File: Mathlib/Analysis/Normed/Field/UnitBall.lean

full_name: Subsemigroup.unitBall
code: /-- Unit ball in a non unital semi normed ring as a bundled `Subsemigroup`. -/
def Subsemigroup.unitBall (𝕜 : Type*) [NonUnitalSeminormedRing 𝕜] : Subsemigroup 𝕜 
goal: Subsemigroup 𝕜

full_name: Metric.unitBall.semigroup
code: instance Metric.unitBall.semigroup [NonUnitalSeminormedRing 𝕜] : Semigroup (ball (0 : 𝕜) 1) 
goal: Semigroup ↑(Metric.ball 0 1)

full_name: Metric.unitBall.continuousMul
code: instance Metric.unitBall.continuousMul [NonUnitalSeminormedRing 𝕜] :
    ContinuousMul (ball (0 : 𝕜) 1) 
goal: ContinuousMul ↑(Metric.ball 0 1)

full_name: Metric.unitBall.commSemigroup
code: instance Metric.unitBall.commSemigroup [SeminormedCommRing 𝕜] : CommSemigroup (ball (0 : 𝕜) 1) 
goal: CommSemigroup ↑(Metric.ball 0 1)

full_name: Metric.unitBall.hasDistribNeg
code: instance Metric.unitBall.hasDistribNeg [NonUnitalSeminormedRing 𝕜] :
    HasDistribNeg (ball (0 : 𝕜) 1) 
goal: HasDistribNeg ↑(Metric.ball 0 1)

full_name: coe_mul_unitBall
code: @[simp, norm_cast]
theorem coe_mul_unitBall [NonUnitalSeminormedRing 𝕜] (x y : ball (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Subsemigroup.unitClosedBall
code: /-- Closed unit ball in a non unital semi normed ring as a bundled `Subsemigroup`. -/
def Subsemigroup.unitClosedBall (𝕜 : Type*) [NonUnitalSeminormedRing 𝕜] : Subsemigroup 𝕜 
goal: Subsemigroup 𝕜

full_name: Metric.unitClosedBall.semigroup
code: instance Metric.unitClosedBall.semigroup [NonUnitalSeminormedRing 𝕜] :
    Semigroup (closedBall (0 : 𝕜) 1) 
goal: Semigroup ↑(Metric.closedBall 0 1)

full_name: Metric.unitClosedBall.hasDistribNeg
code: instance Metric.unitClosedBall.hasDistribNeg [NonUnitalSeminormedRing 𝕜] :
    HasDistribNeg (closedBall (0 : 𝕜) 1) 
goal: HasDistribNeg ↑(Metric.closedBall 0 1)

full_name: Metric.unitClosedBall.continuousMul
code: instance Metric.unitClosedBall.continuousMul [NonUnitalSeminormedRing 𝕜] :
    ContinuousMul (closedBall (0 : 𝕜) 1) 
goal: ContinuousMul ↑(Metric.closedBall 0 1)

full_name: coe_mul_unitClosedBall
code: @[simp, norm_cast]
theorem coe_mul_unitClosedBall [NonUnitalSeminormedRing 𝕜] (x y : closedBall (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Submonoid.unitClosedBall
code: /-- Closed unit ball in a semi normed ring as a bundled `Submonoid`. -/
def Submonoid.unitClosedBall (𝕜 : Type*) [SeminormedRing 𝕜] [NormOneClass 𝕜] : Submonoid 𝕜 
goal: Submonoid 𝕜

full_name: Metric.unitClosedBall.monoid
code: instance Metric.unitClosedBall.monoid [SeminormedRing 𝕜] [NormOneClass 𝕜] :
    Monoid (closedBall (0 : 𝕜) 1) 
goal: Monoid ↑(Metric.closedBall 0 1)

full_name: Metric.unitClosedBall.commMonoid
code: instance Metric.unitClosedBall.commMonoid [SeminormedCommRing 𝕜] [NormOneClass 𝕜] :
    CommMonoid (closedBall (0 : 𝕜) 1) 
goal: CommMonoid ↑(Metric.closedBall 0 1)

full_name: coe_one_unitClosedBall
code: @[simp, norm_cast]
theorem coe_one_unitClosedBall [SeminormedRing 𝕜] [NormOneClass 𝕜] :
    ((1 : closedBall (0 : 𝕜) 1) : 𝕜) = 1 
goal: ↑1 = 1

full_name: coe_pow_unitClosedBall
code: @[simp, norm_cast]
theorem coe_pow_unitClosedBall [SeminormedRing 𝕜] [NormOneClass 𝕜] (x : closedBall (0 : 𝕜) 1)
    (n : ℕ) : ↑(x ^ n) = (x : 𝕜) ^ n 
goal: ↑(x ^ n) = ↑x ^ n

