_cast.lean:lemma b (h g : true) : true ∧ true := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma b_relation : 4 * W.b₈ = W.b₂ * W.b₆ - W.b₄ ^ 2 := by   
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma ball_mem_nhds [TopologicalSpace E] {p : Seminorm 𝕝 E} (hp : Continuous p) {r : ℝ}	
lake-packages/mathlib/test/linarith.lean:lemma bar (x y : Int) (h : 0 ≤ y ∧ 1 ≤ x) : 1 ≤ y + x * x := by linarith [foo h.2]	
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma bar [Category C] [ConcreteCategory C]	
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma bar' {M N K : Mon} {f : M ⟶ N} {g : N ⟶ K} {h : M ⟶ K} (w : f ≫ g = h) (x : M) :	
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma bar'' {M N K : Mon} {f : M ⟶ N} {g : N ⟶ K} {h : M ⟶ K} (w : f ≫ g = h) (x : M) :	
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma bar''' {M N K : Mon} {f : M ⟶ N} {g : N ⟶ K} {h : M ⟶ K} (w : f ≫ g = h) (x : M) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addX :		     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addX_of_baseChange (x₁ x₂ L : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addY :		     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addY' :	    
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addY'_of_baseChange (x₁ x₂ y₁ L : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_addY_of_baseChange (x₁ x₂ y₁ L : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_baseChange : (C.baseChange A).baseChange B = C.baseChange B := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_baseChange : (W.baseChange A).baseChange B = W.baseChange B := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_b₂ : (W.baseChange A).b₂ = algebraMap R A W.b₂ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_b₄ : (W.baseChange A).b₄ = algebraMap R A W.b₄ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_b₆ : (W.baseChange A).b₆ = algebraMap R A W.b₆ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_b₈ : (W.baseChange A).b₈ = algebraMap R A W.b₈ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_comp (C' : VariableChange R) :		
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_c₄ : (W.baseChange A).c₄ = algebraMap R A W.c₄ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_c₆ : (W.baseChange A).c₆ = algebraMap R A W.c₆ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_id : baseChange A (id : VariableChange R) = id := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_injective (h : Function.Injective <| algebraMap R A) :	   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_injective (h : Function.Injective <| algebraMap R A) :	   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_injective (h : Function.Injective <| algebraMap R A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_j : (E.baseChange A).j = algebraMap R A E.j := by  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_negY :	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_negY_of_baseChange (x₁ y₁ : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_self : C.baseChange R = C := 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_self : W.baseChange R = W := by     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_slope :    
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma baseChange_slope_of_baseChange {R : Type u} [CommRing R] (W : WeierstrassCurve R)	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_variableChange (C : VariableChange R) :	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma baseChange_Δ : (W.baseChange A).Δ = algebraMap R A W.Δ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma basis_apply (n : Fin 2) :	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Dimension.lean:lemma basis_finite_of_finite_spans (w : Set M) (hw : w.Finite) (s : span R w = ⊤) {ι : Type w}	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma basis_one : CoordinateRing.basis W 1 = mk W Y := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma basis_zero : CoordinateRing.basis W 0 = 1 := by	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma bddAbove_of_absorbent {p : ι → Seminorm 𝕜 E} {s : Set E} (hs : Absorbent 𝕜 s)	
lake-packages/mathlib/Mathlib/Order/Bounds/Basic.lean:lemma bddAbove_pi {s : Set (∀ a, π a)} :	   
lake-packages/mathlib/Mathlib/Order/Bounds/Basic.lean:lemma bddAbove_range_pi {F : ι → ∀ a, π a} :	   
lake-packages/mathlib/Mathlib/Order/Bounds/Basic.lean:lemma bddBelow_pi {s : Set (∀ a, π a)} :	   
lake-packages/mathlib/Mathlib/Order/Bounds/Basic.lean:lemma bddBelow_range_pi {F : ι → ∀ a, π a} :	   
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma bdd_below_bdd_above_iff_subset_uIcc (s : Set α) :	  
lake-packages/mathlib/Mathlib/LinearAlgebra/BilinearForm.lean:lemma below since the below lemma does not require `V` to be finite dimensional. However,	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma bi_total_eq {α : Type u₁} : Relator.BiTotal (@Eq α) :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn' (h₁ : MapsTo e s t) (h₂ : MapsTo e.symm t s) : BijOn e s t :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_comm {g : β → α} (h : InvOn f g t s) : BijOn f s t ↔ BijOn g t s :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_id (s : Set α) : BijOn id s s := ⟨s.mapsTo_id, s.injOn_id, s.surjOn_id⟩	 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_image : BijOn e s (e '' s) := (e.injective.injOn _).bijOn_image	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_of_subsingleton [Subsingleton α] (f : α → α) (s : Set α) : BijOn f s s :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_of_subsingleton' [Subsingleton α] [Subsingleton β] (f : α → β) 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_swap (ha : a ∈ s) (hb : b ∈ s) : BijOn (swap a b) s s :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma bijOn_symm_image : BijOn e.symm (e '' s) s := e.bijOn_image.symm e.invOn	
lake-packages/mathlib/Mathlib/LinearAlgebra/Dual.lean:lemma bijective_dual_eval [IsReflexive R M] : Bijective $ Dual.eval R M :=   
lake-packages/mathlib/Mathlib/Init/Data/Nat/Bitwise.lean:lemma binaryRec_decreasing (h : n ≠ 0) : div2 n < n := by  
lake-packages/mathlib/Mathlib/RingTheory/WittVector/IsPoly.lean:lemma bind₁_frobenius_poly_wittPolynomial (n : ℕ) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean:lemma biproduct.whiskerEquiv_hom_eq_lift {f : J → C} {g : K → C} (e : J ≃ K)	       
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean:lemma biproduct.whiskerEquiv_inv_eq_lift {f : J → C} {g : K → C} (e : J ≃ K)	       
lake-packages/mathlib/Mathlib/Algebra/Group/Basic.lean:lemma bit0_neg [SubtractionMonoid α] (a : α) : bit0 (-a) = -bit0 a := (neg_add_rev _ _).symm	
lake-packages/mathlib/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean:lemma bound_of_continuous [Nonempty ι] [t : TopologicalSpace E] (hp : WithSeminorms p)	
lake-packages/mathlib/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean:lemma bound_of_continuous_normedSpace (q : Seminorm 𝕜 F)	 
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma bound_of_shell   
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma bound_of_shell_smul	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma bound_of_shell_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι)	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma c_relation : 1728 * W.Δ = W.c₄ ^ 3 - W.c₆ ^ 2 := by    
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean:lemma can be superseded by one that omits the `IsStarNormal` hypothesis. -/			 
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_left_div_gcd (hm : 0 < m) (h : c * a ≡ c * b [MOD m]) :  a ≡ b [MOD m / gcd m c] := by	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_left_div_gcd' (hm : 0 < m) (hcd : c ≡ d [MOD m]) (h : c * a ≡ d * b [MOD m]) :   
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_left_of_coprime (hmc : gcd m c = 1) (h : c * a ≡ c * b [MOD m]) : a ≡ b [MOD m] := by	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_right_div_gcd (hm : 0 < m) (h : a * c ≡ b * c [MOD m]) : a ≡ b [MOD m / gcd m c] := by	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_right_div_gcd' (hm : 0 < m) (hcd : c ≡ d [MOD m]) (h : a * c ≡ b * d [MOD m]) :  
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma cancel_right_of_coprime (hmc : gcd m c = 1) (h : a * c ≡ b * c [MOD m]) : a ≡ b [MOD m] :=	
lake-packages/mathlib/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean:lemma card_commute_odd (hn : Odd n) :  
lake-packages/mathlib/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean:lemma card_conjClasses_odd (hn : Odd n) :	
lake-packages/mathlib/Mathlib/SetTheory/Ordinal/Basic.lean:lemma card_le_of_le_ord {o : Ordinal} {c : Cardinal} (ho : o ≤ c.ord) :	   
lake-packages/mathlib/Mathlib/Topology/Algebra/Module/Cardinality.lean:lemma cardinal_eq_of_mem_nhds_zero   
lake-packages/mathlib/Mathlib/CategoryTheory/Action.lean:lemma cases' ⦃a' b' : ActionCategory G X⦄ (f : a' ⟶ b') : 
lake-packages/mathlib/Mathlib/Data/ZMod/Basic.lean:lemma castHom_comp {m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :	
lake-packages/mathlib/Mathlib/Data/ZMod/Basic.lean:lemma castHom_self : ZMod.castHom dvd_rfl (ZMod n) = RingHom.id (ZMod n) :=	
lake-packages/mathlib/Mathlib/Data/Int/Basic.lean:lemma cast_Nat_cast [AddGroupWithOne R] : (Int.cast (Nat.cast n) : R) = Nat.cast n :=	
lake-packages/mathlib/Mathlib/Data/Int/Basic.lean:lemma cast_eq_cast_iff_Nat (m n : ℕ) : (m : ℤ) = (n : ℤ) ↔ m = n := ofNat_inj	 
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_comap_uniformSpace {u : UniformSpace β} {f : α → β} {l : Filter α} :	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_iInf_uniformSpace {ι : Sort*} [Nonempty ι] {u : ι → UniformSpace β}	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_iInf_uniformSpace' {ι : Sort*} {u : ι → UniformSpace β}	      
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_iff_le {l : Filter α} [hl : l.NeBot] :   
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_inf_uniformSpace {u v : UniformSpace β} {F : Filter β} :	  
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma cauchy_pi_iff [Nonempty ι] {l : Filter (∀ i, α i)} :	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Pi.lean:lemma cauchy_pi_iff' {l : Filter (∀ i, α i)} [l.NeBot] : 
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Cauchy.lean:lemma cauchy_prod_iff [UniformSpace β] {F : Filter (α × β)} : 
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma cdf_eq_toReal [IsProbabilityMeasure μ] (x : ℝ) : cdf μ x = (μ (Iic x)).toReal := by   
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma cdf_le_one (x : ℝ) : cdf μ x ≤ 1 := condCdf_le_one _ _ _			    
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma cdf_measure_stieltjesFunction (f : StieltjesFunction) (hf0 : Tendsto f atBot (𝓝 0))	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma cdf_nonneg (x : ℝ) : 0 ≤ cdf μ x := condCdf_nonneg _ _ _			    
lake-packages/mathlib/Mathlib/Data/List/Chain.lean:lemma chain'_join : ∀ {L : List (List α)}, [] ∉ L →	 
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_def {m : ℕ} (hm : n ∣ m) :		
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_def' {m : ℕ} (hm : n ∣ m) :		
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_eq_cast_of_dvd {m : ℕ} (hm : n ∣ m) (a : Units (ZMod m)) :	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_self : changeLevel (dvd_refl n) χ = χ := by	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_self_toUnitHom : (changeLevel (dvd_refl n) χ).toUnitHom = χ.toUnitHom := by	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma changeLevel_trans {m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :		  
lake-packages/mathlib/Mathlib/Geometry/Manifold/ChartedSpace.lean:lemma chart_mem_atlas (H : Type*) {M : Type*} [TopologicalSpace H] [TopologicalSpace M]		
lake-packages/mathlib/Mathlib/Testing/SlimCheck/Gen.lean:lemma chooseNatLt_aux {lo hi : Nat} (a : Nat) (h : Nat.succ lo ≤ a ∧ a ≤ hi) :	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma ciInf_eq_ite {p : Prop} [Decidable p] {f : p → α} :	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma ciInf_neg {p : Prop} {f : p → α} (hp : ¬ p) :  
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma ciSup_eq_ite {p : Prop} [Decidable p] {f : p → α} :	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma ciSup_neg {p : Prop} {f : p → α} (hp : ¬ p) :  
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma closure_eq_upperClosure {s : Set α} : closure s = upperClosure s :=      
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma closure_iUnion (f : ι → Set α) : closure (⋃ i, f i) = ⋃ i, closure (f i) :=  
lake-packages/mathlib/Mathlib/Topology/Instances/Real.lean:lemma closure_of_rat_image_le_eq {q : ℚ} : closure ((coe : ℚ → ℝ) '' {x | q ≤ x}) = {r | ↑q ≤ r} :=	
lake-packages/mathlib/Mathlib/Topology/Instances/Real.lean:lemma closure_of_rat_image_le_le_eq {a b : ℚ} (hab : a ≤ b) :  
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma closure_sUnion (S : Set (Set α)) : closure (⋃₀ S) = ⋃ s ∈ S, closure s := by 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma cochain_ofHom_homOf_eq_coe (z : Cocycle F G 0) :	     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coeBaseChange_Δ' : ↑(E.baseChange A).Δ' = algebraMap R A E.Δ' :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma coeSubgraph_Adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma coe_add (z₁ z₂ : Cocycle F G n) :	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coe_basis : (CoordinateRing.basis W : Fin 2 → W.CoordinateRing) = ![1, mk W Y] := by	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma coe_comp {X Y Z : CommGroupCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma coe_comp {X Y Z : CommMonCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_comp {X Y Z : CommRingCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_comp {X Y Z : CommSemiRingCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma coe_comp {X Y Z : GroupCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupWithZeroCat.lean:lemma coe_comp {X Y Z : GroupWithZeroCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma coe_comp {X Y Z : MonCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_comp {X Y Z : RingCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_comp {X Y Z : SemiRingCat} {f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g : X → Z) = g ∘ f := rfl	
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma coe_iSup : BddAbove (range f) → ↑(⨆ i, f i) = ⨆ i, (f i : ℕ∞) := WithTop.coe_iSup _	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma coe_id {X : CommGroupCat} : (𝟙 X : X → X) = id := rfl  
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma coe_id {X : CommMonCat} : (𝟙 X : X → X) = id := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_id {X : CommRingCat} : (𝟙 X : X → X) = id := rfl	     
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_id {X : CommSemiRingCat} : (𝟙 X : X → X) = id := rfl 
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma coe_id {X : GroupCat} : (𝟙 X : X → X) = id := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupWithZeroCat.lean:lemma coe_id {X : GroupWithZeroCat} : (𝟙 X : X → X) = id := rfl	  
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma coe_id {X : MonCat} : (𝟙 X : X → X) = id := rfl  
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_id {X : RingCat} : (𝟙 X : X → X) = id := rfl	     
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma coe_id {X : SemiRingCat} : (𝟙 X : X → X) = id := rfl  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coe_inv_baseChange_Δ' : ↑(E.baseChange A).Δ'⁻¹ = algebraMap R A ↑E.Δ'⁻¹ :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coe_inv_variableChange_Δ' :		  
lake-packages/mathlib/Mathlib/Data/List/MinMax.lean:lemma coe_maximum_of_length_pos (h : 0 < l.length) :	   
lake-packages/mathlib/Mathlib/Data/List/MinMax.lean:lemma coe_minimum_of_length_pos (h : 0 < l.length) :	   
lake-packages/mathlib/Mathlib/Algebra/Star/Subalgebra.lean:lemma coe_mk (S : Subalgebra R A) (h) : ((⟨S, h⟩ : StarSubalgebra R A) : Set A) = S := rfl  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma coe_neg (z : Cocycle F G n) :   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coe_norm_smul_basis (p q : R[X]) :	  
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma coe_ofLE (x : RelSeries r) {s : Rel α α} (h : r ≤ s) : 
lake-packages/mathlib/Mathlib/Data/Finsupp/Interval.lean:lemma coe_rangeIcc (f g : ι →₀ α) : rangeIcc f g i = Icc (f i) (g i) := rfl	
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma coe_sInf : s.Nonempty → ↑(sInf s) = ⨅ a ∈ s, (a : ℕ∞) := WithTop.coe_sInf			
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma coe_sSup : BddAbove s → ↑(sSup s) = ⨆ a ∈ s, (a : ℕ∞) := WithTop.coe_sSup			
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma coe_sub (z₁ z₂ : Cocycle F G n) :	 
lake-packages/mathlib/Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean:lemma coe_toMonoidHom [CommMonoid R] (χ : MulChar R R')	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma coe_variableChange_Δ' : (↑(E.variableChange C).Δ' : R) = (↑C.u⁻¹ : R) ^ 12 * E.Δ' := by	
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma coe_withBot (f : LatticeHom α β) : ⇑f.withBot = Option.map f := rfl      
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma coe_withTop (f : LatticeHom α β) : ⇑f.withTop = WithTop.map f := rfl	
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma coe_withTopWithBot (f : LatticeHom α β) : ⇑f.withTopWithBot = Option.map (Option.map f) := rfl	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma coe_zero : (↑(0 : Cocycle F G n) : Cochain F G n) = 0 := by rfl  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma coe_zsmul (z : Cocycle F G n) (x : ℤ) :	
lake-packages/mathlib/Mathlib/Data/Polynomial/Basic.lean:lemma coeff_C_succ {r : R} {n : ℕ} : coeff (C r) (n + 1) = 0 := by simp [coeff_C]	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean:lemma coevaluation_evaluation :	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean:lemma cokernel_π_comp_cokernelToAbelianCoimage :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean:lemma colimit_ι_zero_cokernel_desc {C : Type*} [Category C]  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma comap_apply (f : V ≃ W) (G : SimpleGraph W) (v : V) :	
lake-packages/mathlib/Mathlib/Topology/LocallyConstant/Basic.lean:lemma comap_injective (f : X → Y) (hf: Continuous f) (hfs : f.Surjective) :	
lake-packages/mathlib/Mathlib/RingTheory/Ideal/QuotientOperations.lean:lemma comap_map_mk {I J : Ideal R} (h : I ≤ J) :     
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma comap_symm_apply (f : V ≃ W) (G : SimpleGraph W) (w : W) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma comm (h : HomologyMapData φ h₁ h₂) :  
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:lemma commProb_cons (n : ℕ) (l : List ℕ) : 
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:lemma commProb_nil : commProb (Product []) = 1 := by 
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:lemma commProb_odd {n : ℕ} (hn : Odd n) :      
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma commShiftIso_add (a b : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma commShiftIso_add' {a b c : A} (h : a + b = c) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma commShiftIso_hom_naturality {X Y : C} (f : X ⟶ Y) (a : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma commShiftIso_inv_naturality {X Y : C} (f : X ⟶ Y) (a : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma commShiftIso_zero :	  
lake-packages/mathlib/Mathlib/Algebra/Star/SelfAdjoint.lean:lemma commute_iff {R : Type*} [Mul R] [StarMul R] {x y : R}	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma comp (h : EventuallyConst f l) (g : β → γ) : EventuallyConst (g ∘ f) l := h.map g	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Composition.lean:lemma comp [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] 
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/CommMon_.lean:lemma comp' {A₁ A₂ A₃ : CommMon_ C} (f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient.lean:lemma compClosure_iff_self [h : Congruence r] {X Y : C} (f g : X ⟶ Y) :		
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma comp_app (f : P ⟶ Q) (g : Q ⟶ T) (X : Cᵒᵖ) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma comp_assoc (C C' C'' : VariableChange R) : comp (comp C C') C'' = comp C (comp C' C'') := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_assoc {n₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ}   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_assoc_of_first_is_zero_cochain {n₂ n₃ n₂₃ : ℤ}	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_assoc_of_second_degree_eq_neg_third_degree {n₁ n₂ n₁₂ : ℤ}	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_assoc_of_second_is_zero_cochain {n₁ n₃ n₁₃ : ℤ}	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_assoc_of_third_is_zero_cochain {n₁ n₂ n₁₂ : ℤ}	 
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma comp_def (i : I) (X Y Z : C i) (f : X ⟶ Y) (g : Y ⟶ Z) : comp (mk f) (mk g) = mk (f ≫ g) :=	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma comp_def {X Y Z : CommGroupCat} {f : X ⟶ Y} {g : Y ⟶ Z} : f ≫ g = g.comp f := rfl	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma comp_def {X Y Z : GroupCat} {f : X ⟶ Y} {g : Y ⟶ Z} : f ≫ g = g.comp f := rfl 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma comp_homologyMap_comp [S₁.HasHomology] [S₂.HasHomology] (φ : S₁ ⟶ S₂)	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma comp_id (C : VariableChange R) : comp C id = C := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma comp_id : ∀ {X Y : Σi, C i} (f : X ⟶ Y), f ≫ 𝟙 Y = f  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma comp_left_inv (C : VariableChange R) : comp (inv C) C = id := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma comp_liftCycles {A' : C} (α : A' ⟶ A) :	   
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma comp_mem (W : MorphismProperty C) {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (hf : W f) (hg : W g)	
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient.lean:lemma comp_natTransLift {F G H : Quotient r ⥤ D}	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma comp_tendsto {lb : Filter β} {g : β → γ} (hg : EventuallyConst g lb)    
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_v {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) (h : n₁ + n₂ = n₁₂)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma comp_zero_cochain_v (z₁ : Cochain F G n) (z₂ : Cochain G K 0) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/Order/Compare.lean:lemma compares_eq [LT α] (a b : α) : Compares eq a b = (a = b) := rfl	
lake-packages/mathlib/Mathlib/Order/Compare.lean:lemma compares_gt [LT α] (a b : α) : Compares gt a b = (a > b) := rfl	
lake-packages/mathlib/Mathlib/Order/Compare.lean:lemma compares_lt [LT α] (a b : α) : Compares lt a b = (a < b) := rfl	
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean:lemma compatibility_N₁_N₂ : toKaroubi (SimplicialObject C) ⋙ N₂ = N₁ :=		
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean:lemma compatibility_Γ₂N₁_Γ₂N₂_hom_app (X : SimplicialObject C) :	
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean:lemma compatibility_Γ₂N₁_Γ₂N₂_inv_app (X : SimplicialObject C) :	
lake-packages/mathlib/Mathlib/Data/Set/Prod.lean:lemma compl_prod_eq_union {α β : Type*} (s : Set α) (t : Set β) : 
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by    
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma complete_distinguished_triangle_morphism₁ (T₁ T₂ : Triangle C)	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma complete_distinguished_triangle_morphism₂ (T₁ T₂ : Triangle C)	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma comp₂ {g : α → γ} (hf : EventuallyConst f l) (op : β → γ → δ) (hg : EventuallyConst g l) :	
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma concaveOn_rpow {p : ℝ} (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1) :  
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma concaveOn_rpow {p : ℝ} (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1) :  
lake-packages/mathlib/Mathlib/Probability/Kernel/Condexp.lean:lemma condexpKernel_ae_eq_condexp [IsFiniteMeasure μ]  
lake-packages/mathlib/Mathlib/Probability/Kernel/Condexp.lean:lemma condexpKernel_ae_eq_condexp' [IsFiniteMeasure μ] {s : Set Ω} (hs : MeasurableSet s) :	
lake-packages/mathlib/Mathlib/Probability/Kernel/Condexp.lean:lemma condexpKernel_ae_eq_trim_condexp [IsFiniteMeasure μ]	
lake-packages/mathlib/Mathlib/Probability/Kernel/Condexp.lean:lemma condexpKernel_apply_eq_condDistrib :				  
lake-packages/mathlib/Mathlib/Data/String/Lemmas.lean:lemma congr_append : ∀ (a b : String), a ++ b = String.mk (a.data ++ b.data)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma congr_left_φH {γ₁ γ₂ : HomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma congr_v {z₁ z₂ : Cochain F G n} (h : z₁ = z₂) (p q : ℤ) (hpq : p + n = q) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma congr_φH {γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φH = γ₂.φH := by rw [eq]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma congr_φH {γ₁ γ₂ : RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φH = γ₂.φH := by rw [eq]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma congr_φK {γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φK = γ₂.φK := by rw [eq]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma congr_φQ {γ₁ γ₂ : RightHomologyMapData φ h₁ h₂} (eq : γ₁ = γ₂) : γ₁.φQ = γ₂.φQ := by rw [eq]	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma conjTranspose_fromColumns_eq_fromRows_conjTranspose (A₁ : Matrix m n₁ R)	    
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma conjTranspose_fromRows_eq_fromColumns_conjTranspose (A₁ : Matrix m₁ n R)	    
lake-packages/mathlib/Mathlib/Data/Matrix/Invertible.lean:lemma conjTranspose_invOf [Invertible A] [Invertible Aᴴ] : (⅟A)ᴴ = ⅟(Aᴴ) := star_invOf _	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma conjTranspose_mul_self_eq_zero {A : Matrix m n R} : Aᴴ * A = 0 ↔ A = 0 :=	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma conjTranspose_mul_self_mulVec_eq_zero (A : Matrix m n R) (v : n → R) :   
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) : 
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma connectedSpace_iff_univ : ConnectedSpace α ↔ IsConnected (univ : Set α) :=    
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma connected_iff_exists_forall_reachable : G.Connected ↔ ∃ v, ∀ w, G.Reachable v w := by	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma connected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma connected_induce_iff : (G.induce s).Connected ↔ ((⊤ : G.Subgraph).induce s).Connected := by	
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Path.lean:lemma cons_ne_nil (p : Path a b) (e : b ⟶ a) : p.cons e ≠ Path.nil :=	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/FiniteMeasure.lean:lemma continuous_map {f : Ω → Ω'} (f_cont : Continuous f) : 
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean:lemma continuous_map {f : Ω → Ω'} (f_cont : Continuous f) : 
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma continuous_of_Ici [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Ici a))) :	
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :		
lake-packages/mathlib/Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean:lemma contractRight_contractLeft (x : CliffordAlgebra Q) : (d ⌋ x) ⌊ d' = d ⌋ (x ⌊ d') :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma contractible_distinguished₁ (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma contractible_distinguished₂ (X : C) :	
lake-packages/mathlib/Mathlib/Data/SetLike/Basic.lean:lemma copy_eq (p : MySubobject X) (s : Set X) (hs : s = ↑p) : p.copy s hs = p :=	
lake-packages/mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean:lemma copy_eq {m : MeasurableSpace α} {p : Set α → Prop} (h : ∀ s, p s ↔ MeasurableSet[m] s) :	
lake-packages/mathlib/Mathlib/Data/Finset/Interval.lean:lemma covby_cons {i : α} (hi : i ∉ s) : s ⋖ cons i s hi :=	
lake-packages/mathlib/Mathlib/Data/Finset/Interval.lean:lemma covby_iff : s ⋖ t ↔ ∃ i : α, ∃ hi : i ∉ s, t = cons i s hi := by	
lake-packages/mathlib/Mathlib/Data/Finset/Interval.lean:lemma covby_iff' : s ⋖ t ↔ ∃ i : α, i ∉ s ∧ t = insert i s := by 
lake-packages/mathlib/Mathlib/Data/Finset/Interval.lean:lemma covby_insert {i : α} (hi : i ∉ s) : s ⋖ insert i s := by	 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma coyoneda_exact₁ {X : C} (f : X ⟶ T.obj₁⟦(1 : ℤ)⟧) (hf : f ≫ T.mor₁⟦1⟧' = 0) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma coyoneda_exact₂ {X : C} (f : X ⟶ T.obj₂) (hf : f ≫ T.mor₂ = 0) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma coyoneda_exact₃ {X : C} (f : X ⟶ T.obj₃) (hf : f ≫ T.mor₃ = 0) :	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIsoLeftHomology_hom_inv_id (hf : S.f = 0) :		 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIsoLeftHomology_inv_hom_id (hf : S.f = 0) :		 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIsoX₂_hom_inv_id (hg : S.g = 0) :	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIsoX₂_inv_hom_id (hg : S.g = 0) :	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIso_hom_comp_i : h.cyclesIso.hom ≫ h.i = S.iCycles := by		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesIso_inv_comp_iCycles : h.cyclesIso.inv ≫ S.iCycles = h.i := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap'_add :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃) 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap'_eq : cyclesMap' φ h₁ h₂ = γ.φK :=   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma cyclesMap'_eq : cyclesMap' φ h₁.left h₂.left = γ.left.φK :=     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap'_i : cyclesMap' φ h₁ h₂ ≫ h₂.i = h₁.i ≫ φ.τ₂ :=	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap'_id (h : S.LeftHomologyData) :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap'_neg :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap'_sub :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap'_zero (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :	  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap_add : cyclesMap (φ + φ') = cyclesMap φ + cyclesMap φ' :=  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_comm [S₁.HasLeftHomology] [S₂.HasLeftHomology] :     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_comp [HasLeftHomology S₁] [HasLeftHomology S₂] [HasLeftHomology S₃]   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_eq [S₁.HasLeftHomology] [S₂.HasLeftHomology] :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_i : cyclesMap φ ≫ S₂.iCycles = S₁.iCycles ≫ φ.τ₂ :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_id [HasLeftHomology S] :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap_neg : cyclesMap (-φ) = -cyclesMap φ :=		     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma cyclesMap_sub : cyclesMap (φ - φ') = cyclesMap φ - cyclesMap φ' := 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cyclesMap_zero [HasLeftHomology S₁] [HasLeftHomology S₂] : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cycles_ext (f₁ f₂ : A ⟶ S.cycles) (h : f₁ ≫ S.iCycles = f₂ ≫ S.iCycles) : f₁ = f₂ := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma cycles_ext_iff (f₁ f₂ : A ⟶ S.cycles) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma d_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₂ = p₃) (p₁ : ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma d_comp_XIsoOfEq_inv (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₃ = p₂) (p₁ : ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma d_comp_ofHom_v (φ : F ⟶ G) (p' p q  : ℤ) (hpq : p + 0 = q) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma d_comp_ofHoms_v (ψ : ∀ (p : ℤ), F.X p ⟶ G.X p) (p' p q  : ℤ) (hpq : p + 0 = q) :	
lake-packages/mathlib/Mathlib/Algebra/DirectSum/Decomposition.lean:lemma decomposeAddEquiv_apply (a : M) :			  
lake-packages/mathlib/Mathlib/Algebra/DirectSum/Decomposition.lean:lemma decomposeAddEquiv_symm_apply (a : ⨁ i, ℳ i) :	  
lake-packages/mathlib/Archive/Wiedijk100Theorems/Konigsberg.lean:lemma degree_eq_degree (v : Verts) : graph.degree v = degree v := by cases v <;> rfl	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma degree_norm_ne_one [IsDomain R] (x : W.CoordinateRing) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma degree_norm_smul_basis [IsDomain R] (p q : R[X]) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma degree_polynomial [Nontrivial R] : W.polynomial.degree = 2 := by	
