Algebraic structures on unit balls and spheres

In this file we define algebraic structures (Semigroup, CommSemigroup, Monoid, CommMonoid, Group, CommGroup) on Metric.ball (0 : 𝕜) 1, Metric.closedBall (0 : 𝕜) 1, and Metric.sphere (0 : 𝕜) 1. In each case we use the weakest possible typeclass assumption on 𝕜, from NonUnitalSeminormedRing to NormedField.

def Subsemigroup.unitBall (𝕜 : Type u_2) [NonUnitalSeminormedRing 𝕜] : Subsemigroup 𝕜

Unit ball in a non unital semi normed ring as a bundled Subsemigroup.

Equations

• Subsemigroup.unitBall 𝕜 = { carrier := Metric.ball 0 1, mul_mem' := ⋯ }

instance Metric.unitBall.semigroup {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : Semigroup ↑(Metric.ball 0 1)

Equations

• Metric.unitBall.semigroup = MulMemClass.toSemigroup (Subsemigroup.unitBall 𝕜)

instance Metric.unitBall.continuousMul {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : ContinuousMul ↑(Metric.ball 0 1)

Equations

• ⋯ = ⋯

instance Metric.unitBall.commSemigroup {𝕜 : Type u_1} [SeminormedCommRing 𝕜] : CommSemigroup ↑(Metric.ball 0 1)

Equations

• Metric.unitBall.commSemigroup = MulMemClass.toCommSemigroup (Subsemigroup.unitBall 𝕜)

instance Metric.unitBall.hasDistribNeg {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : HasDistribNeg ↑(Metric.ball 0 1)

Equations

• Metric.unitBall.hasDistribNeg = Function.Injective.hasDistribNeg Subtype.val ⋯ ⋯ ⋯

@[simp]

theorem coe_mul_unitBall {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] (x : ↑(Metric.ball 0 1)) (y : ↑(Metric.ball 0 1)) : ↑(x * y) = ↑x * ↑y

def Subsemigroup.unitClosedBall (𝕜 : Type u_2) [NonUnitalSeminormedRing 𝕜] : Subsemigroup 𝕜

Closed unit ball in a non unital semi normed ring as a bundled Subsemigroup.

Equations

• Subsemigroup.unitClosedBall 𝕜 = { carrier := Metric.closedBall 0 1, mul_mem' := ⋯ }

instance Metric.unitClosedBall.semigroup {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : Semigroup ↑(Metric.closedBall 0 1)

Equations

• Metric.unitClosedBall.semigroup = MulMemClass.toSemigroup (Subsemigroup.unitClosedBall 𝕜)

instance Metric.unitClosedBall.hasDistribNeg {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : HasDistribNeg ↑(Metric.closedBall 0 1)

Equations

• Metric.unitClosedBall.hasDistribNeg = Function.Injective.hasDistribNeg Subtype.val ⋯ ⋯ ⋯

instance Metric.unitClosedBall.continuousMul {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] : ContinuousMul ↑(Metric.closedBall 0 1)

Equations

• ⋯ = ⋯

@[simp]

theorem coe_mul_unitClosedBall {𝕜 : Type u_1} [NonUnitalSeminormedRing 𝕜] (x : ↑(Metric.closedBall 0 1)) (y : ↑(Metric.closedBall 0 1)) : ↑(x * y) = ↑x * ↑y

def Submonoid.unitClosedBall (𝕜 : Type u_2) [SeminormedRing 𝕜] [NormOneClass 𝕜] : Submonoid 𝕜

Closed unit ball in a semi normed ring as a bundled Submonoid.

Equations

• Submonoid.unitClosedBall 𝕜 = { carrier := Metric.closedBall 0 1, mul_mem' := ⋯, one_mem' := ⋯ }

instance Metric.unitClosedBall.monoid {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] : Monoid ↑(Metric.closedBall 0 1)

Equations

• Metric.unitClosedBall.monoid = SubmonoidClass.toMonoid (Submonoid.unitClosedBall 𝕜)

instance Metric.unitClosedBall.commMonoid {𝕜 : Type u_1} [SeminormedCommRing 𝕜] [NormOneClass 𝕜] : CommMonoid ↑(Metric.closedBall 0 1)

