# Lemmas about powers in ordered fields. #

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theorem Even.zpow_nonneg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {n : ℤ} (hn : Even n) (a : α) :
0 ≤ a ^ n
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theorem zpow_two_nonneg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a : α) :
0 ≤ a ^ 2
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theorem zpow_neg_two_nonneg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] (a : α) :
0 ≤ a ^ (-2)
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theorem Even.zpow_pos {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Even n) (ha : a ≠ 0) :
0 < a ^ n
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theorem zpow_two_pos_of_ne_zero {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} (ha : a ≠ 0) :
0 < a ^ 2
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theorem Even.zpow_pos_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Even n) (h : n ≠ 0) :
0 < a ^ n ↔ a ≠ 0
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theorem Odd.zpow_neg_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
a ^ n < 0 ↔ a < 0
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theorem Odd.zpow_nonneg_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
0 ≤ a ^ n ↔ 0 ≤ a
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theorem Odd.zpow_nonpos_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
a ^ n ≤ 0 ↔ a ≤ 0
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theorem Odd.zpow_pos_iff {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
0 < a ^ n ↔ 0 < a
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theorem Odd.zpow_neg {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
a < 0 → a ^ n < 0

Alias of the reverse direction of `Odd.zpow_neg_iff`.

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theorem Odd.zpow_nonpos {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : Odd n) :
a ≤ 0 → a ^ n ≤ 0

Alias of the reverse direction of `Odd.zpow_nonpos_iff`.

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theorem Even.zpow_abs {α : Type u_1} [Field α] [LinearOrder α] {p : ℤ} (hp : Even p) (a : α) :
|a| ^ p = a ^ p
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theorem zpow_eq_zpow_iff_of_ne_zero₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {n : ℤ} (hn : n ≠ 0) :
a ^ n = b ^ n ↔ a = b ∨ a = -b ∧ Even n
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theorem zpow_eq_zpow_iff_cases₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {n : ℤ} :
a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n
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theorem zpow_eq_one_iff_of_ne_zero₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} (hn : n ≠ 0) :
a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n
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theorem zpow_eq_one_iff_cases₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} :
a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n
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theorem zpow_eq_neg_zpow_iff₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {n : ℤ} (hb : b ≠ 0) :
a ^ n = -b ^ n ↔ a = -b ∧ Odd n
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theorem zpow_eq_neg_one_iff₀ {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} {n : ℤ} :
a ^ n = -1 ↔ a = -1 ∧ Odd n

### Bernoulli's inequality #

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theorem Nat.cast_le_pow_sub_div_sub {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} (H : 1 < a) (n : ℕ) :
↑n ≤ (a ^ n - 1) / (a - 1)

Bernoulli's inequality reformulated to estimate `(n : α)`.

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theorem Nat.cast_le_pow_div_sub {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} (H : 1 < a) (n : ℕ) :
↑n ≤ a ^ n / (a - 1)

For any `a > 1` and a natural `n` we have `n ≤ a ^ n / (a - 1)`. See also `Nat.cast_le_pow_sub_div_sub` for a stronger inequality with `a ^ n - 1` in the numerator.

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def Mathlib.Meta.Positivity.evalZPow :
PositivityExt

The `positivity` extension which identifies expressions of the form `a ^ (b : ℤ)`, such that `positivity` successfully recognises both `a` and `b`.

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- One or more equations did not get rendered due to their size.

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