lake-packages/mathlib/Mathlib/Topology/Instances/EReal.lean:lemma denseRange_ratCast : DenseRange (fun r : ℚ ↦ ((r : ℝ) : EReal)) :=   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma derivative_addPolynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma descOpcycles_comp {A' : C} (α : A ⟶ A') :	    
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma descUniq_hom_app (q : (Σi, C i) ⥤ D) (h : ∀ i, incl i ⋙ q ≅ F i) (i : I) (X : C i) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma descUniq_inv_app (q : (Σi, C i) ⥤ D) (h : ∀ i, incl i ⋙ q ≅ F i) (i : I) (X : C i) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma desc_map_mk {i : I} (X Y : C i) (f : X ⟶ Y) : (desc F).map (SigmaHom.mk f) = (F i).map f :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Opposites.lean:lemma desc_op_comp_opCoproductIsoProduct_hom {X : C} (π : (a : α) → Z a ⟶ X) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean:lemma det_ne_zero_of_sum_col_pos [DecidableEq n] {S : Type*} [LinearOrderedCommRing S]	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean:lemma det_ne_zero_of_sum_row_pos [DecidableEq n] {S : Type*} [LinearOrderedCommRing S]	
lake-packages/mathlib/Mathlib/Data/Set/Prod.lean:lemma diagonal_nonempty [Nonempty α] : (diagonal α).Nonempty :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma diff_v (p q : ℤ) (hpq : p + 1 = q) : (diff K).v p q hpq = K.d p q := rfl	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean:lemma differentiableAt_rpow_const_of_ne (p : ℝ) {x : ℝ} (hx : x ≠ 0) :	       
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean:lemma differentiableOn_rpow_const (p : ℝ) :  
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma directed : Directed (· ≤ ·) c := directedOn_range.2 c.isChain_range.directedOn		  
lake-packages/mathlib/Mathlib/Algebra/QuadraticDiscriminant.lean:lemma discrim_eq_sq_of_quadratic_eq_zero {x : R} (h : a * x * x + b * x + c = 0) :  
lake-packages/mathlib/Mathlib/Algebra/QuadraticDiscriminant.lean:lemma discrim_le_zero_of_nonpos (h : ∀ x : K, a * x * x + b * x + c ≤ 0) : discrim a b c ≤ 0 :=	
lake-packages/mathlib/Mathlib/Algebra/QuadraticDiscriminant.lean:lemma discrim_lt_zero_of_neg (ha : a ≠ 0) (h : ∀ x : K, a * x * x + b * x + c < 0) : 
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_iff_forall_ne : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ t → a ≠ b := by		
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_of_subset (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂) : Disjoint s₁ t₁ :=	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_of_subset_left (h : s ⊆ u) (d : Disjoint u t) : Disjoint s t := d.mono_left h	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_of_subset_right (h : t ⊆ u) (d : Disjoint s u) : Disjoint s t := d.mono_right h	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_or_nonempty_inter (s t : Set α) : Disjoint s t ∨ (s ∩ t).Nonempty :=		
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_sdiff_left : Disjoint (t \ s) s := disjoint_sdiff_self_left    
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_sdiff_right : Disjoint s (t \ s) := disjoint_sdiff_self_right  
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_singleton : Disjoint ({a} : Set α) {b} ↔ a ≠ b :=	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_singleton_left : Disjoint {a} s ↔ a ∉ s := by simp [Set.disjoint_iff, subset_def]	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_singleton_right : Disjoint s {a} ↔ a ∉ s :=   
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_union_left : Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u := disjoint_sup_left	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma disjoint_union_right : Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u := disjoint_sup_right	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Unitization.lean:lemma dist_inr (a b : A) : dist (a : Unitization 𝕜 A) (b : Unitization 𝕜 A) = dist a b :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma distinguished_cocone_triangle₁ {Y Z : C} (g : Y ⟶ Z) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma distinguished_cocone_triangle₂ {Z X : C} (h : Z ⟶ X⟦(1 : ℤ)⟧) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma distinguished_iff_of_iso {T₁ T₂ : Triangle C} (e : T₁ ≅ T₂) :	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Injective.lean:lemma divBy_self (n : ℕ) : divBy n n = 0 := by	
lake-packages/mathlib/Mathlib/NumberTheory/Divisors.lean:lemma divisors_filter_dvd_of_dvd {n m : ℕ} (hn : n ≠ 0) (hm : m ∣ n) :	
lake-packages/mathlib/Mathlib/Data/FunLike/Embedding.lean:lemma do_something {F : Type*} [MyEmbeddingClass F A B] (f : F) : sorry := sorry	 
lake-packages/mathlib/Mathlib/Data/FunLike/Basic.lean:lemma do_something {F : Type*} [MyHomClass F A B] (f : F) : sorry := sorry	   
lake-packages/mathlib/Mathlib/Data/FunLike/Equiv.lean:lemma do_something {F : Type*} [MyIsoClass F A B] (f : F) : sorry := sorry	   
lake-packages/mathlib/Mathlib/Data/Set/Semiring.lean:lemma down_imageHom [MulOneClass α] [MulOneClass β] (f : α →* β) (s : SetSemiring α) :	      
lake-packages/mathlib/Mathlib/Data/List/Infix.lean:lemma dropSlice_sublist (n m : ℕ) (l : List α) : l.dropSlice n m <+ l :=			
lake-packages/mathlib/Mathlib/Data/List/Infix.lean:lemma dropSlice_subset (n m : ℕ) (l : List α) : l.dropSlice n m ⊆ l :=				
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma dvd {d : ℕ} (h : FactorsThrough χ d) : d ∣ n := h.1  
lake-packages/mathlib/Mathlib/Data/Nat/Choose/Dvd.lean:lemma dvd_choose (hp : Prime p) (ha : a < p) (hab : b - a < p) (h : p ≤ b) : p ∣ choose b a :=	
lake-packages/mathlib/Mathlib/Data/Nat/Choose/Dvd.lean:lemma dvd_choose_self (hp : Prime p) (hk : k ≠ 0) (hkp : k < p) : p ∣ choose p k :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma edge_firstDart (p : G.Walk v w) (hp : ¬ p.Nil) : 
lake-packages/mathlib/Mathlib/Data/Option/Defs.lean:lemma elim'_eq_elim {α β : Type*} (b : β) (f : α → β) (a : Option α) :	
lake-packages/mathlib/Mathlib/Topology/Separation.lean:lemma embedding_iff_inducing [TopologicalSpace β] [T0Space α] {f : α → β} :			
lake-packages/mathlib/Mathlib/Topology/Algebra/Constructions.lean:lemma embedding_val_mk {M : Type*} [DivisionMonoid M] [TopologicalSpace M]	    
lake-packages/mathlib/Mathlib/Topology/Algebra/Constructions.lean:lemma embedding_val_mk' {M : Type*} [Monoid M] [TopologicalSpace M] {f : M → M}	    
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Ends/Properties.lean:lemma end_componentCompl_infinite (e : G.end) (K : (Finset V)ᵒᵖ) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma end_mem_verts_toSubgraph (p : G.Walk u v) : v ∈ p.toSubgraph.verts := by		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma epi_homologyMap_of_epi_cyclesMap'	    
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Basic.lean:lemma epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma epi₁ (h : T.mor₂ = 0) : Epi T.mor₁ := (T.mor₂_eq_zero_iff_epi₁ hT).1 h	     
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma epi₂ (h : T.mor₃ = 0) : Epi T.mor₂ := (T.mor₃_eq_zero_iff_epi₂ hT).1 h	     
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma epi₃ (h : T.mor₁ = 0) : Epi T.mor₃ := (T.mor₁_eq_zero_iff_epi₃ hT).1 h	     
lake-packages/mathlib/Mathlib/Data/MvPolynomial/Basic.lean:lemma eq_C_of_isEmpty [IsEmpty σ] (p : MvPolynomial σ R) :  
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma eq_changeLevel {d : ℕ} (h : FactorsThrough χ d) : χ = changeLevel h.dvd h.χ₀ :=	
lake-packages/mathlib/Mathlib/Data/Bool/Basic.lean:lemma eq_not_iff : ∀ {a b : Bool}, a = !b ↔ a ≠ b := by decide	   
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma eq_of_abs_lt (h : a ≡ b [MOD m]) (h2 : |(b : ℤ) - a| < m) : a = b := by		
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma eq_of_lt_of_lt (h : a ≡ b [MOD m]) (ha : a < m) (hb : b < m) : a = b :=		
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Stieltjes.lean:lemma eq_of_measure_of_eq (g : StieltjesFunction) {y : ℝ}    
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Stieltjes.lean:lemma eq_of_measure_of_tendsto_atBot (g : StieltjesFunction) {l : ℝ}  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma eq_of_mem_uIcc_of_mem_uIcc (ha : a ∈ [[b, c]]) (hb : b ∈ [[a, c]]) : a = b :=	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma eq_of_mem_uIcc_of_mem_uIcc' : b ∈ [[a, c]] → c ∈ [[a, b]] → b = c := by  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma eq_of_mem_uIoc_of_mem_uIoc : a ∈ Ι b c → b ∈ Ι a c → a = b := by	 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma eq_of_mem_uIoc_of_mem_uIoc' : b ∈ Ι a c → c ∈ Ι a b → b = c := by	 
lake-packages/mathlib/Mathlib/Tactic/Linarith/Lemmas.lean:lemma eq_of_not_lt_of_not_gt {α} [LinearOrder α] (a b : α) (h1 : ¬ a < b) (h2 : ¬ b < a) : a = b :=	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma eq_of_not_mem_uIoc_of_not_mem_uIoc (ha : a ≤ c) (hb : b ≤ c) : 
lake-packages/mathlib/Mathlib/Data/Bool/Basic.lean:lemma eq_or_eq_not : ∀ a b, a = b ∨ a = !b := by decide	
lake-packages/mathlib/Mathlib/Order/Basic.lean:lemma eq_or_eq_or_eq_of_forall_not_lt_lt [LinearOrder α]  
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma eq_singleton_or_nontrivial (ha : a ∈ s) : s = {a} ∨ s.Nontrivial := by	       
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Injective.lean:lemma eq_zero_of_toRatCircle_apply_self	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_add (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_add' (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_add_iff :	    
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_iff (x y : R) :			
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_iff' (x y : R) :			
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_iff_baseChange [Nontrivial A] [NoZeroSMulDivisors R A] (x y : R) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_iff_baseChange_of_baseChange [Nontrivial B] [NoZeroSMulDivisors A B] (x y : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_iff_variableChange (x y : R) : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_neg (h : W.equation x₁ y₁) : W.equation x₁ <| W.negY x₁ y₁ := 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_neg_iff : W.equation x₁ (W.negY x₁ y₁) ↔ W.equation x₁ y₁ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma equation_neg_of (h : W.equation x₁ <| W.negY x₁ y₁) : W.equation x₁ y₁ :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma equation_zero : W.equation 0 0 ↔ W.a₆ = 0 := by	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Equicontinuity.lean:lemma equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :	   
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Equicontinuity.lean:lemma equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :  
lake-packages/mathlib/Mathlib/LinearAlgebra/Dual.lean:lemma equiv [IsReflexive R M] (e : M ≃ₗ[R] N) : IsReflexive R N where	
lake-packages/mathlib/Mathlib/CategoryTheory/Adjunction/Reflective.lean:lemma equivEssImageOfReflective_map_counitIso_app_hom [Reflective i]	
lake-packages/mathlib/Mathlib/CategoryTheory/Adjunction/Reflective.lean:lemma equivEssImageOfReflective_map_counitIso_app_inv [Reflective i]	
lake-packages/mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean:lemma equivFreeAlgebra_symm_ι (b : Basis κ R M) (i : κ) :				 
lake-packages/mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean:lemma equivFreeAlgebra_ι_apply (b : Basis κ R M) (i : κ) :				 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_add [Add α] [Small α] (x y : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_div [Div α] [Small α] (x y : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_inv [Inv α] [Small α] (x : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_mul [Mul α] [Small α] (x y : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_neg [Neg α] [Small α] (x : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_sub [Sub α] [Small α] (x y : α) : 
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_add [Add α] [Small α] (x y : Shrink α) :	    
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_div [Div α] [Small α] (x y : Shrink α) :	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_inv [Inv α] [Small α] (x : Shrink α) :	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_mul [Mul α] [Small α] (x y : Shrink α) :	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_neg [Neg α] [Small α] (x : Shrink α) :	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_one [One α] [Small α] : (equivShrink α).symm 1 = 1 :=	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_sub [Sub α] [Small α] (x y : Shrink α) :	
lake-packages/mathlib/Mathlib/Logic/Small/Group.lean:lemma equivShrink_symm_zero [Zero α] [Small α] : (equivShrink α).symm 0 = 0 :=	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Injective.lean:lemma equivZModSpanAddOrderOf_apply_self : 
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma equiv_compl_fromColumns_mul_fromRows_eq_one_comm (p : n → Prop)[DecidablePred p]	    
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Equivalence.lean:lemma equivalence_counitIso_app (X : C₂) :  
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean:lemma essSup_comp_quotientGroup_mk [μ.IsMulRightInvariant] {g : G ⧸ Γ → ℝ≥0∞}		  
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma eval_modulus_sub (x : ZMod n) : χ (n - x) = χ (-x) := by simp 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma eval_negPolynomial : (W.negPolynomial.eval <| C y₁).eval x₁ = W.negY x₁ y₁ := by	
lake-packages/mathlib/Mathlib/Data/Polynomial/Eval.lean:lemma eval_ofNat (n : ℕ) [n.AtLeastTwo] (a : R) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomial (x y : R) :      
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomialX (x y : R) :	    
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomialX_zero : (W.polynomialX.eval 0).eval 0 = -W.a₄ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomialY (x y : R) :		  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomialY_zero : (W.polynomialY.eval 0).eval 0 = W.a₃ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma eval_polynomial_zero : (W.polynomial.eval 0).eval 0 = -W.a₆ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean:lemma evaluation_coevaluation :	 
lake-packages/mathlib/Mathlib/Data/Polynomial/Eval.lean:lemma eval₂_ofNat {S : Type*} [Semiring S] (n : ℕ) [n.AtLeastTwo] (f : R →+* S) (a : S) :		
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma eventuallyConst_atTop [SemilatticeSup α] [Nonempty α] :	 
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma eventuallyConst_atTop_nat {f : ℕ → α} :  
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma eventuallyConst_iff_tendsto [Nonempty β] :  
lake-packages/mathlib/Mathlib/Topology/MetricSpace/Basic.lean:lemma eventually_isCompact_closedBall [LocallyCompactSpace α] (x : α) :		    
lake-packages/mathlib/Mathlib/Topology/MetricSpace/HausdorffDistance.lean:lemma eventually_not_mem_cthickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) : 
lake-packages/mathlib/Mathlib/Topology/MetricSpace/HausdorffDistance.lean:lemma eventually_not_mem_thickening_of_infEdist_pos {E : Set α} {x : α} (h : x ∉ closure E) :  
lake-packages/mathlib/test/LibrarySearch/basic.lean:lemma ex' (x : ℕ) (_h₁ : x = 0) (h : 2 * 2 ∣ x) : 2 ∣ x := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_homology_iso_zero [S.HasHomology] [HasZeroObject C] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_isZero_homology [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_isZero_leftHomology [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_isZero_rightHomology [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_iff_of_iso (e : S₁ ≅ S₂) : S₁.Exact ↔ S₂.Exact := 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_of_isZero_X₂ (h : IsZero S.X₂) : S.Exact := by 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_of_iso (e : S₁ ≅ S₂) (h : S₁.Exact) : S₂.Exact := by 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_op_iff : S.op.Exact ↔ S.Exact:=  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Exact.lean:lemma exact_unop_iff (S : ShortComplex Cᵒᵖ) : S.unop.Exact ↔ S.Exact :=	
lake-packages/mathlib/Mathlib/Data/Sigma/Basic.lean:lemma exists' {p : ∀ a, β a → Prop} : (∃ a b, p a b) ↔ ∃ x : Σ a, β a, p x.1 x.2 :=   
lake-packages/mathlib/Mathlib/Topology/ExtremallyDisconnected.lean:lemma exists_compact_surjective_zorn_subset [T1Space A] [CompactSpace D] {π : D → A}	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean:lemma exists_eq_one {x : V} (hx : x ≠ 0) :  
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma exists_eventually_atBot [SemilatticeInf α] [Nonempty α] {r : α → β → Prop} :	
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma exists_eventually_atTop [SemilatticeSup α] [Nonempty α] {r : α → β → Prop} :	
lake-packages/mathlib/Mathlib/RingTheory/Finiteness.lean:lemma exists_fin' (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M] [Finite R M] :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Girth.lean:lemma exists_girth_eq_length :	   
lake-packages/mathlib/Mathlib/Topology/MetricSpace/Basic.lean:lemma exists_isCompact_closedBall [LocallyCompactSpace α] (x : α) :   
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/Rolle.lean:lemma exists_isExtrOn_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/Rolle.lean:lemma exists_isLocalExtr_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b))	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Basic.lean:lemma exists_is_open_mem_uniformity_of_forall_mem_eq  
lake-packages/mathlib/Mathlib/Data/Set/Pairwise/Basic.lean:lemma exists_lt_mem_inter_of_not_pairwiseDisjoint [LinearOrder ι]	
lake-packages/mathlib/Mathlib/Data/Set/Pairwise/Basic.lean:lemma exists_lt_mem_inter_of_not_pairwise_disjoint [LinearOrder ι]	
lake-packages/mathlib/Mathlib/Order/LocallyFinite.lean:lemma exists_min_greater [LinearOrder α] [LocallyFiniteOrder α] {x ub : α} (hx : x < ub) :	
lake-packages/mathlib/Mathlib/Data/Set/Pairwise/Basic.lean:lemma exists_ne_mem_inter_of_not_pairwiseDisjoint      
lake-packages/mathlib/Mathlib/Data/Set/Pairwise/Basic.lean:lemma exists_ne_mem_inter_of_not_pairwise_disjoint	       
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean:lemma exists_ne_zero {x : V} (hx : x ≠ 0) : 
lake-packages/mathlib/Mathlib/NumberTheory/WellApproximable.lean:lemma exists_norm_nsmul_le (ξ : 𝕊) {n : ℕ} (hn : 0 < n) : 
lake-packages/mathlib/Mathlib/Data/Finset/Card.lean:lemma exists_of_one_lt_card_pi {ι : Type*} {α : ι → Type*} [∀ i, DecidableEq (α i)]	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma exists_sUnion {p : α → Prop} :   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma exists_smul_basis_eq (x : W.CoordinateRing) : 
lake-packages/mathlib/Mathlib/LinearAlgebra/FiniteDimensional.lean:lemma exists_smul_eq_of_finrank_eq_one	  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ext (z₁ z₂ : Cochain F G n)  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ext (z₁ z₂ : Cocycle F G n) (h : (z₁ : Cochain F G n) = z₂) : z₁ = z₂ :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Endofunctor/Algebra.lean:lemma ext {A B : Algebra F} {f g : A ⟶ B} (w : f.f = g.f := by aesop_cat) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Endofunctor/Algebra.lean:lemma ext {A B : Coalgebra F} {f g : A ⟶ B} (w : f.f = g.f := by aesop_cat) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean:lemma ext {F G : OplaxFunctor B C} {α β : F ⟶ G} {m n : α ⟶ β} (w : ∀ b, m.app b = n.app b) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Basic.lean:lemma ext {M N : ModuleCat.{v} R} {f₁ f₂ : M ⟶ N} (h : ∀ (x : M), f₁ x = f₂ x) : f₁ = f₂ :=	
lake-packages/mathlib/Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean:lemma ext {M N : SemiNormedGroupCat} {f₁ f₂ : M ⟶ N} (h : ∀ (x : M), f₁ x = f₂ x) : f₁ = f₂ :=	
lake-packages/mathlib/Mathlib/Topology/Sheaves/Presheaf.lean:lemma ext {P Q : Presheaf C X} {f g : P ⟶ Q} (w : ∀ U : Opens X, f.app (op U) = g.app (op U)) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Endomorphism.lean:lemma ext {X : C} {φ₁ φ₂ : Aut X} (h : φ₁.hom = φ₂.hom) : φ₁ = φ₂ :=	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma ext {X Y : CommGroupCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma ext {X Y : CommMonCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma ext {X Y : CommRingCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma ext {X Y : CommSemiRingCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g := 
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma ext {X Y : GroupCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g := 
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma ext {X Y : MonCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=	 
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Mon_.lean:lemma ext {X Y : Mon_ C} {f g : X ⟶ Y} (w : f.hom = g.hom) : f = g :=	
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma ext {X Y : RingCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g := 
lake-packages/mathlib/Mathlib/Algebra/Category/Ring/Basic.lean:lemma ext {X Y : SemiRingCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Pi/Basic.lean:lemma ext {X Y : ∀ i, C i} {f g : X ⟶ Y} (w : ∀ i, f i = g i) : f = g := 
lake-packages/mathlib/Mathlib/Topology/FiberBundle/Trivialization.lean:lemma ext {e e' : Pretrivialization F proj} (h₁ : ∀ x, e x = e' x)  
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma ext {x y : RelSeries r} (length_eq : x.length = y.length)	  
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/SingleObj.lean:lemma ext {x y : SingleObj α} : x = y := Unit.ext x y 
lake-packages/mathlib/Mathlib/Topology/FiberBundle/Trivialization.lean:lemma ext' (e e' : Pretrivialization F proj) (h₁ : e.toLocalEquiv = e'.toLocalEquiv)	
lake-packages/mathlib/Mathlib/Topology/FiberBundle/Trivialization.lean:lemma ext' (e e' : Trivialization F proj) (h₁ : e.toLocalHomeomorph = e'.toLocalHomeomorph)	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Braided.lean:lemma ext' {F G : BraidedFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Braided.lean:lemma ext' {F G : LaxBraidedFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean:lemma ext' {F G : LaxMonoidalFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ext_iff (z₁ z₂ : Cocycle F G n) : z₁ = z₂ ↔ (z₁ : Cochain F G n) = z₂ :=	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma extendDomain_pow (n : ℕ) : (e ^ n).extendDomain f = e.extendDomain f ^ n :=		
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma extendDomain_zpow (n : ℤ) : (e ^ n).extendDomain f = e.extendDomain f ^ n :=		
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma extend_finset_to_connected (Gpc : G.Preconnected) {t : Finset V} (tn : t.Nonempty) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_def (s : Set α) : exterior s = ⋂₀ {t : Set α | IsOpen t ∧ s ⊆ t} := 
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_mem_nhdsSet : exterior s ∈ 𝓝ˢ s := isOpen_exterior.mem_nhdsSet.2 subset_exterior	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_minimal (h₁ : s ⊆ t) (h₂ : IsOpen t) : exterior s ⊆ t := by    
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_singleton_eq_ker_nhds (a : α) : exterior {a} = (𝓝 a).ker := by simp [exterior]	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_singleton_subset_iff_mem_nhds : exterior {a} ⊆ t ↔ t ∈ 𝓝 a := by	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_subset_iff : exterior s ⊆ t ↔ ∃ U, IsOpen U ∧ s ⊆ U ∧ U ⊆ t :=	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma exterior_subset_iff_mem_nhdsSet : exterior s ⊆ t ↔ t ∈ 𝓝ˢ s := 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ext₀ (z₁ z₂ : Cochain F G 0) 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma f'_cyclesMap' : h₁.f' ≫ cyclesMap' φ h₁ h₂ = φ.τ₁ ≫ h₂.f' := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma f_pOpcycles : S.f ≫ S.pOpcycles = 0 := S.rightHomologyData.wp  
lake-packages/mathlib/Mathlib/Data/Nat/Factorial/Basic.lean:lemma factorial_le_of_le {m n : ℕ} (h : n ≤ m) : n ! ≤ m ! := monotone_factorial h	
lake-packages/mathlib/Mathlib/Data/Nat/Factorial/Basic.lean:lemma factorial_lt_of_lt {m n : ℕ} (hn : 0 < n) (h : n < m) : n ! < m ! := (factorial_lt hn).mpr h	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma factorsThruAlong_id {X : C} (S T : Presieve X) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma factorsThru_of_le {X : C} (S T : Presieve X) (h : S ≤ T) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma factorsThru_top {X : C} (S : Presieve X) : S.FactorsThru ⊤ :=  
lake-packages/mathlib/Mathlib/CategoryTheory/WithTerminal.lean:lemma false_of_from_star {X : C} (f : star ⟶ of X) : False := (f : PEmpty).elim	
lake-packages/mathlib/Mathlib/CategoryTheory/WithTerminal.lean:lemma false_of_to_star {X : C} (f : of X ⟶ star) : False := (f : PEmpty).elim	
lake-packages/mathlib/Mathlib/NumberTheory/FLT/Basic.lean:lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :			   
lake-packages/mathlib/Mathlib/Data/Nat/Fib.lean:lemma fib_add_one : ∀ {n}, n ≠ 0 → fib (n + 1) = fib (n - 1) + fib n 
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma fib_greatestFib_le (n : ℕ) : fib (greatestFib n) ≤ n := 
lake-packages/mathlib/Mathlib/Data/Nat/Fib.lean:lemma fib_lt_fib {m : ℕ} (hm : 2 ≤ m) : ∀ {n}, fib m < fib n ↔ m < n	
lake-packages/mathlib/Mathlib/Data/Nat/Fib.lean:lemma fib_strictMonoOn : StrictMonoOn fib (Set.Ici 2)	
lake-packages/mathlib/Mathlib/Logic/Equiv/Fin.lean:lemma finRotate_succ (n : ℕ) : finRotate (n + 1) = finAddFlip.trans (finCongr (add_comm 1 n)) := rfl	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma finiteCoproduct.hom_ext {B : CompHaus.{u}} (f g : finiteCoproduct X ⟶ B) 
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma finiteCoproduct.hom_ext {B : Profinite.{u}} (f g : finiteCoproduct X ⟶ B)	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma finiteCoproduct.hom_ext {B : Stonean.{u}} (f g : finiteCoproduct X ⟶ B) 
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma finiteCoproduct.openEmbedding_ι {α : Type} [Fintype α] (Z : α → Stonean.{u}) (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma finiteCoproduct.ι_desc {B : CompHaus.{u}} (e : (a : α) → (X a ⟶ B)) (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma finiteCoproduct.ι_desc {B : Profinite.{u}} (e : (a : α) → (X a ⟶ B)) (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma finiteCoproduct.ι_desc {B : Stonean.{u}} (e : (a : α) → (X a ⟶ B)) (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma finiteCoproduct.ι_desc_apply {B : CompHaus} {π : (a : α) → X a ⟶ B} (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma finiteCoproduct.ι_desc_apply {B : Profinite} {π : (a : α) → X a ⟶ B} (a : α) :	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma finiteCoproduct.ι_injective (a : α) : Function.Injective (finiteCoproduct.ι X a) := by	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma finiteCoproduct.ι_injective (a : α) : Function.Injective (finiteCoproduct.ι X a) := by	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma finiteCoproduct.ι_jointly_surjective (R : finiteCoproduct X) :		  
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma finiteCoproduct.ι_jointly_surjective (R : finiteCoproduct X) :		  
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean:lemma finset_prod_rpow {ι} (s : Finset ι) (f : ι → ℝ≥0) (r : ℝ) :  
lake-packages/mathlib/Mathlib/Control/Fold.lean:lemma foldMap_hom (α β)	    
lake-packages/mathlib/test/solve_by_elim/instances.lean:lemma foo (a b : ℕ) (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 := by	
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma foo [Category C]		
lake-packages/mathlib/test/CategoryTheory/Elementwise.lean:lemma foo' [Category C]		
lake-packages/mathlib/test/toAdditive.lean:lemma foo13 {α β : Type u} [my_has_pow α β] (x : α) (y : β) : x ^ y = x ^ y := rfl	
lake-packages/mathlib/test/toAdditive.lean:lemma foo15 {α β : Type u} [my_has_pow α β] (x : α) (y : β) : foo14 x y = (x ^ y) ^ y := rfl	
lake-packages/mathlib/test/toAdditive.lean:lemma foo16 {α β : Type u} [my_has_pow α β] (x : α) (y : β) : foo14 x y = (x ^ y) ^ y := foo15 x y	
lake-packages/mathlib/test/toAdditive.lean:lemma foo18 [Group α] (x : α) : foo17 x = x ∧ foo17 x = 1 * x :=    
lake-packages/mathlib/test/toAdditive.lean:lemma foo4_test {α β : Type u} : @foo4 α β = @my_has_pow α β := rfl	
lake-packages/mathlib/Mathlib/Data/Sigma/Basic.lean:lemma forall' {p : ∀ a, β a → Prop} : (∀ a b, p a b) ↔ ∀ x : Σ a, β a, p x.1 x.2 :=   
lake-packages/mathlib/Mathlib/Data/Nat/SuccPred.lean:lemma forall_ne_zero_iff (P : ℕ → Prop) :	     
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma forall_sUnion {p : α → Prop} :   
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma forall_uIoc_iff {P : α → Prop} : 
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Basic.lean:lemma forget_ContinuousMap_mk {X Y : Profinite} (f : X → Y) (hf : Continuous f) :	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Basic.lean:lemma forget_map {X Y : CommGroupCat} (f : X ⟶ Y) :  
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma forget_map {X Y : CommMonCat} (f : X ⟶ Y) :  
lake-packages/mathlib/Mathlib/Order/Category/NonemptyFinLinOrd.lean:lemma forget_map_apply {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) (a : A) :   
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Basic.lean:lemma forget₂_map_homMk :   
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma four_ne_zero [OfNat α 4] [NeZero (4 : α)] : (4 : α) ≠ 0 := NeZero.ne (4 : α)	
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma four_ne_zero' [OfNat α 4] [NeZero (4 : α)] : (4 : α) ≠ 0 := four_ne_zero		
lake-packages/mathlib/Mathlib/Algebra/Order/Floor.lean:lemma fract_pos : 0 < fract a ↔ a ≠ ⌊a⌋ :=	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromBlocks_mul_fromRows (A₁ : Matrix n₁ n R) (A₂ : Matrix n₂ n R)	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_apply_inl (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) (i : m) (j : n₁) :   
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_apply_inr (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) (i : m) (j : n₂) :   
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_ext_iff (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) (B₁ : Matrix m n₁ R)	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_fromRows_eq_fromBlocks (B₁₁ : Matrix m₁ n₁ R) (B₁₂ : Matrix m₁ n₂ R)	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_inj : Function.Injective2 (@fromColumns R m n₁ n₂) := by	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_mul_fromBlocks (A₁ : Matrix m m₁ R) (A₂ : Matrix m m₂ R)   
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_mul_fromRows (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R)     
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_mul_fromRows_eq_one_comm (e : n ≃ n₁ ⊕ n₂)	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_toColumns (A : Matrix m (n₁ ⊕ n₂) R) : 
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromColumns_zero : fromColumns (0 : Matrix m n₁ R) (0 : Matrix m n₂ R) = 0 := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma fromOpcycles_naturality : opcyclesMap φ ≫ S₂.fromOpcycles = S₁.fromOpcycles ≫ φ.τ₃ :=	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_apply_inl (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (i : m₁) (j : n) :	   
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_apply_inr (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (i : m₂) (j : n) :	   
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_ext_iff (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (B₁ : Matrix m₁ n R) 
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_fromColumn_eq_fromBlocks (B₁₁ : Matrix m₁ n₁ R) (B₁₂ : Matrix m₁ n₂ R)	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_inj : Function.Injective2 (@fromRows R m₁ m₂ n) := by  
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_mul (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (B : Matrix n m R) :	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_mul_fromColumns (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R)     
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_toRows (A : Matrix (m₁ ⊕ m₂) n R) : fromRows A.toRows₁ A.toRows₂ = A := by	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma fromRows_zero : fromRows (0 : Matrix m₁ n R) (0 : Matrix m₂ n R) = 0 := by   
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean:lemma fst_prod [IsProbabilityMeasure ν] : (μ.prod ν).fst = μ := by  
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient/Preadditive.lean:lemma functor_additive :  
lake-packages/mathlib/Mathlib/Algebra/Module/Zlattice.lean:lemma fundamentalDomain_pi_basisFun [Fintype ι] :     
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma gc_exterior_interior : GaloisConnection (exterior : Set α → Set α) interior :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Girth.lean:lemma girth_anti : Antitone (girth : SimpleGraph α → ℕ∞) :=  