full_name: Submonoid.unitSphere
code: /-- Unit sphere in a normed division ring as a bundled `Submonoid`. -/
def Submonoid.unitSphere (𝕜 : Type*) [NormedDivisionRing 𝕜] : Submonoid 𝕜 
goal: Submonoid 𝕜

full_name: Metric.unitSphere.inv
code: instance Metric.unitSphere.inv [NormedDivisionRing 𝕜] : Inv (sphere (0 : 𝕜) 1) 
goal: Inv ↑(Metric.sphere 0 1)

full_name: coe_inv_unitSphere
code: @[simp, norm_cast]
theorem coe_inv_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) : ↑x⁻¹ = (x⁻¹ : 𝕜) 
goal: ↑x⁻¹ = (↑x)⁻¹

full_name: Metric.unitSphere.div
code: instance Metric.unitSphere.div [NormedDivisionRing 𝕜] : Div (sphere (0 : 𝕜) 1) 
goal: Div ↑(Metric.sphere 0 1)

full_name: coe_div_unitSphere
code: @[simp, norm_cast]
theorem coe_div_unitSphere [NormedDivisionRing 𝕜] (x y : sphere (0 : 𝕜) 1) :
    ↑(x / y) = (x / y : 𝕜) 
goal: ↑(x / y) = ↑x / ↑y

full_name: Metric.unitSphere.pow
code: instance Metric.unitSphere.pow [NormedDivisionRing 𝕜] : Pow (sphere (0 : 𝕜) 1) ℤ 
goal: Pow ↑(Metric.sphere 0 1) ℤ

full_name: coe_zpow_unitSphere
code: @[simp, norm_cast]
theorem coe_zpow_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℤ) :
    ↑(x ^ n) = (x : 𝕜) ^ n 
goal: ↑(x ^ n) = ↑x ^ n

full_name: Metric.unitSphere.monoid
code: instance Metric.unitSphere.monoid [NormedDivisionRing 𝕜] : Monoid (sphere (0 : 𝕜) 1) 
goal: Monoid ↑(Metric.sphere 0 1)

full_name: coe_one_unitSphere
code: @[simp, norm_cast]
theorem coe_one_unitSphere [NormedDivisionRing 𝕜] : ((1 : sphere (0 : 𝕜) 1) : 𝕜) = 1 
goal: ↑1 = 1

full_name: coe_mul_unitSphere
code: @[simp, norm_cast]
theorem coe_mul_unitSphere [NormedDivisionRing 𝕜] (x y : sphere (0 : 𝕜) 1) :
    ↑(x * y) = (x * y : 𝕜) 
goal: ↑(x * y) = ↑x * ↑y

full_name: coe_pow_unitSphere
code: @[simp, norm_cast]
theorem coe_pow_unitSphere [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) (n : ℕ) :
    ↑(x ^ n) = (x : 𝕜) ^ n 
goal: ↑(x ^ n) = ↑x ^ n

full_name: unitSphereToUnits
code: /-- Monoid homomorphism from the unit sphere to the group of units. -/
def unitSphereToUnits (𝕜 : Type*) [NormedDivisionRing 𝕜] : sphere (0 : 𝕜) 1 →* Units 𝕜 
goal: ↑(Metric.sphere 0 1) →* 𝕜ˣ

full_name: unitSphereToUnits_apply_coe
code: @[simp]
theorem unitSphereToUnits_apply_coe [NormedDivisionRing 𝕜] (x : sphere (0 : 𝕜) 1) :
    (unitSphereToUnits 𝕜 x : 𝕜) = x 
goal: ↑((unitSphereToUnits 𝕜) x) = ↑x

full_name: unitSphereToUnits_injective
code: theorem unitSphereToUnits_injective [NormedDivisionRing 𝕜] :
    Function.Injective (unitSphereToUnits 𝕜) 
goal: Function.Injective ⇑(unitSphereToUnits 𝕜)

full_name: Metric.sphere.group
code: instance Metric.sphere.group [NormedDivisionRing 𝕜] : Group (sphere (0 : 𝕜) 1) 
goal: Group ↑(Metric.sphere 0 1)

full_name: Metric.sphere.hasDistribNeg
code: instance Metric.sphere.hasDistribNeg [NormedDivisionRing 𝕜] : HasDistribNeg (sphere (0 : 𝕜) 1) 
goal: HasDistribNeg ↑(Metric.sphere 0 1)

full_name: Metric.sphere.topologicalGroup
code: instance Metric.sphere.topologicalGroup [NormedDivisionRing 𝕜] :
    TopologicalGroup (sphere (0 : 𝕜) 1) 
goal: TopologicalGroup ↑(Metric.sphere 0 1)

full_name: Metric.sphere.commGroup
code: instance Metric.sphere.commGroup [NormedField 𝕜] : CommGroup (sphere (0 : 𝕜) 1) 
goal: CommGroup ↑(Metric.sphere 0 1)