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Mathlib.Analysis.Normed.Group.BallSphere

Negation on spheres and balls #

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

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

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@[simp]
theorem coe_neg_sphere {E : Type u_1} [i : SeminormedAddCommGroup E] {r : } (v : (Metric.sphere 0 r)) :
(-v) = -v

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

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@[simp]
theorem coe_neg_ball {E : Type u_1} [i : SeminormedAddCommGroup E] {r : } (v : (Metric.ball 0 r)) :
(-v) = -v

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

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  • One or more equations did not get rendered due to their size.
@[simp]
theorem coe_neg_closedBall {E : Type u_1} [i : SeminormedAddCommGroup E] {r : } (v : (Metric.closedBall 0 r)) :
(-v) = -v