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma greatestFib_mono : Monotone greatestFib :=    
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma greatestFib_ne_zero : greatestFib n ≠ 0 ↔ n ≠ 0 := greatestFib_eq_zero.not	
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma greatestFib_sub_fib_greatestFib_le_greatestFib (hn : n ≠ 0) :  
lake-packages/mathlib/Mathlib/CategoryTheory/CatCommSq.lean:lemma hInv_hInv (h : CatCommSq T.functor L R B.functor) :	    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma hasCokernel [S.HasLeftHomology] [HasKernel S.g] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasCokernel [S.HasRightHomology] : HasCokernel S.f :=	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Colimits.lean:lemma hasColimit : HasColimit F := ⟨_, Colimits.colimitCoconeIsColimit.{w} F⟩	
lake-packages/mathlib/Mathlib/Algebra/Homology/LocalCohomology.lean:lemma hasColimitDiagram (I : D ⥤ Ideal R) (i : ℕ) :	       
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Colimits.lean:lemma hasColimitsOfShape : HasColimitsOfShape J AddCommGroupCat.{max u v w} where	
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Colimits.lean:lemma hasColimitsOfSize : HasColimitsOfSize.{v, u} AddCommGroupCat.{max u v w} :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma hasHomology_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [HasHomology S₁]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma hasHomology_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [HasHomology S₂]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma hasHomology_of_isIsoLeftRightHomologyComparison [S.HasLeftHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma hasHomology_of_isIso_leftRightHomologyComparison'			
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma hasHomology_of_iso (e : S₁ ≅ S₂) [HasHomology S₁] : HasHomology S₂ :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma hasKernel [S.HasLeftHomology] : HasKernel S.g :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasKernel [S.HasRightHomology] [HasCokernel S.f] : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasLeftHomology_iff_op (S : ShortComplex C) :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasLeftHomology_iff_unop (S : ShortComplex Cᵒᵖ) :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma hasLeftHomology_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [HasLeftHomology S₁]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma hasLeftHomology_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [HasLeftHomology S₂]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma hasLeftHomology_of_iso {S₁ S₂ : ShortComplex C} (e : S₁ ≅ S₂) [HasLeftHomology S₁] :	
lake-packages/mathlib/Mathlib/Algebra/Category/AlgebraCat/Limits.lean:lemma hasLimitsOfSize : HasLimitsOfSize.{v, v} (AlgebraCatMax.{v, w} R) :=	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Limits.lean:lemma hasLimitsOfSize : HasLimitsOfSize.{v, v} (ModuleCatMax.{v, w, u} R) where	    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasRightHomology_iff_op (S : ShortComplex C) :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasRightHomology_iff_unop (S : ShortComplex Cᵒᵖ) :	      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasRightHomology_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [HasRightHomology S₁]   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasRightHomology_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [HasRightHomology S₂]  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma hasRightHomology_of_iso {S₁ S₂ : ShortComplex C}	
lake-packages/mathlib/Mathlib/Data/Subtype.lean:lemma heq_iff_coe_heq {α β : Sort _} {p : α → Prop} {q : β → Prop} {a : {x // p x}} 
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Path.lean:lemma heq_of_cons_eq_cons {p : Path a b} {p' : Path a c} 
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}	
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_comp (f : CostructuredArrow S T) (g : Y' ⟶ f.left) (g' : Y'' ⟶ Y') :   
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_comp (f : StructuredArrow S T) (g : f.right ⟶ Y') (g' : Y' ⟶ Y'') :    
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_id (f : CostructuredArrow S T) : homMk' f (𝟙 f.left) = eqToHom (by aesop_cat) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_id (f : StructuredArrow S T) : homMk' f (𝟙 f.right) = eqToHom (by aesop_cat) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_mk_comp (f : S ⟶ T.obj Y) (g : Y ⟶ Y') (g' : Y' ⟶ Y'') :	
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_mk_comp (f : S.obj Y ⟶ T) (g : Y' ⟶ Y) (g' : Y'' ⟶ Y') : 
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_mk_id (f : S ⟶ T.obj Y) : homMk' (mk f) (𝟙 Y) = eqToHom (by aesop_cat) :=	  
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma homMk'_mk_id (f : S.obj Y ⟶ T) : homMk' (mk f) (𝟙 Y) = eqToHom (by aesop_cat) :=	   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma homOf_ofHom_eq_self (φ : F ⟶ G) : homOf (ofHom φ) = φ := by aesop_cat  
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Preadditive.lean:lemma hom_add_apply {P Q : AddCommGroupCat} (f g : P ⟶ Q) (x : P) : (f + g) x = f x + g x := rfl	
lake-packages/mathlib/Mathlib/Topology/Category/TopCat/Basic.lean:lemma hom_apply {X Y : TopCat} (f : X ⟶ Y) (x : X) : f x = ContinuousMap.toFun f x := rfl  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Basic.lean:lemma hom_ext (f g : S₁ ⟶ S₂) (h₁ : f.τ₁ = g.τ₁) (h₂ : f.τ₂ = g.τ₂) (h₃ : f.τ₃ = g.τ₃) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Comma.lean:lemma hom_ext (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/CommMon_.lean:lemma hom_ext {A B : CommMon_ C} (f g : A ⟶ B) (h : f.hom = g.hom) : f = g :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomologicalComplex.lean:lemma hom_ext {C D : HomologicalComplex V c} (f g : C ⟶ D) 
lake-packages/mathlib/Mathlib/RepresentationTheory/Action.lean:lemma hom_ext {M N : Action V G} (φ₁ φ₂ : M ⟶ N) (h : φ₁.hom = φ₂.hom) : φ₁ = φ₂ :=	 
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Bimod.lean:lemma hom_ext {M N : Bimod A B} (f g : M ⟶ N) (h : f.hom = g.hom) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/Mod_.lean:lemma hom_ext {M N : Mod_ A} (f₁ f₂ : M ⟶ N) (h : f₁.hom = f₂.hom) : f₁ = f₂ :=	  
lake-packages/mathlib/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean:lemma hom_ext {M N : QuadraticModuleCat.{v} R} (f g : M ⟶ N) (h : f.toIsometry = g.toIsometry) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Arrow.lean:lemma hom_ext {X Y : Arrow T} (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) :	
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialObject.lean:lemma hom_ext {X Y : Augmented C} (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) :	
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialObject.lean:lemma hom_ext {X Y : Augmented C} (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) :	
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialObject.lean:lemma hom_ext {X Y : CosimplicialObject C} (f g : X ⟶ Y)     
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma hom_ext {X Y : CostructuredArrow S T} (f g : X ⟶ Y) (h : f.left = g.left) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/FintypeCat.lean:lemma hom_ext {X Y : FintypeCat} (f g : X ⟶ Y) (h : ∀ x, f x = g x) : f = g := by     
lake-packages/mathlib/Mathlib/CategoryTheory/GradedObject.lean:lemma hom_ext {X Y : GradedObject β C} (f g : X ⟶ Y) (h : ∀ x, f x = g x) : f = g := by	
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialSet.lean:lemma hom_ext {X Y : SSet} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Sheaf.lean:lemma hom_ext {X Y : Sheaf J A} (x y : X ⟶ Y) (h : x.val = y.val) : x = y :=  
lake-packages/mathlib/Mathlib/AlgebraicTopology/SimplicialObject.lean:lemma hom_ext {X Y : SimplicialObject C} (f g : X ⟶ Y) 
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma hom_ext {X Y : StructuredArrow S T} (f g : X ⟶ Y) (h : f.right = g.right) : f = g :=	
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Path.lean:lemma hom_heq_of_cons_eq_cons {p : Path a b} {p' : Path a c}	
lake-packages/mathlib/Mathlib/Algebra/Homology/ImageToKernel.lean:lemma homology.π_map_apply [ConcreteCategory.{w} V] (p : α.right = β.left)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap'_eq : homologyMap' φ h₁ h₂ = γ.left.φH :=	      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap'_id (h : S.HomologyData) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap'_op : (homologyMap' φ h₁ h₂).op =	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap'_zero (h₁ : S₁.HomologyData) (h₂ : S₂.HomologyData) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_comm :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_comm :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_comp [HasHomology S₁] [HasHomology S₂] [HasHomology S₃]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_eq :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_eq : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_id [HasHomology S] :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_op [HasHomology S₁] [HasHomology S₂] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyMap_zero [S₁.HasHomology] [S₂.HasHomology] :	      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homology_π_ι :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyι_comp_asIsoHomologyι_inv (hg : S.g = 0) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyι_comp_fromOpcycles :	  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyι_descOpcycles_π_eq_zero_of_boundary [S.HasHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyι_naturality (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyπ_comp_asIsoHomologyπ_inv (hf : S.f = 0) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyπ_comp_leftHomologyIso_inv:	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma homologyπ_naturality (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma iCycles_g : S.iCycles ≫ S.g = 0 := S.leftHomologyData.wi	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndepFun_iff [MeasurableSpace Ω] {β : ι → Type*}	
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndepFun_iff_iIndep [MeasurableSpace Ω] {β : ι → Type*}	  
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndepSet_iff [MeasurableSpace Ω] (s : ι → Set Ω) (μ : Measure Ω) :   
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndepSet_iff_iIndep [MeasurableSpace Ω] (s : ι → Set Ω) (μ : Measure Ω) : 
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndepSets_iff [MeasurableSpace Ω] (π : ι → Set (Set Ω)) (μ : Measure Ω) :   
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndep_iff (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :	 
lake-packages/mathlib/Mathlib/Probability/Independence/Basic.lean:lemma iIndep_iff_iIndepSets (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω) :	
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iInf_coe_eq_top : ⨅ i, (f i : ℕ∞) = ⊤ ↔ IsEmpty ι := WithTop.iInf_coe_eq_top		
lake-packages/mathlib/Mathlib/Data/Real/ENNReal.lean:lemma iInf_coe_eq_top : ⨅ i, (f i : ℝ≥0∞) = ⊤ ↔ IsEmpty ι := WithTop.iInf_coe_eq_top		
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma iInf_coe_eq_top : ⨅ x, (f x : WithTop α) = ⊤ ↔ IsEmpty ι := by simp [isEmpty_iff]	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma iInf_coe_lt_top : ⨅ i, (f i : WithTop α) < ⊤ ↔ Nonempty ι := by	  
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iInf_coe_lt_top : ⨅ i, (f i : ℕ∞) < ⊤ ↔ Nonempty ι := WithTop.iInf_coe_lt_top		
lake-packages/mathlib/Mathlib/Data/Real/ENNReal.lean:lemma iInf_coe_lt_top : ⨅ i, (f i : ℝ≥0∞) < ⊤ ↔ Nonempty ι := WithTop.iInf_coe_lt_top		
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iInf_coe_ne_top : ⨅ i, (f i : ℕ∞) ≠ ⊤ ↔ Nonempty ι := by					
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma iInf_prod' (f : β → γ → α) : (⨅ i, ⨅ j, f i j) = ⨅ x : β × γ, f x.1 x.2 :=	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iInf_sUnion (S : Set (Set α)) (f : α → β) : 
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma iInf_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const  
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=    
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :   
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iSup_coe_eq_top : ⨆ i, (f i : ℕ∞) = ⊤ ↔ ¬ BddAbove (range f) := WithTop.iSup_coe_eq_top	
lake-packages/mathlib/Mathlib/Data/Real/ENNReal.lean:lemma iSup_coe_eq_top : ⨆ i, (f i : ℝ≥0∞) = ⊤ ↔ ¬ BddAbove (range f) := WithTop.iSup_coe_eq_top	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) := by	
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iSup_coe_lt_top : ⨆ i, (f i : ℕ∞) < ⊤ ↔ BddAbove (range f) := WithTop.iSup_coe_lt_top	
lake-packages/mathlib/Mathlib/Data/Real/ENNReal.lean:lemma iSup_coe_lt_top : ⨆ i, (f i : ℝ≥0∞) < ⊤ ↔ BddAbove (range f) := WithTop.iSup_coe_lt_top	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma iSup_coe_lt_top : ⨆ x, (f x : WithTop α) < ⊤ ↔ BddAbove (range f) :=  
lake-packages/mathlib/Mathlib/Data/ENat/Lattice.lean:lemma iSup_coe_ne_top : ⨆ i, (f i : ℕ∞) ≠ ⊤ ↔ BddAbove (range f) := iSup_coe_eq_top.not_left	
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma iSup_inf_principal (f : ι → Filter α) (s : Set α) : ⨆ i, f i ⊓ 𝓟 s = (⨆ i, f i) ⊓ 𝓟 s := by 
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma iSup_prod' (f : β → γ → α) : (⨆ i, ⨆ j, f i j) = ⨆ x : β × γ, f x.1 x.2 :=	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iSup_sUnion (S : Set (Set α)) (f : α → β) : 
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma iSup_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const  
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=    
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Monoidal/CommMon_.lean:lemma id' (A : CommMon_ C) : (𝟙 A : A.toMon_ ⟶ A.toMon_) = 𝟙 (A.toMon_) := rfl	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma id_comp (C : VariableChange R) : comp id C = C := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma id_comp : ∀ {X Y : Σi, C i} (f : X ⟶ Y), 𝟙 X ≫ f = f  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma id_mem (W : MorphismProperty C) [W.ContainsIdentities] (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean:lemma idem_f (P : Karoubi (Karoubi C)) : P.p.f ≫ P.p.f = P.p.f := by 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Localization.lean:lemma iff {X Y : C} (f : X ⟶ Y) (a : A) : W (f⟦a⟧') ↔ W f := by	
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image2_curry (f : α × β → γ) (s : Set α) (t : Set β) :  
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image2_inter_left (hf : Injective2 f) :  
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image2_inter_right (hf : Injective2 f) : 
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image2_left_identity {f : α → β → β} {a : α} (h : ∀ b, f a b = b) (t : Set β) :	
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image2_right_identity {f : α → β → α} {b : β} (h : ∀ a, f a b = a) (s : Set α) :	
lake-packages/mathlib/Mathlib/Data/Set/Semiring.lean:lemma imageHom_def [MulOneClass α] [MulOneClass β] (f : α →* β) (s : SetSemiring α) :	      
lake-packages/mathlib/Mathlib/Algebra/Homology/ImageToKernel.lean:lemma imageToKernel_arrow_apply [ConcreteCategory V] (w : f ≫ g = 0)	
lake-packages/mathlib/Mathlib/Data/Set/NAry.lean:lemma image_prod : (fun x : α × β ↦ f x.1 x.2) '' s ×ˢ t = image2 f s t :=	
lake-packages/mathlib/Mathlib/Topology/ExtremallyDisconnected.lean:lemma image_subset_closure_compl_image_compl_of_isOpen {ρ : E → A} (ρ_cont : Continuous ρ)	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_comp_subtype_selfAdjoint :	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_eq_neg_I_smul_skewAdjointPart (x : A) :   
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_imaginaryPart {x : A} : ℑ (ℑ x : A) = 0 :=	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_ofReal (r : ℝ) : ℑ (r : ℂ) = 0 := by  
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_realPart {x : A} : ℑ (ℜ x : A) = 0 :=	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma imaginaryPart_surjective : Function.Surjective (imaginaryPart (A := A)) :=	
lake-packages/mathlib/Mathlib/NumberTheory/Basic.lean:lemma in the construction of the ring of Witt vectors.	
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma inclDesc_hom_app (i : I) (X : C i) : (inclDesc F i).hom.app X = 𝟙 ((F i).obj X) :=     
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma inclDesc_inv_app (i : I) (X : C i) : (inclDesc F i).inv.app X = 𝟙 ((F i).obj X) :=     
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma incl_obj {i : I} (X : C i) : (incl i).obj X = ⟨i, X⟩ :=	
lake-packages/mathlib/Mathlib/MeasureTheory/Function/LpSpace.lean:lemma indicatorConstLp_univ :	    
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma indicator_eq_sum_attach_single [AddCommMonoid M] {s : Finset α} (f : ∀ a ∈ s, M) :	
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma indicator_eq_sum_single [AddCommMonoid M] (s : Finset α) (f : α → M) : 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma induceHom_injective (hi : Set.InjOn φ s) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_connected_adj_union {s t : Set V}	 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_connected_of_patches {s : Set V} (u : V) (hu : u ∈ s)	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_pair_connected_of_adj {u v : V} (huv : G.Adj u v) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_sUnion_connected_of_pairwise_not_disjoint {S : Set (Set V)} (Sn : S.Nonempty)	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_union_connected {H : G.Subgraph} {s t : Set V}  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma induce_union_connected {s t : Set V}	    
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma infClosed_iInter (hf : ∀ i, InfClosed (f i)) : InfClosed (⋂ i, f i) :=	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma infClosed_sInter (hS : ∀ s ∈ S, InfClosed s) : InfClosed (⋂₀ S) :=   
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma infClosure_idem (s : Set α) : infClosure (infClosure s) = infClosure s :=				
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma infClosure_mono : Monotone (infClosure : Set α → Set α) := infClosure.monotone	
lake-packages/mathlib/Mathlib/Order/Lattice.lean:lemma inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by	 
lake-packages/mathlib/Mathlib/Algebra/Order/LatticeGroup.lean:lemma inf_eq_half_smul_add_sub_abs_sub (x y : β) :	
lake-packages/mathlib/Mathlib/Algebra/Order/LatticeGroup.lean:lemma inf_eq_half_smul_add_sub_abs_sub' (x y : β) : x ⊓ y = (2⁻¹ : α) • (x + y - |y - x|) := by	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma injOn_id (s : Set α) : InjOn id s := injective_id.injOn _	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma injOn_of_subsingleton [Subsingleton α] (f : α → β) (s : Set α) : InjOn f s :=	
lake-packages/mathlib/Mathlib/Tactic/CategoryTheory/Coherence.lean:lemma insert_id_lhs {C : Type*} [Category C] {X Y : C} (f g : X ⟶ Y) (w : f ≫ 𝟙 _ = g) :	
lake-packages/mathlib/Mathlib/Tactic/CategoryTheory/Coherence.lean:lemma insert_id_rhs {C : Type*} [Category C] {X Y : C} (f g : X ⟶ Y) (w : f = g ≫ 𝟙 _) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma interior_iInter (f : ι → Set α) : interior (⋂ i, f i) = ⋂ i, interior (f i) :=	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma interior_sInter (S : Set (Set α)) : interior (⋂₀ S) = ⋂ s ∈ S, interior s := by	
lake-packages/mathlib/Mathlib/Topology/Basic.lean:lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s → U ⊆ t := by    
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma invOn : InvOn e e.symm t s := 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma invOn_id (s : Set α) : InvOn id id s s := ⟨s.leftInvOn_id, s.rightInvOn_id⟩	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma inv_mulLeft : (Equiv.mulLeft a)⁻¹ = Equiv.mulLeft a⁻¹ := Equiv.coe_inj.1 rfl	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma inv_mulRight : (Equiv.mulRight a)⁻¹ = Equiv.mulRight a⁻¹ := Equiv.coe_inj.1 rfl	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma inverseImage_equivalence_functor_eq_map_inverse  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma inverseImage_equivalence_inverse_eq_map_functor  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma inverseImage_map_eq_of_isEquivalence	       
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean:lemma inverts : W₁.IsInvertedBy (Φ.functor ⋙ L₂) :=	
lake-packages/mathlib/Mathlib/Data/Polynomial/Laurent.lean:lemma involutive_invert : Involutive (invert (R := R)) := fun _ ↦ by ext; simp	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma irreducible_polynomial [IsDomain R] : Irreducible W.polynomial := by	
lake-packages/mathlib/Mathlib/Analysis/InnerProductSpace/Projection.lean:lemma is only intended for use in setting up the bundled version	
lake-packages/mathlib/Mathlib/Geometry/Euclidean/Basic.lean:lemma is only intended for use in setting up the bundled version and 
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean:lemma isBigO_deriv_rpow_const_atTop (p : ℝ) :	 
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma isChain_range : IsChain (· ≤ ·) (Set.range c) := Monotone.isChain_range (OrderHomClass.mono c)	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_iInter (hf : ∀ i, IsClopen (f i)) : IsClopen (⋂ i, f i) :=	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_iInter₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsClopen (f i j)) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_iUnion (hf : ∀ i, IsClopen (f i)) : IsClopen (⋃ i, f i) :=	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsClopen (f i j)) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_sInter (hS : ∀ s ∈ S, IsClopen s) : IsClopen (⋂₀ S) :=  
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClopen_sUnion (hS : ∀ s ∈ S, IsClopen s) : IsClopen (⋃₀ S) :=  
lake-packages/mathlib/Mathlib/Topology/Basic.lean:lemma isClosed_biUnion_finset {s : Finset β} {f : β → Set α} (h : ∀ i ∈ s, IsClosed (f i)) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClosed_iUnion (hf : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) :=	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClosed_iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsClosed (f i j)) :	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isClosed_iff_isLower {s : Set α} : IsClosed s ↔ (IsLowerSet s) := by	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isClosed_iff_isUpper {s : Set α} : IsClosed s ↔ (IsUpperSet s) := by     
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isClosed_isLower {s : Set α} : IsClosed s → IsLowerSet s := fun h =>	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isClosed_isUpper {s : Set α} : IsClosed s → IsUpperSet s := fun h =>     
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isClosed_sUnion (hS : ∀ s ∈ S, IsClosed s) : IsClosed (⋃₀ S) := by 
lake-packages/mathlib/Mathlib/CategoryTheory/Filtered/Basic.lean:lemma isCofilteredOrEmpty_of_isFilteredOrEmpty_op [IsFilteredOrEmpty Cᵒᵖ] : IsCofilteredOrEmpty C :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Filtered/Basic.lean:lemma isCofiltered_of_isFiltered_op [IsFiltered Cᵒᵖ] : IsCofiltered C :=    
lake-packages/mathlib/Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean:lemma isEquivalence_whiskeringLeft_obj_toKaroubi_aux : 
lake-packages/mathlib/Mathlib/CategoryTheory/Filtered/Basic.lean:lemma isFilteredOrEmpty_of_isCofilteredOrEmpty_op [IsCofilteredOrEmpty Cᵒᵖ] : IsFilteredOrEmpty C :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Filtered/Basic.lean:lemma isFiltered_of_isCofiltered_op [IsCofiltered Cᵒᵖ] : IsFiltered C :=    
lake-packages/mathlib/Mathlib/CategoryTheory/Abelian/Exact.lean:lemma isIso_cokernel_desc_of_exact_of_epi (ex : Exact f g) [Epi g] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_cyclesMap'_of_isIso_of_mono (φ : S₁ ⟶ S₂) (h₂ : IsIso φ.τ₂) (h₃ : Mono φ.τ₃)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_cyclesMap_of_isIso_of_mono' (φ : S₁ ⟶ S₂) (h₂ : IsIso φ.τ₂) (h₃ : Mono φ.τ₃)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isIso_homologyMap_of_epi_of_isIso_of_mono' (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isIso_homologyMap_of_isIso_cyclesMap_of_epi {φ : S₁ ⟶ S₂}		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isIso_homologyMap_of_isIso_opcyclesMap_of_mono {φ : S₁ ⟶ S₂}	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isIso_homologyι (hg : S.g = 0) [S.HasHomology] :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isIso_homologyπ (hf : S.f = 0) [S.HasHomology] :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_i (hg : S.g = 0) : IsIso h.i :=  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_iCycles (hg : S.g = 0) : IsIso S.iCycles :=	    
lake-packages/mathlib/Mathlib/CategoryTheory/Abelian/Exact.lean:lemma isIso_kernel_lift_of_exact_of_mono (ex : Exact f g) [Mono f] : IsIso (kernel.lift g f ex.w) :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_leftHomologyπ (hf : S.f = 0) : IsIso S.leftHomologyπ :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Basic.lean:lemma isIso_of_isIso (f : S₁ ⟶ S₂) [IsIso f.τ₁] [IsIso f.τ₂] [IsIso f.τ₃] : IsIso f :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma isIso_of_yoneda_map_bijective {X Y : C} (f : X ⟶ Y)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_opcyclesMap'_of_isIso_of_epi (φ : S₁ ⟶ S₂) (h₂ : IsIso φ.τ₂) (h₁ : Epi φ.τ₁)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_opcyclesMap_of_isIso_of_epi' (φ : S₁ ⟶ S₂) (h₂ : IsIso φ.τ₂) (h₁ : Epi φ.τ₁)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_p (hf : S.f = 0) : IsIso h.p :=  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_pOpcycles (hf : S.f = 0) : IsIso S.pOpcycles :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_rightHomologyι (hg : S.g = 0) : IsIso S.rightHomologyι :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma isIso_ι (hg : S.g = 0) : IsIso h.ι := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma isIso_π (hf : S.f = 0) : IsIso h.π := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isIso₁_of_isIso₂₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isIso₂_of_isIso₁₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isIso₃_of_isIso₁₂ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)	
lake-packages/mathlib/Mathlib/RingTheory/Jacobson.lean:lemma isMaximal_comap_C_of_isJacobson' {P : Ideal R[X]} (hP : IsMaximal P) :	
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Nilpotent.lean:lemma isNilpotent_C_mul_pow_X_of_isNilpotent (n : ℕ) (hnil : IsNilpotent r) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:lemma isNilpotent_charpoly_sub_pow_of_isNilpotent (hM : IsNilpotent M) :	
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Nilpotent.lean:lemma isNilpotent_pow_X_mul_C_of_isNilpotent (n : ℕ) (hnil : IsNilpotent r) :	
lake-packages/mathlib/Mathlib/RingTheory/Nilpotent.lean:lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}		 
lake-packages/mathlib/Mathlib/RingTheory/Nilpotent.lean:lemma isNilpotent_toMatrix_iff (b : Basis ι R M) (f : M →ₗ[R] M) :	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:lemma isNilpotent_trace_of_isNilpotent (hM : IsNilpotent M) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Trace.lean:lemma isNilpotent_trace_of_isNilpotent {f : M →ₗ[R] M} (hf : IsNilpotent f) :	
lake-packages/mathlib/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean:lemma isOpen_A_with_param {r s : ℝ} (hf : Continuous f.uncurry) (L : E →L[𝕜] F) :	
lake-packages/mathlib/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean:lemma isOpen_B_with_param {r s t : ℝ} (hf : Continuous f.uncurry) (K : Set (E →L[𝕜] F)) :	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isOpen_iInter (hf : ∀ i, IsOpen (f i)) : IsOpen (⋂ i, f i) :=			      
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isOpen_iInter₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsOpen (f i j)) :		      
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isOpen_iff_forall_specializes : IsOpen s ↔ ∀ x y, x ⤳ y → y ∈ s → x ∈ s := by	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s := by		       
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by			
lake-packages/mathlib/Mathlib/Topology/Basic.lean:lemma isOpen_mk {p h₁ h₂ h₃} {s : Set α} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl  
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma isOpen_sInter : (∀ s ∈ S, IsOpen s) → IsOpen (⋂₀ S) := AlexandrovDiscrete.isOpen_sInter _	
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma isProperMap_iff_isClosedMap_and_tendsto_cofinite [T1Space Y] : 
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma isProperMap_iff_tendsto_cocompact [T2Space Y] [WeaklyLocallyCompactSpace Y] :	
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma isProperMap_iff_ultrafilter : IsProperMap f ↔ Continuous f ∧   
lake-packages/mathlib/Mathlib/Topology/ProperMap.lean:lemma isProperMap_iff_ultrafilter_of_t2 [T2Space Y] : IsProperMap f ↔ Continuous f ∧	
lake-packages/mathlib/Mathlib/LinearAlgebra/PerfectPairing.lean:lemma isReflexive_of_equiv_dual_of_isReflexive : IsReflexive R N := by	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma isSeparable_pi {ι : Type*} [Fintype ι] {α : ∀ (_ : ι), Type*} {s : ∀ i, Set (α i)}	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma isSheafFor_of_factorsThru	 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coherent.lean:lemma isSheaf_coherent [Precoherent C] (P : Cᵒᵖ ⥤ Type w) :    
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean:lemma isTheta_deriv_rpow_const_atTop {p : ℝ} (hp : p ≠ 0) :	     
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:lemma isUnit_charpolyRev_of_isNilpotent (hM : IsNilpotent M) :	
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Nilpotent.lean:lemma isUnit_iff' :		     
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma isZeckendorfRep_zeckendorf : ∀ n, (zeckendorf n).IsZeckendorfRep	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma isZero_homology_of_isZero_X₂ (hS : IsZero S.X₂) [S.HasHomology] :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₁_iff : IsZero T.obj₁ ↔ (T.mor₁ = 0 ∧ T.mor₃ = 0) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₁_iff_isIso₂ :    
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₁_of_isIso₂ (h : IsIso T.mor₂) : IsZero T.obj₁ := (T.isZero₁_iff_isIso₂ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₁_of_isZero₂₃ (h₂ : IsZero T.obj₂) (h₃ : IsZero T.obj₃) : IsZero T.obj₁ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₂_iff : IsZero T.obj₂ ↔ (T.mor₁ = 0 ∧ T.mor₂ = 0) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₂_iff_isIso₃ : IsZero T.obj₂ ↔ IsIso T.mor₃ :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₂_of_isIso₃ (h : IsIso T.mor₃) : IsZero T.obj₂ := (T.isZero₂_iff_isIso₃ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₂_of_isZero₁₃ (h₁ : IsZero T.obj₁) (h₃ : IsZero T.obj₃) : IsZero T.obj₂ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₃_iff : IsZero T.obj₃ ↔ (T.mor₂ = 0 ∧ T.mor₃ = 0) := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₃_iff_isIso₁ : IsZero T.obj₃ ↔ IsIso T.mor₁ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₃_of_isIso₁ (h : IsIso T.mor₁) : IsZero T.obj₃ := (T.isZero₃_iff_isIso₁ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma isZero₃_of_isZero₁₂ (h₁ : IsZero T.obj₁) (h₂ : IsZero T.obj₂) : IsZero T.obj₃ := 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma isoAdd_hom_app {a b : A}	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/CommShift.lean:lemma isoAdd_inv_app {a b : A}	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma isoClosure_isoClosure (P : MorphismProperty C) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma isoClosure_respectsIso (P : MorphismProperty C) :		    
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma isoClosure_subset_iff (P Q : MorphismProperty C) (hQ : RespectsIso Q) :	     
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Unitization.lean:lemma isometry_inr : Isometry ((↑) : A → Unitization 𝕜 A) :=   
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/OperatorNorm.lean:lemma isometry_mul : Isometry (mul 𝕜 𝕜') :=  
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Star/Unitization.lean:lemma isometry_mul_flip : Isometry (mul 𝕜 E).flip :=	 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma isomorphic_distinguished (T₁ : Triangle Cᵒᵖ)  
lake-packages/mathlib/Mathlib/Logic/Function/Iterate.lean:lemma iterate_add_eq_iterate (hf : Injective f) : f^[m + n] a = f^[n] a ↔ f^[m] a = a :=	
lake-packages/mathlib/Mathlib/Logic/Function/Iterate.lean:lemma iterate_cancel (hf : Injective f) (ha : f^[m] a = f^[n] a) : f^[m - n] a = a := by	
lake-packages/mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean:lemma itself is separated below. 