Equations

• Metric.unitClosedBall.commMonoid = SubmonoidClass.toCommMonoid (Submonoid.unitClosedBall 𝕜)

@[simp]

theorem coe_one_unitClosedBall {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] : ↑1 = 1

@[simp]

theorem coe_pow_unitClosedBall {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] (x : ↑(Metric.closedBall 0 1)) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

@[simp]

theorem Submonoid.unitSphere_coe (𝕜 : Type u_2) [NormedDivisionRing 𝕜] : ↑(Submonoid.unitSphere 𝕜) = Metric.sphere 0 1

def Submonoid.unitSphere (𝕜 : Type u_2) [NormedDivisionRing 𝕜] : Submonoid 𝕜

Unit sphere in a normed division ring as a bundled Submonoid.

Equations

• Submonoid.unitSphere 𝕜 = { carrier := Metric.sphere 0 1, mul_mem' := ⋯, one_mem' := ⋯ }

instance Metric.unitSphere.inv {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Inv ↑(Metric.sphere 0 1)

Equations

• Metric.unitSphere.inv = { inv := fun (x : ↑(Metric.sphere 0 1)) => ⟨(↑x)⁻¹, ⋯⟩ }

@[simp]

theorem coe_inv_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) : ↑x⁻¹ = (↑x)⁻¹

instance Metric.unitSphere.div {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Div ↑(Metric.sphere 0 1)

Equations

• Metric.unitSphere.div = { div := fun (x y : ↑(Metric.sphere 0 1)) => ⟨↑x / ↑y, ⋯⟩ }

@[simp]

theorem coe_div_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (y : ↑(Metric.sphere 0 1)) : ↑(x / y) = ↑x / ↑y

instance Metric.unitSphere.pow {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Pow ↑(Metric.sphere 0 1) ℤ

Equations

• Metric.unitSphere.pow = { pow := fun (x : ↑(Metric.sphere 0 1)) (n : ℤ) => ⟨↑x ^ n, ⋯⟩ }

@[simp]

theorem coe_zpow_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (n : ℤ) : ↑(x ^ n) = ↑x ^ n

instance Metric.unitSphere.monoid {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Monoid ↑(Metric.sphere 0 1)

Equations

• Metric.unitSphere.monoid = SubmonoidClass.toMonoid (Submonoid.unitSphere 𝕜)

@[simp]

theorem coe_one_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : ↑1 = 1

@[simp]

theorem coe_mul_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (y : ↑(Metric.sphere 0 1)) : ↑(x * y) = ↑x * ↑y

@[simp]

theorem coe_pow_unitSphere {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

def unitSphereToUnits (𝕜 : Type u_2) [NormedDivisionRing 𝕜] : ↑(Metric.sphere 0 1) →* 𝕜ˣ

Monoid homomorphism from the unit sphere to the group of units.

Equations

• unitSphereToUnits 𝕜 = Units.liftRight (Submonoid.unitSphere 𝕜).subtype (fun (x : ↑(Metric.sphere 0 1)) => Units.mk0 ↑x ⋯) ⋯

@[simp]

theorem unitSphereToUnits_apply_coe {𝕜 : Type u_1} [NormedDivisionRing 𝕜] (x : ↑(Metric.sphere 0 1)) : ↑((unitSphereToUnits 𝕜) x) = ↑x

theorem unitSphereToUnits_injective {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Function.Injective ⇑(unitSphereToUnits 𝕜)

instance Metric.sphere.group {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Group ↑(Metric.sphere 0 1)

Equations

• Metric.sphere.group = Function.Injective.group ⇑(unitSphereToUnits 𝕜) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance Metric.sphere.hasDistribNeg {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : HasDistribNeg ↑(Metric.sphere 0 1)

Equations

• Metric.sphere.hasDistribNeg = Function.Injective.hasDistribNeg Subtype.val ⋯ ⋯ ⋯

instance Metric.sphere.topologicalGroup {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : TopologicalGroup ↑(Metric.sphere 0 1)

Equations

• ⋯ = ⋯

instance Metric.sphere.commGroup {𝕜 : Type u_1} [NormedField 𝕜] : CommGroup ↑(Metric.sphere 0 1)

Equations

• Metric.sphere.commGroup = CommGroup.mk ⋯