# Negation on spheres and balls #

In this file we define `InvolutiveNeg` and `ContinuousNeg` instances for spheres, open balls, and closed balls in a seminormed group.

source
instance instInvolutiveNegElemSphereOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
InvolutiveNeg ↑(Metric.sphere 0 r)

We equip the sphere, in a seminormed group, with a formal operation of negation, namely the antipodal map.

Equations

- instInvolutiveNegElemSphereOfNat = { neg := Subtype.map Neg.neg ⋯, neg_neg := ⋯ }

source
@[simp]
theorem coe_neg_sphere {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.sphere 0 r)) :
↑(-v) = -↑v
source
instance instContinuousNegElemSphereOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
ContinuousNeg ↑(Metric.sphere 0 r)
source
instance instInvolutiveNegElemBallOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
InvolutiveNeg ↑(Metric.ball 0 r)

We equip the ball, in a seminormed group, with a formal operation of negation, namely the antipodal map.

Equations

- instInvolutiveNegElemBallOfNat = { neg := Subtype.map Neg.neg ⋯, neg_neg := ⋯ }

source
@[simp]
theorem coe_neg_ball {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.ball 0 r)) :
↑(-v) = -↑v
source
instance instContinuousNegElemBallOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
ContinuousNeg ↑(Metric.ball 0 r)
source
instance instInvolutiveNegElemClosedBallOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
InvolutiveNeg ↑(Metric.closedBall 0 r)

We equip the closed ball, in a seminormed group, with a formal operation of negation, namely the antipodal map.

Equations

- instInvolutiveNegElemClosedBallOfNat = { neg := Subtype.map Neg.neg ⋯, neg_neg := ⋯ }

source
@[simp]
theorem coe_neg_closedBall {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} (v : ↑(Metric.closedBall 0 r)) :
↑(-v) = -↑v
source
instance instContinuousNegElemClosedBallOfNat {E : Type u_1} [i : SeminormedAddCommGroup E] {r : ℝ} :
ContinuousNeg ↑(Metric.closedBall 0 r)