lake-packages/mathlib/Mathlib/Topology/LocallyConstant/Algebra.lean:lemma ker_comapₗ [Semiring R] [AddCommMonoid Z] [Module R Z] (f : X → Y)  
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma ker_def (f : Filter α) : f.ker = ⋂ s ∈ f, s := sInter_eq_biInter	
lake-packages/mathlib/Mathlib/GroupTheory/Subgroup/Basic.lean:lemma ker_fst : ker (fst G G') = .prod ⊥ ⊤ := SetLike.ext fun _ => (and_true_iff _).symm 
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma ker_mono : Monotone (ker : Filter α → Set α) := gi_principal_ker.gc.monotone_u	
lake-packages/mathlib/Mathlib/GroupTheory/Subgroup/Basic.lean:lemma ker_snd : ker (snd G G') = .prod ⊤ ⊥ := SetLike.ext fun _ => (true_and_iff _).symm 
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma ker_surjective : Surjective (ker : Filter α → Set α) := gi_principal_ker.u_surjective	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by  
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_mono : Monotone (kernImage f) :=       
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_preimage_union {s : Set α} {t : Set β} :    
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma kernImage_union_preimage {s : Set α} {t : Set β} :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean:lemma kernel_ι_comp_cokernel_π_comp_cokernelToAbelianCoimage :	
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma killingForm_eq :	       
lake-packages/mathlib/Mathlib/Computability/Language.lean:lemma kstar_def (l : Language α) : l∗ = {x | ∃ L : List (List α), x = L.join ∧ ∀ y ∈ L, y ∈ l} :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma le_comap_of_subsingleton (f : V → W) [Subsingleton V] : G ≤ G'.comap f := by	
lake-packages/mathlib/Mathlib/Data/Nat/Fib.lean:lemma le_fib_add_one : ∀ n, n ≤ fib n + 1 
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma le_iff_reflTransGen_covby [PartialOrder α] [LocallyFiniteOrder α] {x y : α} :	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma le_iff_transGen_wcovby [Preorder α] [LocallyFiniteOrder α] {x y : α} :   
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by	
lake-packages/mathlib/Mathlib/LinearAlgebra/FreeModule/PID.lean:lemma le_ker_coord_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Portmanteau.lean:lemma le_liminf_measure_open_of_forall_tendsto_measure	    
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma le_normalClosure : K ≤ normalClosure F K L :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma le_of_factorsThru_sieve {X : C} (S : Presieve X) (T : Sieve X) (h : S.FactorsThru T) :	
lake-packages/mathlib/Mathlib/Order/BooleanAlgebra.lean:lemma le_sdiff : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z :=   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftHomologyIso_hom_naturality :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftHomologyIso_inv_naturality :		      
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap'_add :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap'_eq : leftHomologyMap' φ h₁ h₂ = γ.φH :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap'_id (h : S.LeftHomologyData) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap'_neg : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma leftHomologyMap'_op		     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap'_sub :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap'_zero (h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap_add : leftHomologyMap (φ + φ')  = leftHomologyMap φ + leftHomologyMap φ' :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap_comm [S₁.HasLeftHomology] [S₂.HasLeftHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap_comp [HasLeftHomology S₁] [HasLeftHomology S₂] [HasLeftHomology S₃]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap_eq [S₁.HasLeftHomology] [S₂.HasLeftHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap_id [HasLeftHomology S] :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap_neg : leftHomologyMap (-φ)  = -leftHomologyMap φ :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma leftHomologyMap_op (φ : S₁ ⟶ S₂) [S₁.HasLeftHomology] [S₂.HasLeftHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean:lemma leftHomologyMap_sub : leftHomologyMap (φ - φ') = leftHomologyMap φ - leftHomologyMap φ' :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyMap_zero [HasLeftHomology S₁] [HasLeftHomology S₂] :	    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomology_ext (f₁ f₂ : S.leftHomology ⟶ A)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomology_ext_iff (f₁ f₂ : S.leftHomology ⟶ A) :			
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyπ_comp_leftHomologyIso_hom :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyπ_naturality :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma leftHomologyπ_naturality' :	 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma leftInvOn_id (s : Set α) : LeftInvOn id id s := fun _ _ ↦ rfl  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_compatibility (h₁ h₁' : S.LeftHomologyData)		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_eq_descH :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_eq_liftH :	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_fac (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison'_naturality (φ : S₁ ⟶ S₂) (h₁ : S₁.LeftHomologyData)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison_eq [S.HasLeftHomology] [S.HasRightHomology]		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma leftRightHomologyComparison_fac [S.HasHomology] :		 
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:lemma left_ne_zero_of_smul : a • b ≠ 0 → a ≠ 0 := mt $ fun h ↦ smul_eq_zero_of_left h b	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma left_unique_of_rel_eq {eq' : β → β → Prop} (he : (R ⇒ (R ⇒ Iff)) Eq eq') : LeftUnique R :=	
lake-packages/mathlib/Mathlib/Data/String/Lemmas.lean:lemma leftpad_prefix (n : ℕ) (c : Char) : ∀ s, isPrefix (replicate (n - length s) c) (leftpad n c s)	
lake-packages/mathlib/Mathlib/Data/String/Lemmas.lean:lemma leftpad_suffix (n : ℕ) (c : Char) : ∀ s, isSuffix s (leftpad n c s)      
lake-packages/mathlib/test/Explode.lean:lemma lemma_5 : ∀ p q : Prop, (¬q → ¬p) → (p → q) :=	
lake-packages/mathlib/test/Explode.lean:lemma lemma_5' : ∀ p q : Prop, (¬q → ¬p) → (p → q) := 
lake-packages/mathlib/test/Explode.lean:lemma lemma_6 : ∀ p q : Prop, (p → q) → p → q :=  
lake-packages/mathlib/test/Explode.lean:lemma lemma_7 : ∀ p q r : Prop, (p → q) → (p → q → r) → (p → r) :=	
lake-packages/mathlib/Mathlib/Data/String/Lemmas.lean:lemma length_eq_list_length (l : List Char) : (String.mk l).length = l.length := by		
lake-packages/mathlib/Mathlib/Data/List/Basic.lean:lemma length_injective [Subsingleton α] : Injective (length : List α → ℕ) :=		 
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma length_le_length_longestOf [r.FiniteDimensional] (x : RelSeries r) :	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma level_mem_conductorSet : n ∈ conductorSet χ := FactorsThrough.same_level χ	 
lake-packages/mathlib/Mathlib/Data/Prod/Basic.lean:lemma lex_iff : Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2 := lex_def _ _	
lake-packages/mathlib/Mathlib/Algebra/Lie/Weights.lean:lemma lie_mem_maxGenEigenspace_toEndomorphism	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma liftCycles_comp_cyclesMap (φ : S ⟶ S₁) [S₁.HasLeftHomology] :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma liftCycles_homologyπ_eq_zero_of_boundary [S.HasHomology]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma liftCycles_i : S.liftCycles k hk ≫ S.iCycles = k :=  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma liftCycles_leftHomologyπ_eq_zero_of_boundary (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma liftH_ι (k : A ⟶ h.Q) (hk : k ≫ h.g' = 0) : h.liftH k hk ≫ h.ι = k :=   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma liftHomology_ι (k : A ⟶ S.opcycles) (hk : k ≫ S.fromOpcycles = 0) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma liftK_i (k : A ⟶ S.X₂) (hk : k ≫ S.g = 0) : h.liftK k hk ≫ h.i = k :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma liftK_π_eq_zero_of_boundary (k : A ⟶ S.X₂) (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean:lemma lift_desc_condition {W : Stonean} {e : (a : α) → X a ⟶ W}	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean:lemma lift_fst {S : C} (k : IsKernelPair f a b) (p q : S ⟶ X) (w : p ≫ f = q ≫ f) :  
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Basic.lean:lemma lift_lifts {X Y : CompHaus} {Z : Stonean} (e : Z.compHaus ⟶ Y) (f : X ⟶ Y) [Epi f] :	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Basic.lean:lemma lift_lifts {X Y : Profinite} {Z : Stonean} (e : Stonean.toProfinite.obj Z ⟶ Y) (f : X ⟶ Y)	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Basic.lean:lemma lift_mk_le_lift_mk_of_injective {α : Type u} {β : Type v} {f : α → β} (hf : Injective f) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean:lemma lift_snd {S : C} (k : IsKernelPair f a b) (p q : S ⟶ X) (w : p ≫ f = q ≫ f) :  
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma liminf_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma liminf_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]	
lake-packages/mathlib/Mathlib/Topology/Instances/ENNReal.lean:lemma liminf_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) 
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma liminf_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]	
lake-packages/mathlib/Mathlib/Topology/Instances/ENNReal.lean:lemma liminf_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma liminf_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Types.lean:lemma limitCone_pt_ext (F : J ⥤ Type u) {x y : (limitCone F).pt} 
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma limsup_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma limsup_const_add (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]	
lake-packages/mathlib/Mathlib/Topology/Instances/ENNReal.lean:lemma limsup_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) 
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma limsup_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Portmanteau.lean:lemma limsup_measure_closed_le_of_forall_tendsto_measure    
lake-packages/mathlib/Mathlib/Topology/Instances/ENNReal.lean:lemma limsup_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :	
lake-packages/mathlib/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean:lemma limsup_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Adjunction.lean:lemma localization_counit_app (X₂ : C₂) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Adjunction.lean:lemma localization_unit_app (X₁ : C₁) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean:lemma locallyRingedSpaceAdjunction_counit :				   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean:lemma locallyRingedSpaceAdjunction_unit :				   
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_nonneg_of_isNat (h : NormNum.IsNat e n) : 0 ≤ Real.log (e : ℝ) := by	  
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_nonneg_of_isNegNat (h : NormNum.IsInt e (.negOfNat n)) : 0 ≤ Real.log (e : ℝ) := by  
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_nz_of_isRat : (NormNum.IsRat e n d) → (decide ((0 : ℚ) < n / d))	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_nz_of_isRat_neg : (NormNum.IsRat e n d) → (decide (n / d < (0 : ℚ)))	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_pos_of_isNat (h : NormNum.IsNat e n) (w : Nat.blt 1 n = true) : 0 < Real.log (e : ℝ) := by	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_pos_of_isNegNat (h : NormNum.IsInt e (.negOfNat n)) (w : Nat.blt 1 n = true) :   
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_pos_of_isRat :  
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean:lemma log_pos_of_isRat_neg :	
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma longestOf_is_longest [FiniteDimensionalOrder α] (x : LTSeries α) : 
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma longestOf_len_unique [FiniteDimensionalOrder α] (p : LTSeries α) 
lake-packages/mathlib/Mathlib/Data/Finmap.lean:lemma lookup_eq_some_iff {s : Finmap β} {a : α} {b : β a} :	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma lowerSet_LE_lower {t₁ : TopologicalSpace α} [@LowerSetTopology α t₁ _]   
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma lowerSet_dual_iff_upperSet [Preorder α] [TopologicalSpace α] : 
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma lower_dual_iff_upper [Preorder α] [TopologicalSpace α] :	
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma lt_add_one [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α]	
lake-packages/mathlib/Mathlib/Data/Nat/PartENat.lean:lemma lt_coe_succ_iff_le {x : PartENat} {n : ℕ} (hx : x ≠ ⊤) : x < n.succ ↔ x ≤ n := by		
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma lt_fib_greatestFib_add_one (n : ℕ) : n < fib (greatestFib n + 1) :=	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma lt_iff_transGen_covby [Preorder α] [LocallyFiniteOrder α] {x y : α} :   
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma lt_of_le_of_ne' [PartialOrder A] :  
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma lt_one_add [One α] [AddZeroClass α] [PartialOrder α] [ZeroLEOneClass α]	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma mapHomologicalComplex_commShiftIso_eq (n : ℤ) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma mapHomologicalComplex_commShiftIso_hom_app_f (K : CochainComplex C ℤ) (n i : ℤ) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma mapHomologicalComplex_commShiftIso_inv_app_f (K : CochainComplex C ℤ) (n i : ℤ) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma map_apply (f : V ≃ W) (G : SimpleGraph V) (v : V) : 
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/FiniteMeasure.lean:lemma map_apply' (ν : FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean:lemma map_apply' (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/FiniteMeasure.lean:lemma map_apply_of_aemeasurable (ν : FiniteMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean:lemma map_apply_of_aemeasurable (ν : ProbabilityMeasure Ω) {f : Ω → Ω'}		  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean:lemma map_condition : G.map c.ι ≫ G.map f = 0 := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean:lemma map_condition : G.map f ≫ G.map c.π = 0 := by	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_eq_of_iso (P : MorphismProperty C) {F G : C ⥤ D} (e : F ≅ G) :  
lake-packages/mathlib/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean:lemma map_eq_zero_of_norm_zero (q : Seminorm 𝕜 F)		 
lake-packages/mathlib/Mathlib/Data/Equiv/Functor.lean:lemma map_equiv_apply (h : α ≃ β) (x : f α) : (map_equiv f h : f α ≃ f β) x = map h x := rfl 
lake-packages/mathlib/Mathlib/Data/Equiv/Functor.lean:lemma map_equiv_symm_apply (h : α ≃ β) (y : f β) :	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean:lemma map_f' : (h.map F).f' = F.map h.f' := by	
lake-packages/mathlib/Mathlib/RingTheory/WittVector/Frobenius.lean:lemma map_frobeniusPoly (n : ℕ) :	      
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma map_hasShiftOfFullyFaithful_add_hom_app (a b : A) (X : C) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma map_hasShiftOfFullyFaithful_add_inv_app (a b : A) (X : C) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma map_hasShiftOfFullyFaithful_zero_hom_app (X : C) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma map_hasShiftOfFullyFaithful_zero_inv_app (X : C) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_id (P : MorphismProperty C) (hP : RespectsIso P) :	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_id_eq_isoClosure (P : MorphismProperty C) :  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_inverseImage_eq_of_isEquivalence	       
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_inverseImage_subset (P : MorphismProperty D) (F : C ⥤ D) :   
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_isoClosure (P : MorphismProperty C) (F : C ⥤ D) :	     
lake-packages/mathlib/Mathlib/LinearAlgebra/AffineSpace/Ordered.lean:lemma map_le_lineMap_iff_slope_le_slope_left (h : 0 < r * (b - a)) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma map_le_of_subsingleton (f : V ↪ W) [Subsingleton V] : G.map f ≤ G' := by	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_map (P : MorphismProperty C) (F : C ⥤ D) {E : Type*} [Category E] (G : D ⥤ E) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma map_map {j : J} {X Y : C (g j)} (f : X ⟶ Y) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_mem_map (P : MorphismProperty C) (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) (hf : P f) :	
lake-packages/mathlib/Mathlib/Data/Sigma/Basic.lean:lemma map_mk (f₁ : α₁ → α₂) (f₂ : ∀ a, β₁ a → β₂ (f₁ a)) (x : α₁) (y : β₁ x) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma map_obj (j : J) (X : C (g j)) : (Sigma.map C g).obj ⟨j, X⟩ = ⟨g j, X⟩ := 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_respectsIso (P : MorphismProperty C) (F : C ⥤ D) : 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma map_subset_iff (P : MorphismProperty C) (F : C ⥤ D)  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Basic.lean:lemma map_symm_apply (f : V ≃ W) (G : SimpleGraph V) (w : W) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean:lemma map_ι : (c.map G).ι = G.map c.ι := rfl  
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean:lemma map_π : (c.map G).π = G.map c.π := rfl  
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma mapsTo_of_subsingleton [Subsingleton α] (f : α → α) (s : Set α) : MapsTo f s s :=	 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma mapsTo_of_subsingleton' [Subsingleton β] (f : α → β) (h : s.Nonempty → t.Nonempty) :	
lake-packages/mathlib/Mathlib/Data/Option/NAry.lean:lemma map₂_left_identity {f : α → β → β} {a : α} (h : ∀ b, f a b = b) (o : Option β) :	    
lake-packages/mathlib/Mathlib/Data/Option/NAry.lean:lemma map₂_right_identity {f : α → β → α} {b : β} (h : ∀ a, f a b = a) (o : Option α) :	    
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean:lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :	
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean:lemma measurableSet_bddBelow_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :	
lake-packages/mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean:lemma measurableSet_copy {m : MeasurableSpace α} {p : Set α → Prop}	    
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean:lemma measurableSet_of_tendsto_indicator [NeBot L] (As_mble : ∀ i, MeasurableSet (As i))	
lake-packages/mathlib/Mathlib/MeasureTheory/MeasurableSpace/Basic.lean:lemma measurable_indicator_const_iff [Zero β] [MeasurableSingletonClass β] (b : β) [NeZero b] :	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma measure_cdf [IsProbabilityMeasure μ] : (cdf μ).measure = μ := by	 
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean:lemma measure_prod_compl_eq_zero {s : Set α} {t : Set β}   
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/MeasureSpace.lean:lemma measure_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t) :	
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/MeasureSpace.lean:lemma measure_symmDiff_le (s t u : Set α) :   
lake-packages/mathlib/Mathlib/Dynamics/Ergodic/MeasurePreserving.lean:lemma measure_symmDiff_preimage_iterate_le    
lake-packages/mathlib/Mathlib/Geometry/Manifold/ChartedSpace.lean:lemma mem_chart_source (H : Type*) {M : Type*} [TopologicalSpace H] [TopologicalSpace M]	
lake-packages/mathlib/Mathlib/RingTheory/Jacobson.lean:lemma mem_closure_X_union_C {R : Type*} [Ring R] (p : R[X]) : 
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma mem_comap_prod_mk {x : α} {s : Set β} {F : Filter (α × β)} : 
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma mem_conductorSet_dvd {x : ℕ} (hx : x ∈ conductorSet χ) : x ∣ n := hx.dvd 
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma mem_conductorSet_iff {x : ℕ} : x ∈ conductorSet χ ↔ FactorsThrough χ x := Iff.refl _	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma mem_distinguishedTriangles_iff (T : Triangle Cᵒᵖ) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma mem_distinguishedTriangles_iff' (T : Triangle Cᵒᵖ) : 
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma mem_exterior : a ∈ exterior s ↔ ∀ U, IsOpen U → s ⊆ U → a ∈ U := by simp [exterior_def]	
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma mem_generate_of_mem {s : Set <| Set α} {U : Set α} (h : U ∈ s) :	
lake-packages/mathlib/Mathlib/Algebra/Lie/Normalizer.lean:lemma mem_idealizer {x : L} : x ∈ N.idealizer ↔ ∀ m : M, ⁅x, m⁆ ∈ N := Iff.rfl	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma mem_iff (hnm : n + 1 = m) (z : Cochain F G n) :	
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:lemma mem_inf_principal {f : Filter α} {s t : Set α} : s ∈ f ⊓ 𝓟 t ↔ { x | x ∈ t → x ∈ s } ∈ f := by	
lake-packages/mathlib/Mathlib/Data/Finmap.lean:lemma mem_lookup_iff {s : Finmap β} {a : α} {b : β a} :	
lake-packages/mathlib/Mathlib/Data/List/Infix.lean:lemma mem_of_mem_dropSlice {n m : ℕ} {l : List α} {a : α} (h : a ∈ l.dropSlice n m) : a ∈ l :=	
lake-packages/mathlib/Mathlib/Data/List/Basic.lean:lemma mem_pair {a b c : α} : a ∈ [b, c] ↔ a = b ∨ a = c := by	
lake-packages/mathlib/Mathlib/Data/Polynomial/RingDivision.lean:lemma mem_roots_iff_aeval_eq_zero (w : p ≠ 0) : x ∈ roots p ↔ aeval x p = 0 := by 
lake-packages/mathlib/Mathlib/LinearAlgebra/Finsupp.lean:lemma mem_span_set' {m : M} {s : Set M} :	   
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma mem_uIcc : a ∈ [[b, c]] ↔ b ≤ a ∧ a ≤ c ∨ c ≤ a ∧ a ≤ b := by simp [uIcc_eq_union]   
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma mem_uIcc_of_ge (hb : b ≤ x) (ha : x ≤ a) : x ∈ [[a, b]] := Icc_subset_uIcc' ⟨hb, ha⟩	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma mem_uIcc_of_le (ha : a ≤ x) (hb : x ≤ b) : x ∈ [[a, b]] := Icc_subset_uIcc ⟨ha, hb⟩	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma mem_uIoc : a ∈ Ι b c ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma mk' (h : S.LeftHomologyData) : HasLeftHomology S := ⟨Nonempty.intro h⟩	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma mk' (h : S.RightHomologyData) : HasRightHomology S := ⟨Nonempty.intro h⟩	
lake-packages/mathlib/Mathlib/Data/Polynomial/Derivation.lean:lemma mkDerivation_apply (a : A) (f : R[X]) :	
lake-packages/mathlib/Mathlib/Data/Polynomial/Derivation.lean:lemma mkDerivation_one_eq_derivative (f : R[X]) : mkDerivation R (1 : R[X]) f = derivative f := by	
lake-packages/mathlib/Mathlib/Data/Polynomial/Derivation.lean:lemma mkDerivation_one_eq_derivative' : mkDerivation R (1 : R[X]) = derivative' := by	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Basic.lean:lemma mkOfSMul'_smul (r : R) (x : mkOfSMul' φ) : 
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Basic.lean:lemma mkOfSMul_smul (r : R) : (mkOfSMul φ).smul r = φ r := rfl	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma mk_XYIdeal'_mul_mk_XYIdeal' (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma mk_XYIdeal'_mul_mk_XYIdeal'_of_Y_eq :	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Separation.lean:lemma mk_eq_mk {x y : α} : (⟦x⟧ : SeparationQuotient α) = ⟦y⟧ ↔ Inseparable x y :=	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Ordinal.lean:lemma mk_iUnion_Ordinal_le_of_le {β : Type _} {o : Ordinal} {c : Cardinal} 
lake-packages/mathlib/Mathlib/Data/Prod/Basic.lean:lemma mk_inj_left : (a, b₁) = (a, b₂) ↔ b₁ = b₂ := (mk.inj_left _).eq_iff	
lake-packages/mathlib/Mathlib/Data/Prod/Basic.lean:lemma mk_inj_right : (a₁, b) = (a₂, b) ↔ a₁ = a₂ := (mk.inj_right _).eq_iff	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma mk_inv [Inv M] (a : M) : mk (a⁻¹) = (mk a)⁻¹ := rfl  
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/CountableCover.lean:lemma mk_le_of_countable_eventually_mem {α : Type u} {ι : Type v} {a : Cardinal}		
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Basic.lean:lemma mk_map [Quiver V] {obj : V → V} {map} {X Y : V} {f : X ⟶ Y} :  
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma mk_mul [Mul M] (a b : M) : mk (a * b) = mk b * mk a := rfl	
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Basic.lean:lemma mk_obj [Quiver V] {obj : V → V} {map} {X : V} : (Prefunctor.mk obj map).obj X = obj X := rfl	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/CountableCover.lean:lemma mk_of_countable_eventually_mem {α : Type u} {ι : Type v} {a : Cardinal}			
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma mk_one [One M] : mk (1 : M) = 1 := rfl   
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma mk_pow [Monoid M] (a : M) (n : ℕ) : mk (a ^ n) = mk a ^ n := rfl	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Basic.lean:lemma mk_preimage_down {s : Set α} : #(ULift.down.{v} ⁻¹' s) = lift.{v} (#s) := by		
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/CountableCover.lean:lemma mk_subtype_le_of_countable_eventually_mem {α : Type u} {ι : Type v} {a : Cardinal}	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/CountableCover.lean:lemma mk_subtype_le_of_countable_eventually_mem_aux {α ι : Type u} {a : Cardinal} 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma mk_v (v : ∀ (p q : ℤ) (_ : p + n = q), F.X p ⟶ G.X q) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma mk_zpow [DivInvMonoid M] (a : M) (n : ℤ) : mk (a ^ n) = mk a ^ n := rfl 
lake-packages/mathlib/Mathlib/GroupTheory/Submonoid/Operations.lean:lemma mker_fst : mker (fst M N) = .prod ⊥ ⊤ := SetLike.ext fun _ => (and_true_iff _).symm	
lake-packages/mathlib/Mathlib/GroupTheory/Submonoid/Operations.lean:lemma mker_snd : mker (snd M N) = .prod ⊤ ⊥ := SetLike.ext fun _ => (true_and_iff _).symm	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma modEq_one : a ≡ b [MOD 1] := modEq_of_dvd $ one_dvd _	 
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma modEq_sub (h : b ≤ a) : a ≡ b [MOD a - b] := (modEq_of_dvd $ by rw [Int.ofNat_sub h]).symm	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma monic_polynomial : W.polynomial.Monic := by		   
lake-packages/mathlib/Mathlib/CategoryTheory/Abelian/Exact.lean:lemma mono_cokernel_desc_of_exact (h : Exact f g) : Mono (cokernel.desc f g h.w) :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma mono_homologyMap_of_mono_opcyclesMap' 
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma monotone (x : LTSeries α) : Monotone x :=	 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma monotoneOn_or_antitoneOn_iff_uIcc :	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma monotone_cdf : Monotone (cdf μ) := (condCdf _ _).mono  
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma monotone_iff_forall_covby [PartialOrder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma monotone_iff_forall_wcovby [Preorder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Data/Set/Image.lean:lemma monotone_image {f : α → β} : Monotone (image f) := fun _ _ => image_subset _ 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma monotone_isoClosure (P Q : MorphismProperty C) (h : P ⊆ Q) :  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma monotone_map (P Q : MorphismProperty C) (F : C ⥤ D) (h : P ⊆ Q) :	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma monotone_or_antitone_iff_uIcc :	 
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean:lemma monotone_rpow_of_base_ge_one {b : ℝ} (hb : 1 ≤ b) : 
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma monotone_to_lowerTopology_continuous [TopologicalSpace α]		       
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma monotone_to_upperTopology_continuous [TopologicalSpace α]		       
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mono₁ (h : T.mor₃ = 0) : Mono T.mor₁ := (T.mor₃_eq_zero_iff_mono₁ hT).1 h		
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mono₂ (h : T.mor₁ = 0) : Mono T.mor₂ := (T.mor₁_eq_zero_iff_mono₂ hT).1 h		
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mono₃ (h : T.mor₂ = 0) : Mono T.mor₃ := (T.mor₂_eq_zero_iff_mono₃ hT).1 h		
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₁_eq_zero_iff_epi₃ : T.mor₁ = 0 ↔ Epi T.mor₃ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₁_eq_zero_iff_mono₂ : T.mor₁ = 0 ↔ Mono T.mor₂ := by	     
