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.

Algebraic structures on Metric.ball 0 1

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

Unit ball in a non-unital seminormed ring as a bundled Subsemigroup.

Equations

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

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

Equations

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

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

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

Equations

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

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

Equations

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

@[simp]

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

@[deprecated Metric.unitBall.coe_mul (since := "2025-04-18")]

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

Alias of Metric.unitBall.coe_mul.

instance Metric.unitBall.instZero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] : Zero ↑(ball 0 1)

Equations

• Metric.unitBall.instZero = { zero := ⟨0, ⋯⟩ }

@[simp]

theorem Metric.unitBall.coe_zero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] : ↑0 = 0

@[simp]

theorem Metric.unitBall.coe_eq_zero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] {a : ↑(ball 0 1)} : ↑a = 0 ↔ a = 0

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

Equations

• Metric.unitBall.instSemigroupWithZero = { toSemigroup := Metric.unitBall.instSemigroup, toZero := Metric.unitBall.instZero, zero_mul := ⋯, mul_zero := ⋯ }

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

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

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

Algebraic instances for Metric.closedBall 0 1

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

Closed unit ball in a non-unital seminormed ring as a bundled Subsemigroup.

Equations

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

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

Equations

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

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

Equations

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

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

@[simp]

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

@[deprecated Metric.unitClosedBall.coe_mul (since := "2025-04-18")]

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

Alias of Metric.unitClosedBall.coe_mul.

instance Metric.unitClosedBall.instZero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] : Zero ↑(closedBall 0 1)

Equations

• Metric.unitClosedBall.instZero = { zero := ⟨0, ⋯⟩ }

@[simp]

theorem Metric.unitClosedBall.coe_zero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] : ↑0 = 0

@[simp]

theorem Metric.unitClosedBall.coe_eq_zero {𝕜 : Type u_1} [Zero 𝕜] [PseudoMetricSpace 𝕜] {a : ↑(closedBall 0 1)} : ↑a = 0 ↔ a = 0

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

Equations

• Metric.unitClosedBall.instSemigroupWithZero = { toSemigroup := Metric.unitClosedBall.instSemigroup, toZero := Metric.unitClosedBall.instZero, zero_mul := ⋯, mul_zero := ⋯ }

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

Closed unit ball in a seminormed ring as a bundled Submonoid.

Equations

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

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

Equations

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

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

Equations

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

@[simp]

theorem Metric.unitClosedBall.coe_one {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] : ↑1 = 1

@[deprecated Metric.unitClosedBall.coe_one (since := "2025-04-18")]

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

Alias of Metric.unitClosedBall.coe_one.

@[simp]

theorem Metric.unitClosedBall.coe_eq_one {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormOneClass 𝕜] {a : ↑(closedBall 0 1)} : ↑a = 1 ↔ a = 1

@[simp]

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

@[deprecated Metric.unitClosedBall.coe_pow (since := "2025-04-18")]

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

Alias of Metric.unitClosedBall.coe_pow.

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

Equations

• Metric.unitClosedBall.instMonoidWithZero = { toMonoid := Metric.unitClosedBall.instMonoid, toZero := SemigroupWithZero.toZero, zero_mul := ⋯, mul_zero := ⋯ }

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

Equations

• Metric.unitClosedBall.instCancelMonoidWithZero = { toMonoidWithZero := Metric.unitClosedBall.instMonoidWithZero, toIsCancelMulZero := ⋯ }

Algebraic instances on the unit sphere

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

Unit sphere in a seminormed ring (with strictly multiplicative norm) as a bundled Submonoid.

Equations

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

@[simp]

theorem Submonoid.coe_unitSphere (𝕜 : Type u_2) [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : ↑(unitSphere 𝕜) = Metric.sphere 0 1

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

Equations

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

@[simp]

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

@[deprecated Metric.unitSphere.coe_inv (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_inv.

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

Equations

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

@[simp]

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

@[deprecated Metric.unitSphere.coe_div (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_div.

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

Equations

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

@[simp]

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

@[deprecated Metric.unitSphere.coe_zpow (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_zpow.

instance Metric.unitSphere.instMonoid {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : Monoid ↑(sphere 0 1)

Equations

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

instance Metric.unitSphere.instCommMonoid {𝕜 : Type u_1} [SeminormedCommRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : CommMonoid ↑(sphere 0 1)

Equations

• Metric.unitSphere.instCommMonoid = SubmonoidClass.toCommMonoid (Submonoid.unitSphere 𝕜)

@[simp]

theorem Metric.unitSphere.coe_one {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : ↑1 = 1

@[deprecated Metric.unitSphere.coe_one (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_one.

@[simp]

theorem Metric.unitSphere.coe_mul {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] (x y : ↑(sphere 0 1)) : ↑(x * y) = ↑x * ↑y

@[deprecated Metric.unitSphere.coe_mul (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_mul.

@[simp]

theorem Metric.unitSphere.coe_pow {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] (x : ↑(sphere 0 1)) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

@[deprecated Metric.unitSphere.coe_pow (since := "2025-04-18")]

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

Alias of Metric.unitSphere.coe_pow.

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

Monoid homomorphism from the unit sphere in a normed division ring 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.unitSphere.instGroup {𝕜 : Type u_1} [NormedDivisionRing 𝕜] : Group ↑(sphere 0 1)

Equations

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

instance Metric.sphere.instHasDistribNeg {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : HasDistribNeg ↑(sphere 0 1)

Equations

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

instance Metric.sphere.instContinuousMul {𝕜 : Type u_1} [SeminormedRing 𝕜] [NormMulClass 𝕜] [NormOneClass 𝕜] : ContinuousMul ↑(sphere 0 1)

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

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

Equations

• Metric.sphere.instCommGroup = { toGroup := Metric.unitSphere.instGroup, mul_comm := ⋯ }