File: Mathlib/Analysis/NormedSpace/BallAction.lean

full_name: mulActionClosedBallBall
code: instance mulActionClosedBallBall : MulAction (closedBall (0 : 𝕜) 1) (ball (0 : E) r) 
goal: MulAction ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: continuousSMul_closedBall_ball
code: instance continuousSMul_closedBall_ball : ContinuousSMul (closedBall (0 : 𝕜) 1) (ball (0 : E) r) 
goal: ContinuousSMul ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: mulActionClosedBallClosedBall
code: instance mulActionClosedBallClosedBall :
    MulAction (closedBall (0 : 𝕜) 1) (closedBall (0 : E) r) 
goal: MulAction ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: continuousSMul_closedBall_closedBall
code: instance continuousSMul_closedBall_closedBall :
    ContinuousSMul (closedBall (0 : 𝕜) 1) (closedBall (0 : E) r) 
goal: ContinuousSMul ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: mulActionSphereBall
code: instance mulActionSphereBall : MulAction (sphere (0 : 𝕜) 1) (ball (0 : E) r) 
goal: MulAction ↑(Metric.sphere 0 1) ↑(Metric.ball 0 r)

full_name: continuousSMul_sphere_ball
code: instance continuousSMul_sphere_ball : ContinuousSMul (sphere (0 : 𝕜) 1) (ball (0 : E) r) 
goal: ContinuousSMul ↑(Metric.sphere 0 1) ↑(Metric.ball 0 r)

full_name: mulActionSphereClosedBall
code: instance mulActionSphereClosedBall : MulAction (sphere (0 : 𝕜) 1) (closedBall (0 : E) r) 
goal: MulAction ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 r)

full_name: continuousSMul_sphere_closedBall
code: instance continuousSMul_sphere_closedBall :
    ContinuousSMul (sphere (0 : 𝕜) 1) (closedBall (0 : E) r) 
goal: ContinuousSMul ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 r)

full_name: mulActionSphereSphere
code: instance mulActionSphereSphere : MulAction (sphere (0 : 𝕜) 1) (sphere (0 : E) r) 
goal: MulAction ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 r)

full_name: continuousSMul_sphere_sphere
code: instance continuousSMul_sphere_sphere : ContinuousSMul (sphere (0 : 𝕜) 1) (sphere (0 : E) r) 
goal: ContinuousSMul ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 r)

full_name: isScalarTower_closedBall_closedBall_closedBall
code: instance isScalarTower_closedBall_closedBall_closedBall :
    IsScalarTower (closedBall (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: IsScalarTower ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: isScalarTower_closedBall_closedBall_ball
code: instance isScalarTower_closedBall_closedBall_ball :
    IsScalarTower (closedBall (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (ball (0 : E) r) 
goal: IsScalarTower ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: isScalarTower_sphere_closedBall_closedBall
code: instance isScalarTower_sphere_closedBall_closedBall :
    IsScalarTower (sphere (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: isScalarTower_sphere_closedBall_ball
code: instance isScalarTower_sphere_closedBall_ball :
    IsScalarTower (sphere (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (ball (0 : E) r) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: isScalarTower_sphere_sphere_closedBall
code: instance isScalarTower_sphere_sphere_closedBall :
    IsScalarTower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 r)

full_name: isScalarTower_sphere_sphere_ball
code: instance isScalarTower_sphere_sphere_ball :
    IsScalarTower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (ball (0 : E) r) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.ball 0 r)

full_name: isScalarTower_sphere_sphere_sphere
code: instance isScalarTower_sphere_sphere_sphere :
    IsScalarTower (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (sphere (0 : E) r) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 r)

full_name: isScalarTower_sphere_ball_ball
code: instance isScalarTower_sphere_ball_ball :
    IsScalarTower (sphere (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) 
goal: IsScalarTower ↑(Metric.sphere 0 1) ↑(Metric.ball 0 1) ↑(Metric.ball 0 1)

full_name: isScalarTower_closedBall_ball_ball
code: instance isScalarTower_closedBall_ball_ball :
    IsScalarTower (closedBall (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) 
goal: IsScalarTower ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 1) ↑(Metric.ball 0 1)

full_name: instSMulCommClass_closedBall_closedBall_closedBall
code: instance instSMulCommClass_closedBall_closedBall_closedBall :
    SMulCommClass (closedBall (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: SMulCommClass ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: instSMulCommClass_closedBall_closedBall_ball
code: instance instSMulCommClass_closedBall_closedBall_ball :
    SMulCommClass (closedBall (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (ball (0 : E) r) 
goal: SMulCommClass ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: instSMulCommClass_sphere_closedBall_closedBall
code: instance instSMulCommClass_sphere_closedBall_closedBall :
    SMulCommClass (sphere (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.closedBall 0 r)

full_name: instSMulCommClass_sphere_closedBall_ball
code: instance instSMulCommClass_sphere_closedBall_ball :
    SMulCommClass (sphere (0 : 𝕜) 1) (closedBall (0 : 𝕜') 1) (ball (0 : E) r) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 1) ↑(Metric.ball 0 r)

full_name: instSMulCommClass_sphere_ball_ball
code: instance instSMulCommClass_sphere_ball_ball [NormedAlgebra 𝕜 𝕜'] :
    SMulCommClass (sphere (0 : 𝕜) 1) (ball (0 : 𝕜') 1) (ball (0 : 𝕜') 1) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.ball 0 1) ↑(Metric.ball 0 1)

full_name: instSMulCommClass_sphere_sphere_closedBall
code: instance instSMulCommClass_sphere_sphere_closedBall :
    SMulCommClass (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (closedBall (0 : E) r) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.closedBall 0 r)

full_name: instSMulCommClass_sphere_sphere_ball
code: instance instSMulCommClass_sphere_sphere_ball :
    SMulCommClass (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (ball (0 : E) r) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.ball 0 r)

full_name: instSMulCommClass_sphere_sphere_sphere
code: instance instSMulCommClass_sphere_sphere_sphere :
    SMulCommClass (sphere (0 : 𝕜) 1) (sphere (0 : 𝕜') 1) (sphere (0 : E) r) 
goal: SMulCommClass ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 1) ↑(Metric.sphere 0 r)

full_name: ne_neg_of_mem_sphere
code: theorem ne_neg_of_mem_sphere {r : ℝ} (hr : r ≠ 0) (x : sphere (0 : E) r) : x ≠ -x 
goal: x ≠ -x

full_name: ne_neg_of_mem_unit_sphere
code: theorem ne_neg_of_mem_unit_sphere (x : sphere (0 : E) 1) : x ≠ -x 
goal: x ≠ -x