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₁_eq_zero_of_epi₃ (h : Epi T.mor₃) : T.mor₁ = 0 := (T.mor₁_eq_zero_iff_epi₃ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₁_eq_zero_of_mono₂ (h : Mono T.mor₂) : T.mor₁ = 0 := (T.mor₁_eq_zero_iff_mono₂ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₂_eq_zero_iff_epi₁ : T.mor₂ = 0 ↔ Epi T.mor₁ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₂_eq_zero_iff_mono₃ : T.mor₂ = 0 ↔ Mono T.mor₃ := 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₂_eq_zero_of_epi₁ (h : Epi T.mor₁) : T.mor₂ = 0 := (T.mor₂_eq_zero_iff_epi₁ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₂_eq_zero_of_mono₃ (h : Mono T.mor₃) : T.mor₂ = 0 := (T.mor₂_eq_zero_iff_mono₃ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₃_eq_zero_iff_epi₂ : T.mor₃ = 0 ↔ Epi T.mor₂ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₃_eq_zero_iff_mono₁ : T.mor₃ = 0 ↔ Mono T.mor₁ := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₃_eq_zero_of_epi₂ (h : Epi T.mor₂) : T.mor₃ = 0 := (T.mor₃_eq_zero_iff_epi₂ hT).2 h	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma mor₃_eq_zero_of_mono₁ (h : Mono T.mor₁) : T.mor₃ = 0 := (T.mor₃_eq_zero_iff_mono₁ hT).2 h	
lake-packages/mathlib/Mathlib/Tactic/Contrapose.lean:lemma mtr {p q : Prop} : (¬ q → ¬ p) → (p → q) := fun h hp ↦ by_contra (fun h' ↦ h h' hp)	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma mul [Mul β] {g : α → β} (hf : EventuallyConst f l) (hg : EventuallyConst g l) :	  
lake-packages/mathlib/Mathlib/Algebra/Category/SemigroupCat/Basic.lean:lemma mulEquiv_coe_eq {X Y : Type _} [Mul X] [Mul Y] (e : X ≃* Y) : 
lake-packages/mathlib/Mathlib/Algebra/Category/SemigroupCat/Basic.lean:lemma mulEquiv_coe_eq {X Y : Type _} [Semigroup X] [Semigroup Y] (e : X ≃* Y) :	
lake-packages/mathlib/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean:lemma mulIndicator_cthickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :	
lake-packages/mathlib/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean:lemma mulIndicator_thickening_eventually_eq_mulIndicator_closure (f : α → β) (E : Set α) (x : α) :	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma mulLeft_mul : Equiv.mulLeft (a * b) = Equiv.mulLeft a * Equiv.mulLeft b :=	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma mulLeft_one : Equiv.mulLeft (1 : α) = 1 := ext one_mul  
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma mulRight_mul : Equiv.mulRight (a * b) = Equiv.mulRight b * Equiv.mulRight a :=	
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma mulRight_one : Equiv.mulRight (1 : α) = 1 := ext mul_one	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma mul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (B : Matrix p n R) : 
lake-packages/mathlib/Mathlib/GroupTheory/SemidirectProduct.lean:lemma mul_def (a b : SemidirectProduct N G φ) : a * b = ⟨a.1 * φ a.2 b.1, a.2 * b.2⟩ := rfl	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma mul_fromColumns (A : Matrix m n R) (B₁ : Matrix n n₁ R) (B₂ : Matrix n n₂ R) :	
lake-packages/mathlib/Mathlib/Algebra/Order/Field/Basic.lean:lemma mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b := by	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_left_inj_of_invertible [Invertible A] {x y : Matrix m n α} : x * A = y * A ↔ x = y :=		
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_left_injective_of_inv (A : Matrix m n α) (B : Matrix n m α) (h : A * B = 1) :	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_left_injective_of_invertible [Invertible A] :		    
lake-packages/mathlib/Mathlib/Tactic/NoncommRing.lean:lemma mul_nat_lit_eq_nsmul [n.AtLeastTwo] : r * no_index (OfNat.ofNat n) = n • r := by	      
lake-packages/mathlib/Mathlib/Order/Filter/Pointwise.lean:lemma mul_neBot_iff : (f * g).NeBot ↔ f.NeBot ∧ g.NeBot :=	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma mul_of {A : Type*} [Monoid A] (a b : A) :	     
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_right_inj_of_invertible [Invertible A] {x y : Matrix n m α} : A * x = A * y ↔ x = y :=	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_right_injective_of_inv (A : Matrix m n α) (B : Matrix n m α) (h : A * B = 1) : 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean:lemma mul_right_injective_of_invertible [Invertible A] :	    
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma mul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (B : Matrix p m R) : 
lake-packages/mathlib/Mathlib/Tactic/Linarith/Lemmas.lean:lemma mul_zero_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (_ : R a 0) (h : b = 0) :  
lake-packages/mathlib/Mathlib/Algebra/Group/Basic.lean:lemma multiplicative_of_symmetric_of_isTotal 
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean:lemma multiset_prod_map_rpow {ι} (s : Multiset ι) (f : ι → ℝ≥0) (r : ℝ) :	
lake-packages/mathlib/Mathlib/Data/Int/Basic.lean:lemma natAbs_cast (n : ℕ) : natAbs ↑n = n := rfl 
lake-packages/mathlib/Mathlib/Algebra/GroupPower/Lemmas.lean:lemma natAbs_sq (x : ℤ) : (x.natAbs : ℤ) ^ 2 = x ^ 2 := by rw [sq, Int.natAbs_mul_self', sq]	
lake-packages/mathlib/Mathlib/Data/Nat/Cast/NeZero.lean:lemma natCast_ne (n : ℕ) (R) [AddMonoidWithOne R] [h : NeZero (n : R)] : (n : R) ≠ 0 := h.out	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma natDegree_norm_ne_one [IsDomain R] (x : W.CoordinateRing) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma natDegree_polynomial [Nontrivial R] : W.polynomial.natDegree = 2 := by	
lake-packages/mathlib/Mathlib/Data/Int/Cast/Lemmas.lean:lemma natMod_lt {a : ℤ} {b : ℕ} (hb : b ≠ 0) : a.natMod b < b := 
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient.lean:lemma natTransLift_app (F G : Quotient r ⥤ D) 
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient.lean:lemma natTransLift_id (F : Quotient r ⥤ D) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Sigma/Basic.lean:lemma natTrans_app {F G : (Σi, C i) ⥤ D} (h : ∀ i : I, incl i ⋙ F ⟶ incl i ⋙ G) (i : I)	
lake-packages/mathlib/Mathlib/CategoryTheory/Quotient.lean:lemma natTrans_ext {F G : Quotient r ⥤ D} (τ₁ τ₂ : F ⟶ G)  
lake-packages/mathlib/Mathlib/Tactic/NoncommRing.lean:lemma nat_lit_mul_eq_nsmul [n.AtLeastTwo] : no_index (OfNat.ofNat n) * r = n • r := by	      
lake-packages/mathlib/Mathlib/Init/Data/Nat/Lemmas.lean:lemma nat_repr_len_aux (n b e : Nat) (h_b_pos : 0 < b) :  n < b ^ e.succ → n / b < b ^ e := by	
lake-packages/mathlib/Mathlib/Tactic/NormNum/NatSqrt.lean:lemma nat_sqrt_helper {x y r : ℕ} (hr : y * y + r = x) (hle : Nat.ble r (2 * y)) :	
lake-packages/mathlib/Mathlib/Logic/Basic.lean:lemma ne_and_eq_iff_right {α : Sort*} {a b c : α} (h : b ≠ c) : a ≠ b ∧ a = c ↔ a = c :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma ne_of_ne_of_eq' (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab   
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ne_of_not_subset [IsRefl α (· ⊆ ·)] : ¬a ⊆ b → a ≠ b := mt subset_of_eq		
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ne_of_not_superset [IsRefl α (· ⊆ ·)] : ¬a ⊆ b → b ≠ a := mt superset_of_eq	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ne_of_ssubset [IsIrrefl α (· ⊂ ·)] {a b : α} : a ⊂ b → a ≠ b := ne_of_irrefl	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ne_of_ssuperset [IsIrrefl α (· ⊂ ·)] {a b : α} : a ⊂ b → b ≠ a := ne_of_irrefl'	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not			
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma ne_zero_of_eq_one [One α] [NeZero (1 : α)] {a : α} (h : a = 1) : a ≠ 0 := h ▸ one_ne_zero	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_add (n₁ n₂ : ℤ) :	  
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_def (n : ℤ) : n.negOnePow = (-1 : ℤˣ) ^ n := rfl				
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_eq_iff (n₁ n₂ : ℤ) :	     
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_eq_neg_one_iff (n : ℤ) : n.negOnePow = -1 ↔ Odd n := by	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_eq_one_iff (n : ℤ) : n.negOnePow = 1 ↔ Even n := by	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_even (n : ℤ) (hn : Even n) : n.negOnePow = 1 := by  
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_mul_self (n : ℤ) : n.negOnePow * n.negOnePow = 1 := by	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_neg (n : ℤ) : (-n).negOnePow = n.negOnePow := by  
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_odd (n : ℤ) (hn : Odd n) : n.negOnePow = -1 := by	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_one : negOnePow 1 = -1 := rfl	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_sub (n₁ n₂ : ℤ) :	     
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_succ (n : ℤ) : (n + 1).negOnePow = - n.negOnePow := by	
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_two_mul (n : ℤ) : (2 * n).negOnePow = 1 := 
lake-packages/mathlib/Mathlib/Algebra/GroupPower/NegOnePow.lean:lemma negOnePow_zero : negOnePow 0 = 1 := rfl	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma negY_negY : W.negY x₁ (W.negY x₁ y₁) = y₁ := by 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma neg_add_eq_zero (P Q : W.Point) : -P + Q = 0 ↔ P = Q := by	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma neg_app (f : P ⟶ Q) (X : Cᵒᵖ): (-f).app X = -f.app X := rfl 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma neg_def (P : W.Point) : P.neg = -P := 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma neg_some (h : W.nonsingular x₁ y₁) : -some h = some (nonsingular_neg h) :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma neg_v {n : ℤ} (z : Cochain F G n) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma neg_zero : (-0 : W.Point) = 0 :=	 
lake-packages/mathlib/Mathlib/Topology/NhdsSet.lean:lemma nhdsSet_le : 𝓝ˢ s ≤ f ↔ ∀ a ∈ s, 𝓝 a ≤ f := by simp [nhdsSet]	   
lake-packages/mathlib/Mathlib/Topology/Separation.lean:lemma nhdsWithin_compl_singleton_le [T1Space α] (x y : α) : 𝓝[{x}ᶜ] x ≤ 𝓝[{y}ᶜ] x := by		
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma nil_iff_length_eq {p : G.Walk v w} : p.Nil ↔ p.length = 0 := by  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma nil_iff_support_eq {p : G.Walk v w} : p.Nil ↔ p.support = [v] := by	
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Path.lean:lemma nil_ne_cons (p : Path a b) (e : b ⟶ a) : Path.nil ≠ p.cons e :=	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Unitization.lean:lemma nndist_inr (a b : A) : nndist (a : Unitization 𝕜 A) (b : Unitization 𝕜 A) = nndist a b :=	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Unitization.lean:lemma nnnorm_inr (a : A) : ‖(a : Unitization 𝕜 A)‖₊ = ‖a‖₊ :=  
lake-packages/mathlib/Mathlib/Data/Finmap.lean:lemma nodup_entries (f : Finmap β) : f.entries.Nodup := f.nodupKeys.nodup   
lake-packages/mathlib/Mathlib/Data/Finmap.lean:lemma nodup_keys {m : Multiset (Σ a, β a)} : m.keys.Nodup ↔ m.NodupKeys := by	
lake-packages/mathlib/Mathlib/GroupTheory/Submonoid/ZeroDivisors.lean:lemma nonZeroDivisorsLeft_eq_right (M₀ : Type _) [CommMonoidWithZero M₀] :	
lake-packages/mathlib/Mathlib/Topology/SubsetProperties.lean:lemma noncompact_univ (α : Type*) [TopologicalSpace α] [NoncompactSpace α] :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Ends/Properties.lean:lemma nonempty_ends_of_infinite [LocallyFinite G] [Fact G.Preconnected] [Infinite V] :	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :	     
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :	     
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean:lemma nonempty_isEquivalence_iff : Nonempty (IsEquivalence G) ↔ Nonempty (IsEquivalence G') := by	
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma nonempty_of_infiniteDimensional [r.InfiniteDimensional] : Nonempty α :=	
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma nonempty_of_infiniteDimensionalType [InfiniteDimensionalOrder α] : Nonempty α :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma nonneg_of_isNat [OrderedSemiring A] 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma nonneg_of_isRat [LinearOrderedRing A] :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular [Nontrivial R] {x y : R} (h : E.equation x y) : E.nonsingular x y :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_add (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_add' (hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_add_of_eval_derivative_ne_zero	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_iff (x y : R) :		  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_iff' (x y : R) :		  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_iff_baseChange [Nontrivial A] [NoZeroSMulDivisors R A] (x y : R) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_iff_baseChange_of_baseChange [Nontrivial B] [NoZeroSMulDivisors A B] (x y : A) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_iff_variableChange (x y : R) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_neg (h : W.nonsingular x₁ y₁) : W.nonsingular x₁ <| W.negY x₁ y₁ := 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_neg_iff : W.nonsingular x₁ (W.negY x₁ y₁) ↔ W.nonsingular x₁ y₁ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma nonsingular_neg_of (h : W.nonsingular x₁ <| W.negY x₁ y₁) : W.nonsingular x₁ y₁ :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_of_Δ_ne_zero {x y : R} (h : W.equation x y) (hΔ : W.Δ ≠ 0) : W.nonsingular x y :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_zero : W.nonsingular 0 0 ↔ W.a₆ = 0 ∧ (W.a₃ ≠ 0 ∨ W.a₄ ≠ 0) := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma nonsingular_zero_of_Δ_ne_zero (h : W.equation 0 0) (hΔ : W.Δ ≠ 0) : W.nonsingular 0 0 := by	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma nontrivial_iff_ne_singleton (ha : a ∈ s) : s.Nontrivial ↔ s ≠ {a} :=	       
lake-packages/mathlib/Mathlib/Data/Finset/Prod.lean:lemma nontrivial_prod_iff : Nontrivial (s ×ˢ t) ↔     
lake-packages/mathlib/test/linarith.lean:lemma norm_eq_zero_iff {x y : ℚ} : x * x + y * y = 0 ↔ x = 0 ∧ y = 0 := by	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Unitization.lean:lemma norm_inr (a : A) : ‖(a : Unitization 𝕜 A)‖ = ‖a‖ := by   
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/OperatorNorm.lean:lemma norm_mkContinuous₂_aux (f : E →ₛₗ[σ₁₃] F →ₛₗ[σ₂₃] G) (C : ℝ)     
lake-packages/mathlib/test/linarith.lean:lemma norm_nonpos_left (x y : ℚ) (h1 : x * x + y * y ≤ 0) : x = 0 := by		
lake-packages/mathlib/test/linarith.lean:lemma norm_nonpos_right {x y : ℚ} (h1 : x * x + y * y ≤ 0) : y = 0 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma norm_smul_basis (p q : R[X]) :		  
lake-packages/mathlib/test/linarith.lean:lemma norm_zero_left {x y : ℚ} (h1 : x * x + y * y = 0) : x = 0 := by	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_def : normalClosure F K L = ⨆ f : K →ₐ[F] L, f.fieldRange :=    
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_def' : normalClosure F K L = ⨆ f : L →ₐ[F] L, K.map f := by	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_def'' : normalClosure F K L = ⨆ f : L ≃ₐ[F] L, K.map f := by	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_le_iff {K' : IntermediateField F L} :     
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_mono (h : K ≤ K') : normalClosure F K L ≤ normalClosure F K' L := by	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normalClosure_of_normal [Normal F K] : normalClosure F K L = K :=	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_fieldRange_eq : Normal F K ↔ ∀ σ : K →ₐ[F] L, σ.fieldRange = K :=	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_fieldRange_le : Normal F K ↔ ∀ σ : K →ₐ[F] L, σ.fieldRange ≤ K :=	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_map_eq : Normal F K ↔ ∀ σ : L →ₐ[F] L, K.map σ = K :=	   
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_map_eq' : Normal F K ↔ ∀ σ : L ≃ₐ[F] L, K.map ↑σ = K :=	   
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_map_le : Normal F K ↔ ∀ σ : L →ₐ[F] L, K.map σ ≤ K :=	   
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_forall_map_le' : Normal F K ↔ ∀ σ : L ≃ₐ[F] L, K.map ↑σ ≤ K :=	   
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_normalClosure_eq : Normal F K ↔ normalClosure F K L = K := 
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma normal_iff_normalClosure_le : Normal F K ↔ normalClosure F K L ≤ K := 
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma not_disjoint_iff : ¬Disjoint s t ↔ ∃ x, x ∈ s ∧ x ∈ t :=	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma not_disjoint_iff_nonempty_inter : ¬ Disjoint s t ↔ (s ∩ t).Nonempty := not_disjoint_iff	
lake-packages/mathlib/Mathlib/Data/Bool/Basic.lean:lemma not_eq_iff : ∀ {a b : Bool}, !a = b ↔ a ≠ b := by decide	   
lake-packages/mathlib/Archive/Wiedijk100Theorems/Konigsberg.lean:lemma not_even_degree_iff (w : Verts) : ¬Even (degree w) ↔ w = V1 ∨ w = V2 ∨ w = V3 ∨ w = V4 := by	
lake-packages/mathlib/Mathlib/Order/Basic.lean:lemma not_lt_of_subsingleton [Subsingleton α] : ¬a < b := (Subsingleton.elim a b).not_lt	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma not_mem_uIcc_of_gt (ha : a < c) (hb : b < c) : c ∉ [[a, b]] :=	 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma not_mem_uIcc_of_lt (ha : c < a) (hb : c < b) : c ∉ [[a, b]] :=	 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma not_mem_uIoc : a ∉ Ι b c ↔ a ≤ b ∧ a ≤ c ∨ c < a ∧ b < a := by	
lake-packages/mathlib/Mathlib/Order/Monotone/Basic.lean:lemma not_monotone_not_antitone_iff_exists_le_le :	   
lake-packages/mathlib/Mathlib/Order/Monotone/Basic.lean:lemma not_monotone_not_antitone_iff_exists_lt_lt :	   
lake-packages/mathlib/Mathlib/Data/Bool/Basic.lean:lemma not_ne_self : ∀ b : Bool, (!b) ≠ b := by decide	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma not_nil_iff {p : G.Walk v w} :  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma not_nil_of_ne {p : G.Walk v w} : v ≠ w → ¬ p.Nil := mt Nil.eq   
lake-packages/mathlib/test/toAdditive.lean:lemma npowRec_zero [One M] [Mul M] (x : M) : npowRec 0 x = 1 :=	
lake-packages/mathlib/Mathlib/Data/Polynomial/RingDivision.lean:lemma nthRootsFinset_def (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] [DecidableEq R] :	
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean:lemma nullMeasurableSet_of_tendsto_indicator [NeBot L] {μ : Measure α}		 
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean:lemma nullMeasurableSet_regionBetween (μ : Measure α)	      
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean:lemma nullMeasurableSet_region_between_cc (μ : Measure α)     
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean:lemma nullMeasurableSet_region_between_co (μ : Measure α)     
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean:lemma nullMeasurableSet_region_between_oc (μ : Measure α)     
lake-packages/mathlib/Mathlib/Data/Rat/Defs.lean:lemma num_eq_zero {q : ℚ} : q.num = 0 ↔ q = 0 := by    
lake-packages/mathlib/Mathlib/Data/Rat/Defs.lean:lemma num_ne_zero {q : ℚ} : q.num ≠ 0 ↔ q ≠ 0 := num_eq_zero.not	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma nz_of_isNegNat [StrictOrderedRing A]	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma nz_of_isRat [LinearOrderedRing A] :   
lake-packages/mathlib/Mathlib/Combinatorics/Quiver/Path.lean:lemma obj_eq_of_cons_eq_cons {p : Path a b} {p' : Path a c}	   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma ofBaseChange_injective : Function.Injective <| ofBaseChange W F K := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Abelian/Projective.lean:lemma ofComplex_d_shape (Z : C) {i j : ℕ} (w : i = j + 1) : 
lake-packages/mathlib/Mathlib/Data/Nat/Digits.lean:lemma ofDigits_div_eq_ofDigits_tail (hpos : 0 < p) (digits : List ℕ)   
lake-packages/mathlib/Mathlib/Data/Nat/Digits.lean:lemma ofDigits_div_pow_eq_ofDigits_drop	  
lake-packages/mathlib/Mathlib/Data/Nat/PartENat.lean:lemma ofENat_none : ofENat Option.none = ⊤ := rfl 
lake-packages/mathlib/Mathlib/Data/Nat/PartENat.lean:lemma ofENat_some (n : ℕ) : ofENat (Option.some n) = ↑n := rfl	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma ofEpiOfIsIsoOfMono'_f' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₂)	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma ofGrothendieck_iff {X : C} {S : Presieve X} (J : GrothendieckTopology C) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_add (φ₁ φ₂ : F ⟶ G) :    
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma ofHom_apply {X Y : Type u} [CommMonoid X] [CommMonoid Y] (f : X →* Y) (x : X) :	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma ofHom_apply {X Y : Type u} [Monoid X] [Monoid Y] (f : X →* Y) (x : X) :	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_comp (f : F ⟶ G) (g : G ⟶ K) :    
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_homOf_eq_self (z : Cocycle F G 0) : ofHom (homOf z) = z := by aesop_cat	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_neg (φ : F ⟶ G) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_sub (φ₁ φ₂ : F ⟶ G) :    
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_v (φ : F ⟶ G) (p : ℤ) : (ofHom φ).v p p (add_zero p) = φ.f p := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_v_comp_d (φ : F ⟶ G) (p q q' : ℤ) (hpq : p + 0 = q) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHom_zero : ofHom (0 : F ⟶ G) = 0 := by	  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHomotopy_ofEq {φ₁ φ₂ : F ⟶ G} (h : φ₁ = φ₂) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHomotopy_refl (φ : F ⟶ G) :   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHoms_comp (φ : ∀ (p : ℤ), F.X p ⟶ G.X p) (ψ : ∀ (p : ℤ), G.X p ⟶ K.X p) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHoms_v (ψ : ∀ (p : ℤ), F.X p ⟶ G.X p) (p : ℤ) :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHoms_v_comp_d (ψ : ∀ (p : ℤ), F.X p ⟶ G.X p) (p q q' : ℤ) (hpq : p + 0 = q) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma ofHoms_zero : ofHoms (fun p => (0 : F.X p ⟶ G.X p)) = 0 := by aesop_cat	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ'_j (j : R) [Invertible j] [Invertible (j - 1728)] : (ofJ' j).j = j := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ0_c₄ : (ofJ0 R).c₄ = 0 := by  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ0_j [Invertible (3 : R)] : (ofJ0 R).j = 0 := by   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ0_Δ : (ofJ0 R).Δ = -27 := by  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ1728_c₄ : (ofJ1728 R).c₄ = -48 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ1728_j [Invertible (2 : R)] : (ofJ1728 R).j = 1728 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ1728_Δ : (ofJ1728 R).Δ = -64 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_0_of_three_eq_zero (h3 : (3 : F) = 0) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_0_of_three_ne_zero [h3 : NeZero (3 : F)] :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_0_of_two_eq_zero (h2 : (2 : F) = 0) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_1728_of_three_eq_zero (h3 : (3 : F) = 0) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_1728_of_two_eq_zero (h2 : (2 : F) = 0) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_1728_of_two_ne_zero [h2 : NeZero (2 : F)] :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_c₄ : (ofJ j).c₄ = j * (j - 1728) ^ 3 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_j : (ofJ j).j = j := by   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_ne_0_ne_1728 (h0 : j ≠ 0) (h1728 : j ≠ 1728) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma ofJ_Δ : (ofJ j).Δ = j ^ 2 * (j - 1728) ^ 9 := by	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma ofReal_cdf [IsProbabilityMeasure μ] (x : ℝ) : ENNReal.ofReal (cdf μ x) = μ (Iic x) := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma ofZeros_g' (hf : S.f = 0) (hg : S.g = 0) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Composition.lean:lemma of_comp (W₃ : MorphismProperty C₁)	     
lake-packages/mathlib/Mathlib/RingTheory/FreeCommRing.lean:lemma of_cons (a : α) (m : Multiset α) : (FreeAbelianGroup.of (Multiplicative.ofAdd (a ::ₘ m))) =	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma of_dvd (d : m ∣ n) (h : a ≡ b [MOD n]) : a ≡ b [MOD m] := modEq_of_dvd $ d.natCast.trans h.dvd	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Equivalence.lean:lemma of_equivalence_source (L₁ : C₁ ⥤ D) (W₁ : MorphismProperty C₁)	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Equivalence.lean:lemma of_equivalences (L₁ : C₁ ⥤ D₁) (W₁ : MorphismProperty C₁) [L₁.IsLocalization W₁]	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma of_mulIndicator_const (h : EventuallyConst (s.mulIndicator fun _ ↦ c) l) (hc : c ≠ 1) :	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma of_mul_left (m : ℕ) (h : a ≡ b [MOD m * n]) : a ≡ b [MOD n] := by	     
lake-packages/mathlib/Mathlib/Data/Int/ModEq.lean:lemma of_mul_left (m : ℤ) (h : a ≡ b [ZMOD m * n]) : a ≡ b [ZMOD n] := by	
lake-packages/mathlib/Mathlib/Data/Nat/ModEq.lean:lemma of_mul_right (m : ℕ) : a ≡ b [MOD n * m] → a ≡ b [MOD n] := mul_comm m n ▸ of_mul_left _	
lake-packages/mathlib/Mathlib/Data/Int/ModEq.lean:lemma of_mul_right (m : ℤ) : a ≡ b [ZMOD n * m] → a ≡ b [ZMOD n] := 
lake-packages/mathlib/Mathlib/Data/Nat/Cast/NeZero.lean:lemma of_neZero_natCast (R) [AddMonoidWithOne R] {n : ℕ} [h : NeZero (n : R)] : NeZero n :=	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma of_op (W : MorphismProperty C) [IsMultiplicative W.op] : IsMultiplicative W :=   
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma of_op (W : MorphismProperty C) [W.op.ContainsIdentities] :  
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma of_subset (P Q : MorphismProperty C) (F : C ⥤ D) (hQ : Q.IsInvertedBy F) (h : P ⊆ Q) :	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma of_subsingleton_left [Subsingleton α] : EventuallyConst f l :=		  
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma of_subsingleton_right [Subsingleton β] : EventuallyConst f l := .of_subsingleton 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma of_unop (W : MorphismProperty Cᵒᵖ) [IsMultiplicative W.unop] : IsMultiplicative W :=	
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma of_unop (W : MorphismProperty Cᵒᵖ) [W.unop.ContainsIdentities] :	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Ordinal.lean:lemma omega_lt_omega1 : ω < ω₁ := ord_aleph0.symm.trans_lt (ord_lt_ord.mpr (aleph0_lt_aleph_one))	
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma oneHom_apply (X Y : MonCat.{u}) (x : X) : (1 : X ⟶ Y) x = 1 := rfl 
lake-packages/mathlib/test/toAdditive.lean:lemma one_fooClass [One α] : FooClass α := by infer_instance  
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma one_le_two [LE α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :	    
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma one_le_two' [LE α] [ZeroLEOneClass α] [CovariantClass α α (swap (·+·)) (·≤·)] :	
lake-packages/mathlib/Mathlib/Data/Nat/Basic.lean:lemma one_lt_mul_iff : 1 < m * n ↔ 0 < m ∧ 0 < n ∧ (1 < m ∨ 1 < n) := by	
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma one_lt_two [CovariantClass α α (·+·) (·<·)] : (1 : α) < 2 := by	
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma one_ne_zero' [One α] [NeZero (1 : α)] : (1 : α) ≠ 0 := one_ne_zero   
lake-packages/mathlib/Mathlib/Algebra/Category/MonCat/Basic.lean:lemma one_of {A : Type*} [Monoid A] : (1 : MonCat.of A) = (1 : A) := rfl 
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Opposites.lean:lemma opCoproductIsoProduct_inv_comp_ι (b : α) :   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Basic.lean:lemma opMap_id : opMap (𝟙 S) = 𝟙 S.op := rfl	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Opposites.lean:lemma opProductIsoCoproduct_inv_comp_π_op {X : C} (π : (a : α) → X ⟶ Z a) :	    
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma opShiftFunctorEquivalence_counitIso_hom_app_shift (X : Cᵒᵖ) (n : ℤ) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma opShiftFunctorEquivalence_counitIso_inv_app_shift (X : Cᵒᵖ) (n : ℤ) : 
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/OperatorNorm.lean:lemma op_nnnorm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖₊ = ‖x‖₊ :=	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Star/Unitization.lean:lemma op_nnnorm_mul_flip_apply (a : E) : ‖(mul 𝕜 E).flip a‖₊ = ‖a‖₊ :=	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/OperatorNorm.lean:lemma op_norm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ = ‖x‖ :=	
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/Star/Unitization.lean:lemma op_norm_mul_flip_apply (a : E) : ‖(mul 𝕜 E).flip a‖ = ‖a‖ := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesIsoRightHomology_hom_inv_id (hg : S.g = 0) :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesIsoRightHomology_inv_hom_id (hg : S.g = 0) :	   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesIsoX₂_hom_inv_id (hf : S.f = 0) :		 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesIsoX₂_inv_hom_id (hf : S.f = 0) :		 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap'_eq : opcyclesMap' φ h₁ h₂ = γ.φQ := 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma opcyclesMap'_eq : opcyclesMap' φ h₁.right h₂.right = γ.right.φQ :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap'_g' : opcyclesMap' φ h₁ h₂ ≫ h₂.g' = h₁.g' ≫ φ.τ₃ := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap'_id (h : S.RightHomologyData) :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap'_zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_comm [S₁.HasRightHomology] [S₂.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_comp [HasRightHomology S₁] [HasRightHomology S₂] [HasRightHomology S₃]  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_comp_descOpcycles (φ : S₁ ⟶ S) [S₁.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_eq [S₁.HasRightHomology] [S₂.HasRightHomology] :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_id [HasRightHomology S] : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcyclesMap_zero [HasRightHomology S₁] [HasRightHomology S₂] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcycles_ext (f₁ f₂ : S.opcycles ⟶ A)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma opcycles_ext_iff (f₁ f₂ : S.opcycles ⟶ A) :	  
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean:lemma opensMeasurableSpace_iff_forall_measurableSet 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorAdd'_hom_app :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorAdd'_inv_app :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorAdd_hom_app :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorAdd_inv_app :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorZero_hom_app (X : OppositeShift C A) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Opposite.lean:lemma oppositeShiftFunctorZero_inv_app (X : OppositeShift C A) :  
lake-packages/mathlib/test/toAdditive.lean:lemma optiontest (x : Option α) : x.elim .none Option.some = x :=	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/SubMulAction.lean:lemma orbit_of_sub_mul {p : SubMulAction R M} (m : p) :		 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma pOpcycles_comp_opcyclesIso_hom : S.pOpcycles ≫ h.opcyclesIso.hom = h.p := by   
lake-packages/mathlib/Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean:lemma p_comm_f {P Q : Karoubi (Karoubi C)} (f : P ⟶ Q) : P.p.f ≫ f.f.f = f.f.f ≫ Q.p.f := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_comp_opcyclesIso_inv : h.p ≫ h.opcyclesIso.inv = S.pOpcycles := by	     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_descOpcycles : S.pOpcycles ≫ S.descOpcycles k hk = k :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_descQ (k : S.X₂ ⟶ A) (hk : S.f ≫ k = 0) : h.p ≫ h.descQ k hk = k :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_fromOpcycles : S.pOpcycles ≫ S.fromOpcycles = S.g := S.rightHomologyData.p_g'	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_opcyclesMap : S₁.pOpcycles ≫ opcyclesMap φ = φ.τ₂ ≫ S₂.pOpcycles :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma p_opcyclesMap' : h₁.p ≫ opcyclesMap' φ h₁ h₂ = φ.τ₂ ≫ h₂.p :=	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma periodic {m : ℕ} (hm : n ∣ m) : Function.Periodic χ m := by   
lake-packages/mathlib/Mathlib/Logic/Equiv/Basic.lean:lemma piCongrLeft'_symm_apply_apply (P : α → Sort*) (e : α ≃ β) (g : ∀ b, P (e.symm b)) (b : β) :	
lake-packages/mathlib/Mathlib/Logic/Equiv/Basic.lean:lemma piCongrLeft_apply (f : ∀ a, P (e a)) (b : β) :   
lake-packages/mathlib/Mathlib/Logic/Equiv/Basic.lean:lemma piCongrLeft_apply_apply (f : ∀ a, P (e a)) (a : α) : 
lake-packages/mathlib/Mathlib/Logic/Equiv/Basic.lean:lemma piCongrLeft_symm_apply (g : ∀ b, P b) (a : α) :  
lake-packages/mathlib/Mathlib/LinearAlgebra/StdBasis.lean:lemma piEquiv_apply_apply (v : ι → M) (w : ι → R) :				
lake-packages/mathlib/Mathlib/GroupTheory/Submonoid/Pointwise.lean:lemma pointwise_isCentralScalar [DistribMulAction αᵐᵒᵖ A] [IsCentralScalar α A] :  
lake-packages/mathlib/Mathlib/GroupTheory/Submonoid/Pointwise.lean:lemma pointwise_isCentralScalar [MulDistribMulAction αᵐᵒᵖ M] [IsCentralScalar α M] :		
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma polynomial_eq :			
lake-packages/mathlib/Mathlib/FieldTheory/Perfect.lean:lemma polynomial_expand_eq (f : R[X]) :	 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma polynomial_ne_zero [Nontrivial R] : W.polynomial ≠ 0 := by	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma pos_of_isNat [StrictOrderedSemiring A]	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Core.lean:lemma pos_of_isRat [LinearOrderedRing A] :	
lake-packages/mathlib/Mathlib/Data/Nat/Cast/NeZero.lean:lemma pos_of_neZero_natCast (R) [AddMonoidWithOne R] {n : ℕ} [NeZero (n : R)] : 0 < n := 
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma pow_mulLeft (n : ℕ) : Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n) := by	  
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma pow_mulRight (n : ℕ) : Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n) := by  
lake-packages/mathlib/Mathlib/Algebra/Order/LatticeGroup.lean:lemma pow_two_semiclosed    
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma preconnectedSpace_iff_univ : PreconnectedSpace α ↔ IsPreconnected (univ : Set α) :=				
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma preconnected_iff_forall_exists_walk_subgraph (H : G.Subgraph) :	
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma pred_lt_pred_of_not_isMin (h : a < b) (ha : ¬ IsMin a) : pred a < pred b :=	
lake-packages/mathlib/Mathlib/Data/Set/Image.lean:lemma preimage_symmDiff {f : α → β} (s t : Set β) : f ⁻¹' (s ∆ t) = (f ⁻¹' s) ∆ (f ⁻¹' t) :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/Scheme.lean:lemma presheaf_map_eqToHom_op (X : Scheme) (U V : Opens X) (i : U = V) :	
lake-packages/mathlib/Mathlib/NumberTheory/Divisors.lean:lemma prime_divisors_filter_dvd_of_dvd {m n : ℕ} (hn : n ≠ 0) (hmn : m ∣ n) :	
lake-packages/mathlib/test/rewrites.lean:lemma prime_of_prime (n : ℕ) : Prime n ↔ Nat.Prime n := by  
lake-packages/mathlib/test/LibrarySearch/basic.lean:lemma prime_of_prime (n : ℕ) : Prime n ↔ Nat.Prime n := by 
lake-packages/mathlib/Mathlib/Algebra/FreeMonoid/Basic.lean:lemma prodAux_eq : ∀ l : List M, FreeMonoid.prodAux l = l.prod	
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma prod_indicator_index [Zero M] [CommMonoid N] 
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma prod_indicator_index_eq_prod_attach [Zero M] [CommMonoid N] 
lake-packages/mathlib/Mathlib/Data/List/BigOperators/Basic.lean:lemma prod_int_mod (l : List ℤ) (n : ℤ) : l.prod % n = (l.map (· % n)).prod % n := by	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:lemma prod_mk {g : α → γ} (hf : EventuallyConst f l) (hg : EventuallyConst g l) :	  
lake-packages/mathlib/Mathlib/Data/List/BigOperators/Basic.lean:lemma prod_nat_mod (l : List ℕ) (n : ℕ) : l.prod % n = (l.map (· % n)).prod % n := by	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Basic.lean:lemma projective_of_extrDisc {X : Profinite.{u}} (hX : ExtremallyDisconnected X) :   
lake-packages/mathlib/Mathlib/Analysis/NormedSpace/FiniteDimension.lean:lemma properSpace_of_locallyCompactSpace (𝕜 : Type*) [NontriviallyNormedField 𝕜]     
lake-packages/mathlib/Mathlib/RingTheory/MvPolynomial/Symmetric.lean:lemma psum_def (n : ℕ) : psum σ R n = ∑ i, X i ^ n := rfl		 
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma pullback.condition : pullback.fst f g ≫ f = pullback.snd f g ≫ g := by	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma pullback.condition : pullback.fst f g ≫ f = pullback.snd f g ≫ g := by	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma pullback.condition {X Y Z : Stonean.{u}} (f : X ⟶ Z) {i : Y ⟶ Z}	 
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma pullback.hom_ext {X Y Z W : Stonean} (f : X ⟶ Z) {i : Y ⟶ Z} (hi : OpenEmbedding i)	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma pullback.hom_ext {Z : CompHaus.{u}} (a b : Z ⟶ pullback f g) 
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma pullback.hom_ext {Z : Profinite.{u}} (a b : Z ⟶ pullback f g) 
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma pullback.lift_fst {W : Stonean} (a : W ⟶ X) (b : W ⟶ Y) (w : a ≫ f = b ≫ i) :	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma pullback.lift_fst {Z : CompHaus.{u}} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) :	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma pullback.lift_fst {Z : Profinite.{u}} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) :	
lake-packages/mathlib/Mathlib/Topology/Category/Stonean/Limits.lean:lemma pullback.lift_snd {X Y Z W : Stonean} (f : X ⟶ Z) {i : Y ⟶ Z} (hi : OpenEmbedding i)	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/Limits.lean:lemma pullback.lift_snd {Z : CompHaus.{u}} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) :	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/Limits.lean:lemma pullback.lift_snd {Z : Profinite.{u}} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/CoverLifting.lean:lemma pullbackSheafificationCompatibility_hom_app_val  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean:lemma pullbackShiftFunctorAdd'_hom_app :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean:lemma pullbackShiftFunctorAdd'_inv_app :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean:lemma pullbackShiftFunctorZero_hom_app :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean:lemma pullbackShiftFunctorZero_inv_app :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff (φ : S₁ ⟶ S₂) :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_comp_left (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ : QuasiIso φ] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_comp_right (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) [hφ' : QuasiIso φ'] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_isIso_descOpcycles (φ : S₁ ⟶ S₂)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_isIso_homologyMap' (φ : S₁ ⟶ S₂)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_isIso_leftHomologyMap' (φ : S₁ ⟶ S₂)	  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_isIso_liftCycles (φ : S₁ ⟶ S₂) 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_isIso_rightHomologyMap' (φ : S₁ ⟶ S₂)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_iff_of_arrow_mk_iso (φ : S₁ ⟶ S₂) (φ' : S₃ ⟶ S₄) (e : Arrow.mk φ ≅ Arrow.mk φ') :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_of_arrow_mk_iso (φ : S₁ ⟶ S₂) (φ' : S₃ ⟶ S₄) (e : Arrow.mk φ ≅ Arrow.mk φ')	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_of_comp_left (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃)	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_of_comp_right (φ : S₁ ⟶ S₂) (φ' : S₂ ⟶ S₃) 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_opMap (φ : S₁ ⟶ S₂) [QuasiIso φ] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_opMap_iff (φ : S₁ ⟶ S₂) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean:lemma quasiIso_unopMap {S₁ S₂ : ShortComplex Cᵒᵖ} [S₁.HasHomology] [S₂.HasHomology]	
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Quotient.lean:lemma quotientEquivQuotientMvPolynomial_leftInverse (I : Ideal R) :  
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Quotient.lean:lemma quotientEquivQuotientMvPolynomial_rightInverse (I : Ideal R) : 
lake-packages/mathlib/Mathlib/GroupTheory/Subgroup/Basic.lean:lemma rangeRestrict_injective_iff {f : G →* N} : Injective f.rangeRestrict ↔ Injective f := by	
lake-packages/mathlib/Mathlib/Algebra/Algebra/Subalgebra/Tower.lean:lemma range_isScalarTower_toAlgHom [CommSemiring R] [CommSemiring A] 
lake-packages/mathlib/Mathlib/GroupTheory/Coset.lean:lemma range_mk : range (QuotientGroup.mk (s := s)) = univ := range_iff_surjective.mpr mk_surjective	
lake-packages/mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean:lemma rank_eq [Nontrivial R] [Module.Free R M] :   
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Spectrum.lean:lemma rank_eq_card_non_zero_eigs : A.rank = Fintype.card {i // hA.eigenvalues i ≠ 0} := by	
lake-packages/mathlib/Mathlib/LinearAlgebra/Dimension.lean:lemma rank_eq_mk_of_infinite_lt [Infinite K] (h_lt : lift.{v} #K < lift.{u} #V) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Finrank.lean:lemma rank_eq_ofNat_iff_finrank_eq_ofNat (n : ℕ) [Nat.AtLeastTwo n] :  
lake-packages/mathlib/Mathlib/LinearAlgebra/Finrank.lean:lemma rank_eq_one_iff_finrank_eq_one : Module.rank K V = 1 ↔ finrank K V = 1 :=	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Spectrum.lean:lemma rank_eq_rank_diagonal : A.rank = (Matrix.diagonal hA.eigenvalues).rank := by	
lake-packages/mathlib/Mathlib/Data/Matrix/Rank.lean:lemma rank_mul_eq_left_of_isUnit_det [DecidableEq n]	
lake-packages/mathlib/Mathlib/Data/Matrix/Rank.lean:lemma rank_mul_eq_right_of_isUnit_det [DecidableEq m]	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma realPart_comp_subtype_selfAdjoint : 
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma realPart_idem {x : A} : ℜ (ℜ x : A) = ℜ x := 
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma realPart_imaginaryPart {x : A} : ℜ (ℑ x : A) = ℑ x :=	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma realPart_ofReal (r : ℝ) : (ℜ (r : ℂ) : ℂ) = r := by 
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma realPart_surjective : Function.Surjective (realPart (A := A)) :=	
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:lemma reciprocalFactors_even {n : ℕ} (h0 : n ≠ 0) (h2 : Even n) : 
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:lemma reciprocalFactors_odd {n : ℕ} (h1 : n ≠ 1) (h2 : Odd n) :	
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflGen_eq_self (hr : Reflexive r) : ReflGen r = r := by	
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflGen_minimal {r' : α → α → Prop} (hr' : Reflexive r') (h : ∀ x y, r x y → r' x y) {x y : α}	
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflTransGen_eq_reflGen (hr : Transitive r) :	     
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflTransGen_eq_transGen (hr : Reflexive r) :	     
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflTransGen_minimal {r' : α → α → Prop} (hr₁ : Reflexive r') (hr₂ : Transitive r') 
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma reflTransGen_wcovby_eq_reflTransGen_covby [PartialOrder α] :	
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma reflexive_reflGen : Reflexive (ReflGen r) := fun _ ↦ .refl	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_and : ((·↔·) ⇒ (·↔·) ⇒ (·↔·)) (·∧·) (·∧·) := 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_eq {r : α → β → Prop} (hr : BiUnique r) : (r ⇒ r ⇒ (·↔·)) (·=·) (·=·) :=	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_iff : ((·↔·) ⇒ (·↔·) ⇒ (·↔·)) (·↔·) (·↔·) := 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_imp : (Iff ⇒ (Iff ⇒ Iff)) (· → ·) (· → ·) := 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_not : (Iff ⇒ Iff) Not Not := 
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma rel_of_lt [IsTrans α r] (x : RelSeries r) {i j : Fin (x.length + 1)} (h : i < j) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:lemma rel_or : ((·↔·) ⇒ (·↔·) ⇒ (·↔·)) (·∨·) (·∨·) :=  
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma rel_or_eq_of_le [IsTrans α r] (x : RelSeries r) {i j : Fin (x.length + 1)} (h : i ≤ j) :	
lake-packages/mathlib/Mathlib/Data/List/Basic.lean:lemma replicate_one (a : α) : replicate 1 a = [a] := rfl  
lake-packages/mathlib/Mathlib/LinearAlgebra/FreeModule/PID.lean:lemma repr_eq_zero_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :	
lake-packages/mathlib/Mathlib/Init/Data/Nat/Lemmas.lean:lemma repr_length (n e : Nat) : 0 < e → n < 10 ^ e → (Nat.repr n).length <= e := by	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖) 
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell_semi_normed {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) 
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell_semi_normed_zpow {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ} (εpos : 0 < ε) {x : E}	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell_zpow (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖) {ε : ℝ}	
lake-packages/mathlib/Mathlib/Analysis/Seminorm.lean:lemma rescale_to_shell_zpow [NormedAddCommGroup F] [NormedSpace 𝕜 F] {c : 𝕜} (hc : 1 < ‖c‖)	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma restrictPreimage_bijective (hf : Bijective f) : Bijective (t.restrictPreimage f) := 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma restrictPreimage_injective (hf : Injective f) : Injective (t.restrictPreimage f) :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma restrictPreimage_surjective (hf : Surjective f) : Surjective (t.restrictPreimage f) :=	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean:lemma restrictScalarsComp'_hom_apply (M : ModuleCat R₃) (x : M) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean:lemma restrictScalarsComp'_inv_apply (M : ModuleCat R₃) (x : M) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean:lemma restrictScalarsId'_hom_apply (M : ModuleCat R) (x : M) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean:lemma restrictScalarsId'_inv_apply (M : ModuleCat R) (x : M) :	
lake-packages/mathlib/Mathlib/FieldTheory/NormalClosure.lean:lemma restrictScalars_eq :		  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma restrict_Adj {G' G'' : G.Subgraph} (v w : G'.verts) :	
lake-packages/mathlib/Mathlib/Data/Polynomial/Reverse.lean:lemma revAt_eq_self_of_lt {N i : ℕ} (h : N < i) : revAt N i = i := by simp [revAt, Nat.not_le.mpr h]	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean:lemma reverse_charpoly (M : Matrix n n R) :   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma rightHomologyIso_hom_comp_homologyι :	 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyIso_hom_comp_ι :	       
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma rightHomologyIso_hom_naturality :			
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyIso_inv_comp_rightHomologyι :   
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma rightHomologyIso_inv_naturality :			
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap'_comp (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃)     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap'_eq : rightHomologyMap' φ h₁ h₂ = γ.φH :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap'_id (h : S.RightHomologyData) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap'_op  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap'_zero (h₁ : S₁.RightHomologyData) (h₂ : S₂.RightHomologyData) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_comm [S₁.HasRightHomology] [S₂.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_comp [HasRightHomology S₁] [HasRightHomology S₂] [HasRightHomology S₃]	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_eq [S₁.HasRightHomology] [S₂.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_id [HasRightHomology S] :     
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_op (φ : S₁ ⟶ S₂) [S₁.HasRightHomology] [S₂.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyMap_zero [HasRightHomology S₁] [HasRightHomology S₂] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomology_ext (f₁ f₂ : A ⟶ S.rightHomology)	  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomology_ext_iff (f₁ f₂ : A ⟶ S.rightHomology) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyι_comp_fromOpcycles :		
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyι_descOpcycles_π_eq_zero_of_boundary (x : S.X₃ ⟶ A) (hx : k = S.g ≫ x) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyι_naturality :	    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma rightHomologyι_naturality' :   
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma rightInvOn_id (s : Set α) : RightInvOn id id s := fun _ _ ↦ rfl	
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:lemma right_ne_zero_of_smul : a • b ≠ 0 → b ≠ 0 := mt $ smul_eq_zero_of_right a	    
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma sInf_iUnion_Ici (f : ι → α) : sInf (⋃ (i : ι), Ici (f i)) = ⨅ i, f i := 
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma sInf_prod [InfSet α] [InfSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty) :	
lake-packages/mathlib/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean:lemma sSup_iUnion_Iic (f : ι → α) : sSup (⋃ (i : ι), Iic (f i)) = ⨆ i, f i := by	
lake-packages/mathlib/Mathlib/Order/CompleteLattice.lean:lemma sSup_prod [SupSet α] [SupSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty) :	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma same_level : FactorsThrough χ n := ⟨dvd_refl n, χ, (changeLevel_self χ).symm⟩	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma saturate_of_superset (K : Coverage C) {X : C} {S T : Sieve X} (h : S ≤ T)	
lake-packages/mathlib/Mathlib/Topology/Bases.lean:lemma secondCountableTopology_iInf {ι} [Countable ι] {t : ι → TopologicalSpace α}	
lake-packages/mathlib/Mathlib/CategoryTheory/Types.lean:lemma sections_property {F : J ⥤ Type w} (s : (F.sections : Type _))   
lake-packages/mathlib/Mathlib/Analysis/Convex/Segment.lean:lemma segment_inter_eq_endpoint_of_linearIndependent_of_ne [OrderedCommRing 𝕜] [NoZeroDivisors 𝕜]	
lake-packages/mathlib/Mathlib/Analysis/Convex/Segment.lean:lemma segment_inter_eq_endpoint_of_linearIndependent_sub   
lake-packages/mathlib/Mathlib/Data/Nat/Digits.lean:lemma self_div_pow_eq_ofDigits_drop (i n : ℕ) (h : 2 ≤ p):	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma self_mul_conjTranspose_eq_zero {A : Matrix m n R} : A * Aᴴ = 0 ↔ A = 0 :=	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma self_mul_conjTranspose_mulVec_eq_zero (A : Matrix m n R) (v : m → R) :   
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma self_mul_conjTranspose_mul_eq_zero (A : Matrix m n R) (B : Matrix m p R) : 
lake-packages/mathlib/Mathlib/Data/Bool/Basic.lean:lemma self_ne_not : ∀ b : Bool, b ≠ !b := by decide	
lake-packages/mathlib/Archive/Wiedijk100Theorems/Konigsberg.lean:lemma setOf_odd_degree_eq :  
lake-packages/mathlib/Mathlib/Topology/Category/TopCat/Opens.lean:lemma set_range_forget_map_inclusion {X : TopCat} (U : Opens X) :	       
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Whiskering.lean:lemma sheafCompose_comp :		 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Whiskering.lean:lemma sheafCompose_id : sheafCompose_map (F := F) J (𝟙 _) = 𝟙 _ := rfl	 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_add_zero (a : A) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_add_zero_hom_app (a : A) (X : C) :		  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_add_zero_inv_app (a : A) (X : C) :		  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_assoc (a₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A)	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_assoc_hom_app (a₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A)	  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_assoc_inv_app (a₁ a₂ a₃ a₁₂ a₂₃ a₁₂₃ : A)	  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd'_eq (a b c : ℤ) (h : a + b = c) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_eq_shiftFunctorAdd (i j : A) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd'_hom_app_f' (K : CochainComplex C ℤ) (a b ab : ℤ) (h : a + b = ab) (n : ℤ) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd'_inv_app_f' (K : CochainComplex C ℤ) (a b ab : ℤ) (h : a + b = ab) (n : ℤ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shiftFunctorAdd'_op_hom_app (X : Cᵒᵖ) (a₁ a₂ a₃ : ℤ) (h : a₁ + a₂ = a₃)	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shiftFunctorAdd'_op_inv_app (X : Cᵒᵖ) (a₁ a₂ a₃ : ℤ) (h : a₁ + a₂ = a₃)	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_zero_add (a : A) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_zero_add_hom_app (a : A) (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd'_zero_add_inv_app (a : A) (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_add_zero_hom_app (a : A) (X : C) : (shiftFunctorAdd C a 0).hom.app X =	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_add_zero_inv_app (a : A) (X : C) : (shiftFunctorAdd C a 0).inv.app X =	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_assoc (a₁ a₂ a₃ : A) :   
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_assoc_hom_app (a₁ a₂ a₃ : A) (X : C) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_assoc_inv_app (a₁ a₂ a₃ : A) (X : C) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd_eq (a b : ℤ) :  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd_hom_app_f (K : CochainComplex C ℤ) (a b n : ℤ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma shiftFunctorAdd_hom_app_obj_of_induced (a b : A) (X : C) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorAdd_inv_app_f (K : CochainComplex C ℤ) (a b n : ℤ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma shiftFunctorAdd_inv_app_obj_of_induced (a b : A) (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_zero_add_hom_app (a : A) (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorAdd_zero_add_inv_app (a : A) (X : C) : (shiftFunctorAdd C 0 a).inv.app X =	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorComm_eq (i j k : A) (h : i + j = k) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorComm_eq_refl (i : A) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorComm_hom_app_comp_shift_shiftFunctorAdd_hom_app (m₁ m₂ m₃ : A) (X : C) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorComm_hom_app_f (K : CochainComplex C ℤ) (a b p : ℤ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorComm_symm (i j : A) :	 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorComm_zero_hom_app (a : A) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorZero_eq : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorZero_hom_app_f (K : CochainComplex C ℤ) (n : ℤ) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma shiftFunctorZero_hom_app_obj_of_induced (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorZero_hom_app_shift (n : A) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctorZero_inv_app_f (K : CochainComplex C ℤ) (n : ℤ) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma shiftFunctorZero_inv_app_obj_of_induced (X : C) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma shiftFunctorZero_inv_app_shift (n : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shiftFunctorZero_op_hom_app (X : Cᵒᵖ) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shiftFunctorZero_op_inv_app (X : Cᵒᵖ) :  
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Localization.lean:lemma shiftFunctor_comp_inverts (a : A) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctor_map_f' {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n p : ℤ) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean:lemma shiftFunctor_obj_d' (K : CochainComplex C ℤ) (n i j : ℤ) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma shiftFunctor_of_induced (a : A) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shiftFunctor_op_map (n m : ℤ) (hnm : n + m = 0) {K L : Cᵒᵖ} (φ : K ⟶ L) :	
lake-packages/mathlib/Mathlib/Init/Data/Nat/Bitwise.lean:lemma shiftLeft_eq' (m n : Nat) : shiftLeft m n = m <<< n := rfl	 
lake-packages/mathlib/Mathlib/Init/Data/Nat/Bitwise.lean:lemma shiftRight_eq (m n : Nat) : shiftRight m n = m >>> n := rfl	 
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma shift_distinguished (n : ℤ) :	   
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shift_unop_opShiftFunctorEquivalence_counitIso_hom_app (X : Cᵒᵖ) (n : ℤ) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Opposite.lean:lemma shift_unop_opShiftFunctorEquivalence_counitIso_inv_app (X : Cᵒᵖ) (n : ℤ) :	
lake-packages/mathlib/Mathlib/Data/Set/Image.lean:lemma sigma_mk_preimage_image' (h : i ≠ j) : Sigma.mk j ⁻¹' (Sigma.mk i '' s) = ∅ := by	
lake-packages/mathlib/Mathlib/Data/Set/Image.lean:lemma sigma_mk_preimage_image_eq_self : Sigma.mk i ⁻¹' (Sigma.mk i '' s) = s := by   
lake-packages/mathlib/Mathlib/Data/Finsupp/Indicator.lean:lemma single_eq_indicator (b : α) : single i b = indicator {i} (fun _ _ => b) := by 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Sieves.lean:lemma singleton.mk {f : Y ⟶ X} : singleton f f := singleton'.mk	   
lake-packages/mathlib/test/rewrites.lean:lemma six_eq_seven : 6 = 7 := test_sorry 
lake-packages/mathlib/Mathlib/Data/Fin/Basic.lean:lemma size_positive : Fin n → 0 < n	
lake-packages/mathlib/Mathlib/Data/Fin/Basic.lean:lemma size_positive' [Nonempty (Fin n)] : 0 < n :=	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma skewAdjointPart_eq_I_smul_imaginaryPart (x : A) :	      
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma slope_of_X_ne (hx : x₁ ≠ x₂) : W.slope x₁ x₂ y₁ y₂ = (y₁ - y₂) / (x₁ - x₂) := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma slope_of_Y_eq (hx : x₁ = x₂) (hy : y₁ = W.negY x₂ y₂) : W.slope x₁ x₂ y₁ y₂ = 0 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma slope_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.negY x₂ y₂) : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma slope_of_Y_ne_eq_eval (hx : x₁ = x₂) (hy : y₁ ≠ W.negY x₂ y₂) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma smul (x : R[X]) (y : W.CoordinateRing) : x • y = mk W (C x) * y :=     
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma smul_basis_eq_zero {p q : R[X]} (hpq : p • (1 : W.CoordinateRing) + q • mk W Y = 0) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma smul_basis_mul_C (p q : R[X]) :		  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma smul_basis_mul_Y (p q : R[X]) :		  
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:lemma smul_eq_zero_of_left (h : a = 0) (b : M) : a • b = 0 := h.symm ▸ zero_smul _ b	
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:lemma smul_eq_zero_of_right (a : R) (h : b = 0) : a • b = 0 := h.symm ▸ smul_zero a	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Basic.lean:lemma smul_naturality {M N : ModuleCat.{v} R} (f : M ⟶ N) (r : R) : 
lake-packages/mathlib/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean:lemma snd_prod [IsProbabilityMeasure μ] : (μ.prod ν).snd = ν := by  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_self_of_Y_eq (hy : y₁ = W.negY x₁ y₁) : some h₁ + some h₁ = 0 :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_self_of_Y_ne (hy : y₁ ≠ W.negY x₁ y₁) : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_self_of_Y_ne' (hy : y₁ ≠ W.negY x₁ y₁) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_some_of_X_ne (hx : x₁ ≠ x₂) :   
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_some_of_X_ne' (hx : x₁ ≠ x₂) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_some_of_Y_eq (hx : x₁ = x₂) (hy : y₁ = W.negY x₂ y₂) : some h₁ + some h₂ = 0 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_some_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.negY x₂ y₂) : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma some_add_some_of_Y_ne' (hx : x₁ = x₂) (hy : y₁ ≠ W.negY x₂ y₂) :	
lake-packages/mathlib/Mathlib/Tactic/CategoryTheory/Elementwise.lean:lemma some_lemma {C : Type*} [Category C]  
lake-packages/mathlib/Mathlib/Tactic/CategoryTheory/Elementwise.lean:lemma some_lemma_apply {C : Type*} [Category C]	
lake-packages/mathlib/Mathlib/LinearAlgebra/Finsupp.lean:lemma span_eq_iUnion_nat (s : Set M) :	
lake-packages/mathlib/Mathlib/Data/Complex/Module.lean:lemma span_selfAdjoint : span ℂ (selfAdjoint A : Set A) = ⊤ := by   
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma specializes_iff_exterior_subset : x ⤳ y ↔ exterior {x} ⊆ exterior {y} := by 
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Spectrum.lean:lemma spectral_theorem' : 
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:lemma sqrt.iter_sq_le (n guess : ℕ) : sqrt.iter n guess * sqrt.iter n guess ≤ n := by	
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:lemma sqrt.lt_iter_succ_sq (n guess : ℕ) (hn : n < (guess + 1) * (guess + 1)) :	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_asymm [IsAsymm α (· ⊂ ·)] {a b : α} : a ⊂ b → ¬b ⊂ a := asymm 
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_irrefl [IsIrrefl α (· ⊂ ·)] (a : α) : ¬a ⊂ a := irrefl _	 
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_irrfl [IsIrrefl α (· ⊂ ·)] {a : α} : ¬a ⊂ a := irrefl _	 
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_of_eq_of_ssubset (hab : a = b) (hbc : b ⊂ c) : a ⊂ c := by rwa [hab]		
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_of_ssubset_of_eq (hab : a ⊂ b) (hbc : b = c) : a ⊂ c := by rwa [←hbc]		
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma ssubset_trans [IsTrans α (· ⊂ ·)] {a b c : α} : a ⊂ b → b ⊂ c → a ⊂ c := _root_.trans	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma start_mem_verts_toSubgraph (p : G.Walk u v) : u ∈ p.toSubgraph.verts := by	
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma strictAnti_iff_forall_covby [Preorder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean:lemma strictAnti_rpow_of_base_lt_one {b : ℝ} (hb₀ : 0 < b) (hb₁ : b < 1) :	
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma strictConcaveOn_rpow {p : ℝ} (hp₀ : 0 < p) (hp₁ : p < 1) :    
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma strictConcaveOn_rpow {p : ℝ} (hp₀ : 0 < p) (hp₁ : p < 1) :  
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma strictConcaveOn_sqrt : StrictConcaveOn ℝ (Set.Ici 0) Real.sqrt := by	
lake-packages/mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean:lemma strictConcaveOn_sqrt : StrictConcaveOn ℝ≥0 univ NNReal.sqrt := by	
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma strictMono (x : LTSeries α) : StrictMono x :=   
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma strictMono_iff_forall_covby [Preorder α] [LocallyFiniteOrder α] [Preorder β]	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean:lemma strictMono_rpow_of_base_gt_one {b : ℝ} (hb : 1 < b) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma sub_app (f g : P ⟶ Q) (X : Cᵒᵖ) : (f - g).app X = f.app X - g.app X := rfl	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma sub_v {n : ℤ} (z₁ z₂ : Cochain F G n) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean:lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) :	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_antisymm [IsAntisymm α (· ⊆ ·)] : a ⊆ b → b ⊆ a → a = b := antisymm		
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:lemma subset_diff : s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u := le_iff_subset.symm.trans le_sdiff	
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma subset_exterior : s ⊆ exterior s := subset_exterior_iff.2 $ λ _ _ ↦ id  
lake-packages/mathlib/Mathlib/Topology/AlexandrovDiscrete.lean:lemma subset_exterior_iff : s ⊆ exterior t ↔ ∀ U, IsOpen U → t ⊆ U → s ⊆ U := by		
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma subset_iff_le (P Q : MorphismProperty C) : P ⊆ Q ↔ P ≤ Q := Iff.rfl    
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma subset_isoClosure (P : MorphismProperty C) : P ⊆ P.isoClosure :=	
lake-packages/mathlib/Mathlib/Data/Set/Lattice.lean:lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_of_eq [IsRefl α (· ⊆ ·)] : a = b → a ⊆ b := fun h => h ▸ subset_rfl	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_of_eq_of_subset (hab : a = b) (hbc : b ⊆ c) : a ⊆ c := by rwa [hab]     
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_of_subset_of_eq (hab : a ⊆ b) (hbc : b = c) : a ⊆ c := by rwa [←hbc]    
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_refl [IsRefl α (· ⊆ ·)] (a : α) : a ⊆ a := refl _ 
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_rfl [IsRefl α (· ⊆ ·)] : a ⊆ a := refl _	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma subset_trans [IsTrans α (· ⊆ ·)] {a b c : α} : a ⊆ b → b ⊆ c → a ⊆ c := _root_.trans	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/Basic.lean:lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) :=    
lake-packages/mathlib/Mathlib/Data/FunLike/Basic.lean:lemma subsingleton_cod [∀ a, Subsingleton (β a)] : Subsingleton F := 
lake-packages/mathlib/Mathlib/Data/FunLike/Equiv.lean:lemma subsingleton_dom [Subsingleton β] : Subsingleton F :=	   
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma subsingleton_of_disjoint_isClopen	       
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma subsingleton_of_disjoint_isClosed_iUnion_eq_univ [Finite ι]	
lake-packages/mathlib/Mathlib/Topology/Connected.lean:lemma subsingleton_of_disjoint_isOpen_iUnion_eq_univ	
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_add {α} [AddCommMonoid α] (l r tl tr t)	
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_addg {α} [AddCommGroup α] (l r tl tr t)	
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_negg {α} [AddCommGroup α] (a ta t : α)		
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_smul {α} [AddCommMonoid α] 
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_smul_upcast {α} [AddCommGroup α]	
lake-packages/mathlib/Mathlib/Tactic/Abel.lean:lemma subst_into_smulg {α} [AddCommGroup α] 
lake-packages/mathlib/Mathlib/Order/SuccPred/Basic.lean:lemma succ_lt_succ_of_not_isMax (h : a < b) (hb : ¬ IsMax b) : succ a < succ b :=	
lake-packages/mathlib/Mathlib/RepresentationTheory/Maschke.lean:lemma sumOfConjugates_apply (v : W) : π.sumOfConjugates G v = ∑ g : G, π.conjugate g v :=	
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma sum_cons [AddCommMonoid M] (n : ℕ) (σ : Fin n →₀ M) (i : M) :	
lake-packages/mathlib/Mathlib/Algebra/BigOperators/Finsupp.lean:lemma sum_cons' [AddCommMonoid M] [AddCommMonoid N] (n : ℕ) (σ : Fin n →₀ M) (i : M)	
lake-packages/mathlib/Mathlib/Data/List/BigOperators/Basic.lean:lemma sum_int_mod (l : List ℤ) (n : ℤ) : l.sum % n = (l.map (· % n)).sum % n := by 
lake-packages/mathlib/Mathlib/Data/List/BigOperators/Basic.lean:lemma sum_nat_mod (l : List ℕ) (n : ℕ) : l.sum % n = (l.map (· % n)).sum % n := by	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma supClosed_iInter (hf : ∀ i, SupClosed (f i)) : SupClosed (⋂ i, f i) :=	
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma supClosed_sInter (hS : ∀ s ∈ S, SupClosed s) : SupClosed (⋂₀ S) :=   
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma supClosure_idem (s : Set α) : supClosure (supClosure s) = supClosure s :=				
lake-packages/mathlib/Mathlib/Order/SupClosed.lean:lemma supClosure_mono : Monotone (supClosure : Set α → Set α) := supClosure.monotone	
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/Coverage.lean:lemma sup_covering (x y : Coverage C) (B : C) :	 
lake-packages/mathlib/Mathlib/Algebra/Order/LatticeGroup.lean:lemma sup_eq_half_smul_add_add_abs_sub (x y : β) :	
lake-packages/mathlib/Mathlib/Algebra/Order/LatticeGroup.lean:lemma sup_eq_half_smul_add_add_abs_sub' (x y : β) : x ⊔ y = (2⁻¹ : α) • (x + y + |y - x|) := by	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Cofinality.lean:lemma sup_sequence_lt_omega1 {α} [Countable α] (o : α → Ordinal) (ho : ∀ n, o n < ω₁) :	
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma superset_antisymm [IsAntisymm α (· ⊆ ·)] : a ⊆ b → b ⊆ a → b = a := antisymm'		
lake-packages/mathlib/Mathlib/Order/RelClasses.lean:lemma superset_of_eq [IsRefl α (· ⊆ ·)] : a = b → b ⊆ a := fun h => h ▸ subset_rfl	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma surjOn_id (s : Set α) : SurjOn id s s := by simp [SurjOn, subset_rfl]		
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma surjOn_of_subsingleton [Subsingleton α] (f : α → α) (s : Set α) : SurjOn f s s :=	 
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:lemma surjOn_of_subsingleton' [Subsingleton β] (f : α → β) (h : t.Nonempty → s.Nonempty) :	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma symm_mk_inv [Inv M] (a : Mᵈᵐᵃ) : mk.symm (a⁻¹) = (mk.symm a)⁻¹ := rfl	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma symm_mk_mul [Mul M] (a b : Mᵈᵐᵃ) : mk.symm (a * b) = mk.symm b * mk.symm a := rfl	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma symm_mk_one [One M] : mk.symm (1 : Mᵈᵐᵃ) = 1 := rfl	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma symm_mk_pow [Monoid M] (a : Mᵈᵐᵃ) (n : ℕ) : mk.symm (a ^ n) = mk.symm a ^ n := rfl	
lake-packages/mathlib/Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean:lemma symm_mk_zpow [DivInvMonoid M] (a : Mᵈᵐᵃ) (n : ℤ) : mk.symm (a ^ n) = mk.symm a ^ n := rfl	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma tendsto_cdf_atBot : Tendsto (cdf μ) atBot (𝓝 0) := tendsto_condCdf_atBot _ _	
lake-packages/mathlib/Mathlib/Probability/Cdf.lean:lemma tendsto_cdf_atTop : Tendsto (cdf μ) atTop (𝓝 1) := tendsto_condCdf_atTop _ _	
lake-packages/mathlib/Mathlib/Topology/DiscreteSubset.lean:lemma tendsto_cofinite_cocompact_iff :	  
lake-packages/mathlib/Mathlib/Topology/DiscreteSubset.lean:lemma tendsto_cofinite_cocompact_of_discrete [DiscreteTopology X]   
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma tendsto_div_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :	
lake-packages/mathlib/Mathlib/Order/Filter/AtTopBot.lean:lemma tendsto_div_const_atTop_of_pos (hr : 0 < r) :  
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean:lemma tendsto_map_of_tendsto_of_continuous {ι : Type*} {L : Filter ι}	  
lake-packages/mathlib/Mathlib/MeasureTheory/Measure/FiniteMeasure.lean:lemma tendsto_map_of_tendsto_of_continuous {ι : Type*} {L : Filter ι} 
lake-packages/mathlib/Mathlib/MeasureTheory/Integral/Lebesgue.lean:lemma tendsto_measure_of_ae_tendsto_indicator {μ : Measure α} (A_mble : MeasurableSet A)	
lake-packages/mathlib/Mathlib/MeasureTheory/Integral/Lebesgue.lean:lemma tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure [IsCountablyGenerated L]	
lake-packages/mathlib/Mathlib/MeasureTheory/Integral/Indicator.lean:lemma tendsto_measure_of_tendsto_indicator [NeBot L] {μ : Measure α}		       
lake-packages/mathlib/Mathlib/MeasureTheory/Integral/Indicator.lean:lemma tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure [NeBot L]		       
lake-packages/mathlib/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean:lemma tendsto_mulIndicator_cthickening_mulIndicator_closure (f : α → β) (E : Set α) :  
lake-packages/mathlib/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean:lemma tendsto_mulIndicator_thickening_mulIndicator_closure (f : α → β) (E : Set α) :   
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean:lemma tendsto_rpow_atBot_of_base_gt_one (b : ℝ) (hb : 1 < b) : Tendsto (rpow b) atBot (𝓝 0) := by	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean:lemma tendsto_rpow_atBot_of_base_lt_one (b : ℝ) (hb₀ : 0 < b) (hb₁ : b < 1) :	
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean:lemma tendsto_rpow_atTop_of_base_gt_one (b : ℝ) (hb : 1 < b) : 
lake-packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean:lemma tendsto_rpow_atTop_of_base_lt_one (b : ℝ) (hb₀ : -1 < b) (hb₁ : b < 1) :		    
lake-packages/mathlib/test/rewrites.lean:lemma test : f n = f m := by 
lake-packages/mathlib/Mathlib/NumberTheory/LucasLehmer.lean:lemma testFalseHelper (p : ℕ) (hp : Nat.blt 1 p = true)	  
lake-packages/mathlib/Mathlib/NumberTheory/LucasLehmer.lean:lemma testTrueHelper (p : ℕ) (hp : Nat.blt 1 p = true) (h : sMod' (2 ^ p - 1) (p - 2) = 0) :	
lake-packages/mathlib/Mathlib/Order/LocallyFinite.lean:lemma this : (Icc (toDual (toDual a)) (toDual (toDual b)) : _) = (Icc a b : _) := rfl	 
lake-packages/mathlib/Mathlib/Order/LocallyFinite.lean:lemma this : (Ici (toDual (toDual a)) : _) = (Ici a : _) := rfl	 
lake-packages/mathlib/Mathlib/Order/LocallyFinite.lean:lemma this : (Iic (toDual (toDual a)) : _) = (Iic a : _) := rfl	 
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma three_ne_zero [OfNat α 3] [NeZero (3 : α)] : (3 : α) ≠ 0 := NeZero.ne (3 : α)	
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma three_ne_zero' [OfNat α 3] [NeZero (3 : α)] : (3 : α) ≠ 0 := three_ne_zero	
lake-packages/mathlib/Mathlib/Data/Matrix/Block.lean:lemma toBlocks₁₁_diagonal (v : l ⊕ m → α) :  
lake-packages/mathlib/Mathlib/Data/Matrix/Block.lean:lemma toBlocks₁₂_diagonal (v : l ⊕ m → α) : toBlocks₁₂ (diagonal v) = 0 := rfl	
lake-packages/mathlib/Mathlib/Data/Matrix/Block.lean:lemma toBlocks₂₁_diagonal (v : l ⊕ m → α) : toBlocks₂₁ (diagonal v) = 0 := rfl	
lake-packages/mathlib/Mathlib/Data/Matrix/Block.lean:lemma toBlocks₂₂_diagonal (v : l ⊕ m → α) :  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma toClass_eq_zero (P : W.Point) : toClass P = 0 ↔ P = 0 := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma toClass_injective : Function.Injective <| @toClass _ _ W := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma toClass_some : toClass (some h₁) = ClassGroup.mk (XYIdeal' h₁) :=	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma toClass_zero : toClass (0 : W.Point) = 0 :=  
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toColumns₁_apply (A : Matrix m (n₁ ⊕ n₂) R) (i : m) (j : n₁) :	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toColumns₁_fromColumns  (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toColumns₂_apply (A : Matrix m (n₁ ⊕ n₂) R) (i : m) (j : n₂) :	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toColumns₂_fromColumns  (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :	
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma toCostructuredArrow_comp_proj (G : E ⥤ C) (F : C ⥤ D) (X : D)  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma toCycles_comp_homologyπ :		  
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma toCycles_comp_leftHomologyπ : S.toCycles ≫ S.leftHomologyπ = 0 :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma toCycles_i : S.toCycles ≫ S.iCycles = S.f := S.leftHomologyData.f'_i	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma toCycles_naturality : S₁.toCycles ≫ cyclesMap φ = φ.τ₁ ≫ S₂.toCycles :=	
lake-packages/mathlib/Mathlib/Topology/Homeomorph.lean:lemma toHomeomorph_apply (e : α ≃ β) (he) (a : α) : e.toHomeomorph he a = e a := rfl	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Equiv.lean:lemma toHomeomorph_apply (e : α ≃ᵤ β) : (e.toHomeomorph : α → β) = e := rfl	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Equiv.lean:lemma toHomeomorph_symm_apply (e : α ≃ᵤ β) : (e.toHomeomorph.symm : β → α) = e.symm := rfl	
lake-packages/mathlib/Mathlib/Topology/Homeomorph.lean:lemma toHomeomorph_trans (e : α ≃ β) (f : β ≃ γ) (he hf) :	
lake-packages/mathlib/Mathlib/Data/Polynomial/Laurent.lean:lemma toLaurent_reverse (p : R[X]) :   
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma toList_chain' (x : RelSeries r) : x.toList.Chain' r := by	
lake-packages/mathlib/Mathlib/Order/RelSeries.lean:lemma toList_ne_empty (x : RelSeries r) : x.toList ≠ [] := fun m =>	
lake-packages/mathlib/Mathlib/Topology/FiberBundle/Trivialization.lean:lemma toLocalEquiv_injective [Nonempty F] : 
lake-packages/mathlib/Mathlib/LinearAlgebra/FreeModule/PID.lean:lemma toMatrix_restrict_eq_toMatrix [Fintype ι] [DecidableEq ι]	
lake-packages/mathlib/Mathlib/Data/Int/Cast/Lemmas.lean:lemma toNat_lt' {a : ℤ} {b : ℕ} (hb : b ≠ 0) : a.toNat < b ↔ a < b := by	       
lake-packages/mathlib/Mathlib/Algebra/Category/GroupCat/Injective.lean:lemma toNext_inj : Function.Injective <| toNext A_ :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Over.lean:lemma toOver_comp_forget (F : S ⥤ T) (X : T) (f : (Y : S) → F.obj Y ⟶ X)	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Basic.lean:lemma toPartENat_eq_iff_of_le_aleph0 {c c' : Cardinal} (hc : c ≤ ℵ₀) (hc' : c' ≤ ℵ₀) :	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Basic.lean:lemma toPartENat_le_iff_of_le_aleph0 {c c' : Cardinal} (h : c ≤ ℵ₀) :	
lake-packages/mathlib/Mathlib/SetTheory/Cardinal/Basic.lean:lemma toPartENat_le_iff_of_lt_aleph0 {c c' : Cardinal} (hc' : c' < ℵ₀) :	
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma toPresheaf_map_app {P Q : PresheafOfModules R}	 
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toRows₁_apply (A : Matrix (m₁ ⊕ m₂) n R) (i : m₁) (j : n) :  
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toRows₁_fromRows  (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toRows₂_apply (A : Matrix (m₁ ⊕ m₂) n R) (i : m₂) (j : n) :  
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma toRows₂_fromRows  (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :	
lake-packages/mathlib/Mathlib/Topology/DiscreteQuotient.lean:lemma toSetoid_injective : Function.Injective (@toSetoid X _) 
lake-packages/mathlib/Mathlib/CategoryTheory/Sites/CoverLifting.lean:lemma toSheafify_pullbackSheafificationCompatibility   
lake-packages/mathlib/Mathlib/CategoryTheory/StructuredArrow.lean:lemma toStructuredArrow_comp_proj (G : E ⥤ C) (X : D) (F : C ⥤ D)    
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma toSubgraph_le_induce_support (p : G.Walk u v) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Over.lean:lemma toUnder_comp_forget (F : S ⥤ T) (X : T) (f : (Y : S) → X ⟶ F.obj Y)	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma toUnitHom_eq_char' {a : ZMod n} (ha : IsUnit a) : χ a = χ.toUnitHom ha.unit := by simp	
lake-packages/mathlib/Mathlib/NumberTheory/DirichletCharacter/Basic.lean:lemma toUnitHom_eq_iff (ψ : DirichletCharacter R n) : toUnitHom χ = toUnitHom ψ ↔ χ = ψ := by simp	
lake-packages/mathlib/Mathlib/Init/Data/Nat/Lemmas.lean:lemma to_digits_core_length (b : Nat) (h : 2 <= b) (f n e : Nat)  
lake-packages/mathlib/Mathlib/Init/Data/Nat/Lemmas.lean:lemma to_digits_core_lens_eq (b f : Nat) : ∀ (n : Nat) (c : Char) (tl : List Char),	
lake-packages/mathlib/Mathlib/Init/Data/Nat/Lemmas.lean:lemma to_digits_core_lens_eq_aux (b f : Nat) : 
lake-packages/mathlib/Mathlib/CategoryTheory/MorphismProperty.lean:lemma top_apply {X Y : C} (f : X ⟶ Y) : (⊤ : MorphismProperty C) f := by     
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma top_connected [Nonempty V] : (⊤ : SimpleGraph V).Connected where	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:lemma top_induce_pair_connected_of_adj {u v : V} (huv : G.Adj u v) :  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:lemma top_preconnected : (⊤ : SimpleGraph V).Preconnected := fun x y => by 
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma topology_eq : ‹_› = generateFrom { s | ∃ a : α, (Ici a)ᶜ = s } := topology_eq_lowerTopology	
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma topology_eq : ‹_› = generateFrom { s | ∃ a : α, (Iic a)ᶜ = s } := topology_eq_upperTopology	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma topology_eq : ‹_› = lowerSetTopology' α := topology_eq_lowerSetTopology  
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma topology_eq : ‹_› = upperSetTopology' α := topology_eq_upperSetTopology	
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma traceForm_apply_lie_apply (x y z : L) : 
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma traceForm_comm (x y : L) : traceForm R L M x y = traceForm R L M y x :=	
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma traceForm_eq_of_le_idealizer :			
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma traceForm_eq_zero_of_isTrivial [LieModule.IsTrivial I N] :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Trace.lean:lemma trace_comp_cycle (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Trace.lean:lemma trace_comp_cycle' (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Trace.lean:lemma trace_diagonal {o} [Fintype o] [DecidableEq o] (d : o → R) :	      
lake-packages/mathlib/Mathlib/Algebra/Lie/Killing.lean:lemma trace_eq_trace_restrict_of_le_idealizer		
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/Trace.lean:lemma trace_eq_zero_of_isEmpty [IsEmpty n] (A : Matrix n n R) : trace A = 0 := by simp [trace]	
lake-packages/mathlib/Mathlib/LinearAlgebra/Trace.lean:lemma trace_mul_cycle (f g h : M →ₗ[R] M) :	 
lake-packages/mathlib/Mathlib/LinearAlgebra/Trace.lean:lemma trace_mul_cycle' (f g h : M →ₗ[R] M) :	 
lake-packages/mathlib/Mathlib/LinearAlgebra/PID.lean:lemma trace_restrict_eq_of_forall_mem [IsDomain R] [IsPrincipalIdealRing R]	       
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma transGen_covby_of_lt [Preorder α] [LocallyFiniteOrder α] {x y : α} (hxy : x < y) :	
lake-packages/mathlib/Mathlib/Logic/Relation.lean:lemma transGen_minimal {r' : α → α → Prop} (hr' : Transitive r') (h : ∀ x y, r x y → r' x y)  
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma transGen_wcovby_eq_reflTransGen_covby [PartialOrder α] : 
lake-packages/mathlib/Mathlib/Data/Finset/LocallyFinite.lean:lemma transGen_wcovby_of_le [Preorder α] [LocallyFiniteOrder α] {x y : α} (hxy : x ≤ y) :	
lake-packages/mathlib/Mathlib/LinearAlgebra/PerfectPairing.lean:lemma trans_dualMap_symm_flip : e.trans e.flip.symm.dualMap = Dual.eval R N := by ext; simp	
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma transpose_fromColumns (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :	      
lake-packages/mathlib/Mathlib/Data/Matrix/ColumnRowPartitioned.lean:lemma transpose_fromRows (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :  
lake-packages/mathlib/Mathlib/Data/Matrix/Invertible.lean:lemma transpose_invOf [Invertible A] [Invertible Aᵀ] : (⅟A)ᵀ = ⅟(Aᵀ) := by	
lake-packages/mathlib/Mathlib/RingTheory/PowerSeries/Basic.lean:lemma trunc_X_of {n : ℕ} (hn : 2 ≤ n) : trunc n X = (Polynomial.X : R[X]) := by	
lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma tsum_const_smul' {γ : Type*} [Group γ] [DistribMulAction γ α] [ContinuousConstSMul γ α]	
lake-packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean:lemma tsum_const_smul'' {γ : Type*} [DivisionRing γ] [Module γ α] [ContinuousConstSMul γ α]	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma twoTorsionPolynomial_disc : W.twoTorsionPolynomial.disc = 16 * W.Δ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma twoTorsionPolynomial_disc_isUnit [Invertible (2 : R)] : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma twoTorsionPolynomial_disc_ne_zero [Nontrivial R] [Invertible (2 : R)] (hΔ : IsUnit W.Δ) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma twoTorsionPolynomial_disc_ne_zero [Nontrivial R] [Invertible (2 : R)] :	
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma two_ne_zero [OfNat α 2] [NeZero (2 : α)] : (2 : α) ≠ 0 := NeZero.ne (2 : α) 
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma two_ne_zero' [OfNat α 2] [NeZero (2 : α)] : (2 : α) ≠ 0 := two_ne_zero	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_comm (a b : α) : [[a, b]] = [[b, a]] := by simp_rw [uIcc, inf_comm, sup_comm]	  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_eq_union : [[a, b]] = Icc a b ∪ Icc b a := by rw [Icc_union_Icc', max_comm] <;> rfl	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_injective_left (a : α) : Injective (uIcc a) := by 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_injective_right (a : α) : Injective fun b => uIcc b a := fun b c h => by	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_ge (h : b ≤ a) : [[a, b]] = Icc b a := by rw [uIcc, inf_eq_right.2 h, sup_eq_left.2 h]	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_gt (h : b < a) : [[a, b]] = Icc b a := uIcc_of_ge h.le 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_le (h : a ≤ b) : [[a, b]] = Icc a b := by rw [uIcc, inf_eq_left.2 h, sup_eq_right.2 h]	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_lt (h : a < b) : [[a, b]] = Icc a b := uIcc_of_le h.le 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_not_ge (h : ¬b ≤ a) : [[a, b]] = Icc a b := uIcc_of_lt $ lt_of_not_ge h	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_of_not_le (h : ¬a ≤ b) : [[a, b]] = Icc b a := uIcc_of_gt $ lt_of_not_ge h	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_self : [[a, a]] = {a} := by simp [uIcc]   
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_Icc (ha : a₁ ∈ Icc a₂ b₂) (hb : b₁ ∈ Icc a₂ b₂) :  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc (h₁ : a₁ ∈ [[a₂, b₂]]) (h₂ : b₁ ∈ [[a₂, b₂]]) :		  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_iff_le :	 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_iff_le' : 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_iff_mem : 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_left (h : x ∈ [[a, b]]) : [[a, x]] ⊆ [[a, b]] :=		
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_right (h : x ∈ [[a, b]]) : [[x, b]] ⊆ [[a, b]] :=	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIcc_subset_uIcc_union_uIcc : [[a, c]] ⊆ [[a, b]] ∪ [[b, c]] := fun x => by	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIoc_comm (a b : α) : Ι a b = Ι b a := by simp only [uIoc, min_comm a b, max_comm a b]	
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIoc_eq_union : Ι a b = Ioc a b ∪ Ioc b a := by  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIoc_injective_left (a : α) : Injective (Ι a) := by 
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIoc_injective_right (a : α) : Injective fun b => Ι b a := by  
lake-packages/mathlib/Mathlib/Data/Set/Intervals/UnorderedInterval.lean:lemma uIoc_subset_uIoc_of_uIcc_subset_uIcc {a b c d : α}	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Basic.lean:lemma uniformContinuous_inl : UniformContinuous (Sum.inl : α → α ⊕ β) := le_sup_left	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Basic.lean:lemma uniformContinuous_inr : UniformContinuous (Sum.inr : β → α ⊕ β) := le_sup_right	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/Equicontinuity.lean:lemma uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/UniformEmbedding.lean:lemma uniformInducing_iff' {f : α → β} :	   
lake-packages/mathlib/Mathlib/Topology/Instances/Real.lean:lemma uniform_embedding_add_rat {r : ℚ} : uniform_embedding (fun p : ℚ => p + r) :=	
lake-packages/mathlib/Mathlib/Topology/Instances/Real.lean:lemma uniform_embedding_mul_rat {q : ℚ} (hq : q ≠ 0) : uniform_embedding ((*) q) :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Predicate.lean:lemma uniq_symm : (uniq L₁ L₂ W').symm = uniq L₂ L₁ W' := rfl	
lake-packages/mathlib/Mathlib/Algebra/Group/UniqueProds.lean:lemma uniqueMul_of_twoUniqueMul {G} [Mul G] {A B : Finset G} (h : 1 < A.card * B.card →	
lake-packages/mathlib/Mathlib/Topology/ContinuousFunction/Units.lean:lemma unitsLift_apply_inv_apply (f : C(X, Mˣ)) (x : X) :	    
lake-packages/mathlib/Mathlib/Topology/ContinuousFunction/Units.lean:lemma unitsLift_symm_apply_apply_inv' (f : C(X, M)ˣ) (x : X) :	    
lake-packages/mathlib/Mathlib/Data/ZMod/Units.lean:lemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :	  
lake-packages/mathlib/Mathlib/Data/ZMod/Units.lean:lemma unitsMap_def (hm : n ∣ m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl	    
lake-packages/mathlib/Mathlib/Data/ZMod/Units.lean:lemma unitsMap_self (n : ℕ) : unitsMap (dvd_refl n) = MonoidHom.id _ := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Basic.lean:lemma unopMap_id (S : ShortComplex Cᵒᵖ) : unopMap (𝟙 S) = 𝟙 S.unop := rfl	
lake-packages/mathlib/Mathlib/Logic/Function/Basic.lean:lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not			
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma upperSet_LE_upper {t₁ : TopologicalSpace α} [@UpperSetTopology α t₁ _]   
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:lemma upperSet_dual_iff_lowerSet [Preorder α] [TopologicalSpace α] : 
lake-packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean:lemma upper_dual_iff_lower [Preorder α] [TopologicalSpace α] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma v_comp_XIsoOfEq_hom  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma v_comp_XIsoOfEq_inv  
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_b₂ : (W.variableChange C).b₂ = (↑C.u⁻¹ : R) ^ 2 * (W.b₂ + 12 * C.r) := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_b₄ : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_b₆ : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_b₈ : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_comp (C C' : VariableChange R) (W : WeierstrassCurve R) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_comp (C C' : WeierstrassCurve.VariableChange R) (E : EllipticCurve R) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_c₄ : (W.variableChange C).c₄ = (↑C.u⁻¹ : R) ^ 4 * W.c₄ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_c₆ : (W.variableChange C).c₆ = (↑C.u⁻¹ : R) ^ 6 * W.c₆ := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_id : E.variableChange WeierstrassCurve.VariableChange.id = E := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_id : W.variableChange VariableChange.id = W := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_j : (E.variableChange C).j = E.j := by	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:lemma variableChange_Δ : (W.variableChange C).Δ = (↑C.u⁻¹ : R) ^ 12 * W.Δ := by		
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma vecMul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (v : n → R) :   
lake-packages/mathlib/Mathlib/LinearAlgebra/Matrix/DotProduct.lean:lemma vecMul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (v : m → R) :   
lake-packages/mathlib/Mathlib/Order/Cover.lean:lemma wcovby_eq_reflGen_covby [PartialOrder α] : ((· : α) ⩿ ·) = ReflGen (· ⋖ ·) := by	
lake-packages/mathlib/Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean:lemma whiskerLeft_toKaroubi_N₂Γ₂_hom :			
lake-packages/mathlib/Mathlib/AlgebraicTopology/CechNerve.lean:lemma wideCospan.limitIsoPi_hom_comp_pi [Finite ι] (X : C) (j : ι) :	
lake-packages/mathlib/Mathlib/AlgebraicTopology/CechNerve.lean:lemma wideCospan.limitIsoPi_inv_comp_pi [Finite ι] (X : C) (j : ι) :	
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma withBot_apply (f : LatticeHom α β) (a : WithBot α) : f.withBot a = a.map f := rfl	
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma withTopWithBot_apply (f : LatticeHom α β) (a : WithTop <| WithBot α) :   
lake-packages/mathlib/Mathlib/Order/Hom/Lattice.lean:lemma withTop_apply (f : LatticeHom α β) (a : WithTop α) : f.withTop a = a.map f := rfl	
lake-packages/mathlib/test/linarith.lean:lemma works {a b : ℕ} (hab : a ≤ b) (h : b < a) : false := by 
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaEquiv_comp {X : C} {F G : Cᵒᵖ ⥤ Type v₁} (α : yoneda.obj X ⟶ F) (β : F ⟶ G)  :	
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaEquiv_comp' {X : Cᵒᵖ} {F G : Cᵒᵖ ⥤ Type v₁} (α : yoneda.obj (unop X) ⟶ F) (β : F ⟶ G)  :	
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaEquiv_naturality' {X Y : Cᵒᵖ} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj (unop X) ⟶ F)	
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaEquiv_symm_map {X Y : Cᵒᵖ} (f : X ⟶ Y) {F : Cᵒᵖ ⥤ Type v₁} (t : F.obj X) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaEquiv_yoneda_map {X Y : C} (f : X ⟶ Y) : yonedaEquiv (yoneda.map f) = f := by  
lake-packages/mathlib/Mathlib/CategoryTheory/Yoneda.lean:lemma yonedaPairingExt {x y : (yonedaPairing C).obj X} (w : ∀ Y, x.app Y = y.app Y) : x = y :=	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma yoneda_exact₂ {X : C} (f : T.obj₂ ⟶ X) (hf : T.mor₁ ≫ f = 0) :	
lake-packages/mathlib/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean:lemma yoneda_exact₃ {X : C} (f : T.obj₃ ⟶ X) (hf : T.mor₂ ≫ f = 0) :	
lake-packages/mathlib/Archive/ZagierTwoSquares.lean:lemma zagierSet_lower_bound {x y z : ℕ} (h : (x, y, z) ∈ zagierSet k) : 0 < x ∧ 0 < y ∧ 0 < z := by	
lake-packages/mathlib/Archive/ZagierTwoSquares.lean:lemma zagierSet_subset : zagierSet k ⊆ Ioc 0 (k + 1) ×ˢ Ioc 0 k ×ˢ Ioc 0 k := by	
lake-packages/mathlib/Archive/ZagierTwoSquares.lean:lemma zagierSet_upper_bound {x y z : ℕ} (h : (x, y, z) ∈ zagierSet k) :	
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:lemma zeckendorf_sum_fib : ∀ {l}, IsZeckendorfRep l → zeckendorf (l.map fib).sum = l	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Basic.lean:lemma zero_add_inv_app (h : ShiftMkCore C A) (n : A) (X : C) :	 
lake-packages/mathlib/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean:lemma zero_app (X : Cᵒᵖ) : (0 : P ⟶ Q).app X = 0 := rfl	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma zero_cochain_comp_v (z₁ : Cochain F G 0) (z₂ : Cochain G K n) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:lemma zero_def : (zero : W.Point) = 0 :=		
lake-packages/mathlib/test/toAdditive.lean:lemma zero_fooClass [Zero α] : FooClass α := by infer_instance	
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma zero_hom_app_obj (X : C) : 
lake-packages/mathlib/Mathlib/CategoryTheory/Shift/Induced.lean:lemma zero_inv_app_obj (X : C) : 
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_le_four [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :	
lake-packages/mathlib/Mathlib/Algebra/Order/ZeroLEOne.lean:lemma zero_le_one' (α) [Zero α] [One α] [LE α] [ZeroLEOneClass α] : (0 : α) ≤ 1 := 
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_le_three [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :	
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_le_two [Preorder α] [ZeroLEOneClass α] [CovariantClass α α (·+·) (·≤·)] :	
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_lt_four' : (0 : α) < 4 := zero_lt_four		
lake-packages/mathlib/Mathlib/Algebra/Order/ZeroLEOne.lean:lemma zero_lt_one' : (0 : α) < 1 := zero_lt_one			
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_lt_three' : (0 : α) < 3 := zero_lt_three		
lake-packages/mathlib/Mathlib/Algebra/Order/Monoid/NatCast.lean:lemma zero_lt_two' : (0 : α) < 2 := zero_lt_two			
lake-packages/mathlib/Mathlib/Tactic/Linarith/Lemmas.lean:lemma zero_mul_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (h : a = 0) (_ : R b 0) :  
lake-packages/mathlib/Mathlib/Algebra/NeZero.lean:lemma zero_ne_one' [One α] [NeZero (1 : α)] : (0 : α) ≠ 1 := zero_ne_one   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma zero_v {n : ℤ} (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/Order/Zorn.lean:lemma zorny_lemma : zorny_statement :=	   
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma zpow_mulLeft (n : ℤ) : Equiv.mulLeft a ^ n = Equiv.mulLeft (a ^ n) :=   
lake-packages/mathlib/Mathlib/GroupTheory/Perm/Basic.lean:lemma zpow_mulRight : ∀ n : ℤ, Equiv.mulRight a ^ n = Equiv.mulRight (a ^ n)  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma zsmul_v {n k : ℤ} (z : Cochain F G n) (p q : ℤ) (hpq : p + n = q) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_comp {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) (h : n₁ + n₂ = n₁₂)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_comp_zero_cochain {n₁ : ℤ} (z₁ : Cochain F G n₁) (z₂ : Cochain G K 0) 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_eq_zero {n : ℤ} (z : Cocycle F G n) (m : ℤ) : δ n m (z : Cochain F G n) = 0 := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_neg_one_cochain (z : Cochain F G (-1)) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_ofHom {p : ℤ} (φ : F ⟶ G) : δ 0 p (Cochain.ofHom φ) = 0 := by  
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_ofHomotopy {φ₁ φ₂ : F ⟶ G} (h : Homotopy φ₁ φ₂) :   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_shape (hnm : ¬ n + 1 = m) (z : Cochain F G n) : δ n m z = 0 := by   
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_v (hnm : n + 1 = m) (z : Cochain F G n) (p q : ℤ) (hpq : p + m = q) (q₁ q₂ : ℤ)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_zero_cochain_comp {n₂ : ℤ} (z₁ : Cochain F G 0) (z₂ : Cochain G K n₂) 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_zero_cochain_v (z : Cochain F G 0) (p q : ℤ) (hpq : p + 1 = q) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:lemma δ_δ (n₀ n₁ n₂ : ℤ) (z : Cochain F G n₀) : δ n₁ n₂ (δ n₀ n₁ z) = 0 := by 
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Adjunction.lean:lemma ε_app (X₁ : C₁) :	  
lake-packages/mathlib/Mathlib/CategoryTheory/Localization/Adjunction.lean:lemma η_app (X₂ : C₂) :	  
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean:lemma ιFun_continuous : Continuous (ιFun π) := by    
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean:lemma ιFun_continuous : Continuous (ιFun π) := by 
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean:lemma ιFun_injective : (ιFun π).Injective := by	     
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean:lemma ιFun_injective : (ιFun π).Injective := by	  
lake-packages/mathlib/Mathlib/CategoryTheory/Preadditive/Mat.lean:lemma ι_additiveObjIsoBiproduct_inv (F : Mat_ C ⥤ D) [Functor.Additive F] (M : Mat_ C) (i : M.ι) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean:lemma ι_descQ_eq_zero_of_boundary (k : S.X₂ ⟶ A) (x : S.X₃ ⟶ A) (hx : k = S.g ≫ x) :	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean:lemma π'_comp_ι_hom (a : α) : π' π a ≫ (ι _ surj).hom = π a := by ext; rfl	
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean:lemma π'_comp_ι_hom (a : α) : π' π surj a ≫ (ιIso π surj).hom = π a := by	
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean:lemma π_comp_hom (K₁ K₂ : Multicofork I) (f : K₁ ⟶ K₂) (b : I.R) : K₁.π b ≫ f.hom = K₂.π b :=	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma π_comp_leftHomologyIso_inv :	     
lake-packages/mathlib/Mathlib/CategoryTheory/Limits/Opposites.lean:lemma π_comp_opProductIsoCoproduct (b : α) : (Pi.π Z b).op ≫ (opProductIsoCoproduct Z).hom =	
lake-packages/mathlib/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean:lemma π_comp_ι_inv (a : α) : π a ≫ (ι _ surj).inv = π' π a := by  
lake-packages/mathlib/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean:lemma π_comp_ι_inv (a : α) : π a ≫ (ιIso π surj).inv = π' π surj a := by	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma π_descH (k : h.K ⟶ A) (hk : h.f' ≫ k = 0) : h.π ≫ h.descH k hk = k := 
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma π_descHomology (k : S.cycles ⟶ A) (hk : S.toCycles ≫ k = 0) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma π_homologyMap_ι [S₁.HasHomology] [S₂.HasHomology] (φ : S₁ ⟶ S₂) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma π_leftRightHomologyComparison'_ι :    
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Homology.lean:lemma π_leftRightHomologyComparison_ι [S.HasLeftHomology] [S.HasRightHomology] :	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/Basic.lean:lemma π₁Toπ₂_comp_π₂Toπ₃ : (π₁Toπ₂ : (_ : _ ⥤ C) ⟶ _) ≫ π₂Toπ₃ = 0 := by aesop_cat	
lake-packages/mathlib/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean:lemma τ₁_ofEpiOfIsIsoOfMono_f' (φ : S₁ ⟶ S₂) (h : LeftHomologyData S₁)	
lake-packages/mathlib/Mathlib/Order/OmegaCompletePartialOrder.lean:lemma ωSup_eq_of_IsLUB {c : Chain α} {a : α} (h : IsLUB (Set.range c) a) : a = ωSup c := by	  
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:private lemma AM_GM : {a b : ℕ} → (4 * a * b ≤ (a + b) * (a + b))	
lake-packages/mathlib/Mathlib/Topology/ExtremallyDisconnected.lean:private lemma ExtremallyDisconnected.homeoCompactToT2_injective [ExtremallyDisconnected A]	
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Point.lean:private lemma XYIdeal'_mul_inv :  
lake-packages/mathlib/Mathlib/Order/LocallyFinite.lean:private lemma aux (x : α) (p : α → Prop) : 
lake-packages/mathlib/Mathlib/RingTheory/Jacobson.lean:private lemma aux_IH {R : Type u} {S : Type v} {T : Type w} 
lake-packages/mathlib/Mathlib/Probability/Variance.lean:private lemma coe_two : ENNReal.toReal 2 = (2 : ℝ) := rfl   
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:private lemma div_four_lt : {n : ℕ} → (h0 : n ≠ 0) → (h1 : n ≠ 1) → n / 4 + 1 < n	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma div_ne_zero_of_ne_zero_of_pos (ha : a ≠ 0) (hb : 0 < b) : a / b ≠ 0 :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma div_ne_zero_of_pos_of_ne_zero (ha : 0 < a) (hb : b ≠ 0) : a / b ≠ 0 :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma div_nonneg_of_nonneg_of_pos (ha : 0 ≤ a) (hb : 0 < b) : 0 ≤ a / b := 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma div_nonneg_of_pos_of_nonneg (ha : 0 < a) (hb : 0 ≤ b) : 0 ≤ a / b := 
lake-packages/mathlib/Mathlib/GroupTheory/CommutingProbability.lean:private lemma div_two_lt {n : ℕ} (h0 : n ≠ 0) : n / 2 < n :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma int_div_nonneg_of_nonneg_of_pos {a b : ℤ} (ha : 0 ≤ a) (hb : 0 < b) : 0 ≤ a / b :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma int_div_nonneg_of_pos_of_nonneg {a b : ℤ} (ha : 0 < a) (hb : 0 ≤ b) : 0 ≤ a / b :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma int_div_nonneg_of_pos_of_pos {a b : ℤ} (ha : 0 < a) (hb : 0 < b) : 0 ≤ a / b := 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma int_div_self_pos {a : ℤ} (ha : 0 < a) : 0 < a / a :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : ite p a b ≠ 0 := by by_cases p <;> simp [*]	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_ne_zero_of_ne_zero_of_pos [Preorder α] (ha : a ≠ 0) (hb : 0 < b) : 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_ne_zero_of_pos_of_ne_zero [Preorder α] (ha : 0 < a) (hb : b ≠ 0) : 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_nonneg [LE α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ ite p a b :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_nonneg_of_nonneg_of_pos [Preorder α] (ha : 0 ≤ a) (hb : 0 < b) : 0 ≤ ite p a b :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_nonneg_of_pos_of_nonneg [Preorder α] (ha : 0 < a) (hb : 0 ≤ b) : 0 ≤ ite p a b :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma ite_pos [LT α] (ha : 0 < a) (hb : 0 < b) : 0 < ite p a b :=	
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:private lemma iter_fp_bound (n k : ℕ) :	
lake-packages/mathlib/Mathlib/Data/Sign.lean:private lemma le_antisymm (a b : SignType) (_ : a ≤ b) (_: b ≤ a) : a = b := by	  
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma le_min_of_le_of_lt (ha : a ≤ b) (hb : a < c) : a ≤ min b c := le_min ha hb.le	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma le_min_of_lt_of_le (ha : a < b) (hb : a ≤ c) : a ≤ min b c := le_min ha.le hb	
lake-packages/mathlib/Mathlib/Data/Sign.lean:private lemma le_trans (a b c : SignType) (_ : a ≤ b) (_: b ≤ c) : a ≤ c := by	  
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma max_ne (ha : a ≠ c) (hb : b ≠ c) : max a b ≠ c :=	 
lake-packages/mathlib/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean:private lemma mem_pairs (k : ℕ) (t : Finset σ × σ) :  
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma min_ne (ha : a ≠ c) (hb : b ≠ c) : min a b ≠ c :=	 
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma min_ne_of_lt_of_ne (ha : c < a) (hb : b ≠ c) : min a b ≠ c := min_ne ha.ne' hb	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma min_ne_of_ne_of_lt (ha : a ≠ c) (hb : c < b) : min a b ≠ c := min_ne ha hb.ne'	
lake-packages/mathlib/Mathlib/Data/Sign.lean:private lemma mul_assoc : ∀ (a b c : SignType), (a * b) * c = a * (b * c) := by	  
lake-packages/mathlib/Mathlib/Data/Sign.lean:private lemma mul_comm : ∀ (a b : SignType), a * b = b * a := by rintro ⟨⟩ ⟨⟩ <;> rfl	
lake-packages/mathlib/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean:private lemma pairMap_ne_self (t : Finset σ × σ) : pairMap σ t ≠ t := by	
lake-packages/mathlib/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean:private lemma pairMap_of_snd_mem_fst {t : Finset σ × σ} (h : t.snd ∈ t.fst) :	
lake-packages/mathlib/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean:private lemma pairMap_of_snd_nmem_fst {t : Finset σ × σ} (h : t.snd ∉ t.fst) :	
lake-packages/mathlib/Mathlib/SetTheory/ZFC/Basic.lean:private lemma toSet_equiv_aux {s : Set ZFSet.{u}} (hs : Small.{u} s) : 
lake-packages/mathlib/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean:private lemma two_or_three_ne_zero : (2 : F) ≠ 0 ∨ (3 : F) ≠ 0 :=   
lake-packages/mathlib/Mathlib/Data/Nat/Fib/Zeckendorf.lean:private lemma zeckendorf_aux (hm : 0 < m) : m - fib (greatestFib m) < m :=	
lake-packages/mathlib/Mathlib/Tactic/Positivity/Basic.lean:private lemma zpow_zero_pos [LinearOrderedSemifield R] (a : R) : 0 < a ^ (0 : ℤ) :=	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma BijOn.extendDomain (h : BijOn g s t) : 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma Connected.mono {G G' : SimpleGraph V} (h : G ≤ G') 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma Connected.mono {H H' : G.Subgraph} (hle : H ≤ H') (hv : H.verts = H'.verts)	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma Connected.mono' {H H' : G.Subgraph}	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma Connected.nonempty {H : G.Subgraph} (h : H.Connected) : H.verts.Nonempty := by		
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma Connected.preconnected {H : G.Subgraph} (h : H.Connected) : H.Preconnected := by	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma Connected.sup {H K : G.Subgraph}	
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:protected lemma Directed.iSup_le (hf : Directed (. ≤ .) f) (ha : ∀ i, f i ≤ a) :  ⨆ i, f i ≤ a :=	
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:protected lemma Directed.le_iSup (hf : Directed (. ≤ .) f) (i : ι) : f i ≤ ⨆ j, f j := 
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:protected lemma DirectedOn.isLUB_sSup : DirectedOn (. ≤ .) d → IsLUB d (sSup d) :=	
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:protected lemma DirectedOn.le_sSup (hd : DirectedOn (. ≤ .) d) (ha : a ∈ d) : a ≤ sSup d :=	
lake-packages/mathlib/Mathlib/Order/CompletePartialOrder.lean:protected lemma DirectedOn.sSup_le (hd : DirectedOn (. ≤ .) d) (ha : ∀ b ∈ d, b ≤ a) : sSup d ≤ a :=	
lake-packages/mathlib/Mathlib/Topology/UniformSpace/UniformEmbedding.lean:protected lemma Filter.HasBasis.uniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:protected lemma Finset.prod_empty {β α : Type*} [CommSemiring β] (f : α → β) :	
lake-packages/mathlib/Mathlib/Tactic/NormNum/BigOperators.lean:protected lemma Finset.sum_empty {β α : Type*} [CommSemiring β] (f : α → β) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma InvOn.extendDomain (h : InvOn g₁ g₂ s t) : 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma IsCircuit.isTrail (h : IsCircuit p) : IsTrail p := h.toIsTrail	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma IsCycle.isCircuit (h : IsCycle p) : IsCircuit p := h.toIsCircuit	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma IsPath.cons (hp : p.IsPath) (hu : u ∉ p.support) {h : G.Adj u v} :	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma IsPath.isTrail (h : IsPath p) : IsTrail p := h.toIsTrail	
lake-packages/mathlib/Mathlib/Topology/Separation.lean:protected lemma IsPreirreducible.subsingleton [T2Space α] {S : Set α} (h : IsPreirreducible S) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma LeftInvOn.extendDomain (h : LeftInvOn g₁ g₂ s) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma MapsTo.extendDomain (h : MapsTo g s t) : 
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:protected lemma MulActionWithZero.nontrivial	
lake-packages/mathlib/Mathlib/Algebra/SMulWithZero.lean:protected lemma MulActionWithZero.subsingleton	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma Nil.eq {p : G.Walk v w} : p.Nil → v = w | .nil => rfl 
lake-packages/mathlib/Mathlib/Data/Finmap.lean:protected lemma NodupKeys.nodup {m : Multiset (Σ a, β a)} (h : m.NodupKeys) : m.Nodup :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma Preconnected.mono  {G G' : SimpleGraph V} (h : G ≤ G') (hG : G.Preconnected) : 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean:protected lemma Reachable.mono  {G G' : SimpleGraph V} (h : G ≤ G') (Guv : G.Reachable u v) :  
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma RightInvOn.extendDomain (h : RightInvOn g₁ g₂ t) :	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma SurjOn.extendDomain (h : SurjOn g s t) : 
lake-packages/mathlib/Mathlib/Topology/UniformSpace/UniformEmbedding.lean:protected lemma UniformInducing.comap_uniformSpace {f : α → β} (hf : UniformInducing f) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma add_comp {n₁ n₂ n₁₂ : ℤ} (z₁ z₁' : Cochain F G n₁) (z₂ : Cochain G K n₂)	
lake-packages/mathlib/Mathlib/Data/Set/Function.lean:protected lemma bijOn (h : ∀ a, e a ∈ t ↔ a ∈ s) : BijOn e s t := 
lake-packages/mathlib/Mathlib/Data/Int/Basic.lean:protected lemma coe_nat_sub {n m : ℕ} : n ≤ m → (↑(m - n) : ℤ) = ↑m - ↑n := ofNat_sub	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_add {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ z₂' : Cochain G K n₂)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_id {n : ℤ} (z₁ : Cochain F G n) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_neg {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_sub {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ z₂' : Cochain G K n₂)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_zero {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁)	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma comp_zsmul {n₁ n₂ n₁₂ : ℤ} (k : ℤ) (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂)	
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:protected lemma congr {g} (h : EventuallyConst f l) (hg : f =ᶠ[l] g) : EventuallyConst g l :=	
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma connected_iff {H : G.Subgraph} :  
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma connected_iff' {H : G.Subgraph} : 
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:protected lemma const (c : β) : EventuallyConst (fun _ ↦ c) l :=	
lake-packages/mathlib/Mathlib/Order/Filter/Basic.lean:protected lemma disjoint_iff {f g : Filter α} : Disjoint f g ↔ ∃ s ∈ f, ∃ t ∈ g, Disjoint s t := by	
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:protected lemma div_mul_div_le (a b c d : ℕ) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma id_comp {n : ℤ} (z₂ : Cochain F G n) : 
lake-packages/mathlib/Mathlib/Order/Filter/EventuallyConst.lean:protected lemma inv [Inv β] (h : EventuallyConst f l) : EventuallyConst (f⁻¹) l := h.comp Inv.inv	
lake-packages/mathlib/Mathlib/RingTheory/Polynomial/Nilpotent.lean:protected lemma isNilpotent_iff :   
lake-packages/mathlib/Mathlib/RingTheory/Nilpotent.lean:protected lemma isNilpotent_mul_left_iff (hy : y ∈ nonZeroDivisorsLeft R) :	
lake-packages/mathlib/Mathlib/RingTheory/Nilpotent.lean:protected lemma isNilpotent_mul_right_iff (hx : x ∈ nonZeroDivisorsRight R) :	
lake-packages/mathlib/Mathlib/RingTheory/Nilpotent.lean:protected lemma isNilpotent_sum {ι : Type _} {s : Finset ι} {f : ι → R}	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:protected lemma monotone_iff_continuous [TopologicalSpace α] [LowerSetTopology α]	
lake-packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean:protected lemma monotone_iff_continuous [TopologicalSpace α] [UpperSetTopology α]	
lake-packages/mathlib/Mathlib/Data/Nat/ForSqrt.lean:protected lemma mul_le_of_le_div (k x y : ℕ) (h : x ≤ y / k) : x * k ≤ y := by		 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma neg_comp {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) 
lake-packages/mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean:protected lemma preconnected_iff {H : G.Subgraph} :  
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma refl (hr : ∀ a : α, r₁₁ a a) :	 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma refl (hr : ∀ a : α, r₁₁ a a) :	 
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma refl (hr : ∀ a : α, r₁₁ a a) : 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma sub_comp {n₁ n₂ n₁₂ : ℤ} (z₁ z₁' : Cochain F G n₁) (z₂ : Cochain G K n₂)	
lake-packages/mathlib/Mathlib/Data/Set/Basic.lean:protected lemma subsingleton_or_nontrivial (s : Set α) : s.Subsingleton ∨ s.Nontrivial := by	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma symm (hr : ∀ (a : α) (b : β), r₁₂ a b → r₂₁ b a) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma symm (hr : ∀ (a : α) (b : β), r₁₂ a b → r₂₁ b a) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma symm (hr : ∀ (a : α) (b : β), r₁₂ a b → r₂₁ b a) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma trans (hr : ∀ (a : α) (b : β) (c : γ), r₁₂ a b → r₂₃ b c → r₁₃ a c) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma trans (hr : ∀ (a : α) (b : β) (c : γ), r₁₂ a b → r₂₃ b c → r₁₃ a c) :	
lake-packages/mathlib/Mathlib/Logic/Relator.lean:protected lemma trans (hr : ∀ (a : α) (b : β) (c : γ), r₁₂ a b → r₂₃ b c → r₁₃ a c) :	
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma zero_comp {n₁ n₂ n₁₂ : ℤ} (z₂ : Cochain G K n₂)	 
lake-packages/mathlib/Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean:protected lemma zsmul_comp {n₁ n₂ n₁₂ : ℤ} (k : ℤ) (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂)