Affine isometries

In this file we define AffineIsometry 𝕜 P P₂ to be an affine isometric embedding of normed add-torsors P into P₂ over normed 𝕜-spaces and AffineIsometryEquiv to be an affine isometric equivalence between P and P₂.

We also prove basic lemmas and provide convenience constructors. The choice of these lemmas and constructors is closely modelled on those for the LinearIsometry and AffineMap theories.

Since many elementary properties don't require ‖x‖ = 0 → x = 0 we initially set up the theory for SeminormedAddCommGroup and specialize to NormedAddCommGroup only when needed.

Notation

We introduce the notation P →ᵃⁱ[𝕜] P₂ for AffineIsometry 𝕜 P P₂, and P ≃ᵃⁱ[𝕜] P₂ for AffineIsometryEquiv 𝕜 P P₂. In contrast with the notation →ₗᵢ for linear isometries, ≃ᵢ for isometric equivalences, etc., the "i" here is a superscript. This is for aesthetic reasons to match the superscript "a" (note that in mathlib →ᵃ is an affine map, since →ₐ has been taken by algebra-homomorphisms.)

structure AffineIsometry (𝕜 : Type u_1) {V : Type u_2} {V₂ : Type u_4} (P : Type u_8) (P₂ : Type u_9) [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] extends AffineMap : Type (max (max (max u_2 u_4) u_8) u_9)

A 𝕜-affine isometric embedding of one normed add-torsor over a normed 𝕜-space into another.

• toFun : P → P₂
• linear : V →ₗ[𝕜] V₂
• map_vadd' : ∀ (p : P) (v : V), self.toAffineMap.toFun (v +ᵥ p) = self.linear v +ᵥ self.toAffineMap.toFun p
• norm_map : ∀ (x : V), ‖self.linear x‖ = ‖x‖

def «term_→ᵃⁱ[_]_» : Lean.TrailingParserDescr

A 𝕜-affine isometric embedding of one normed add-torsor over a normed 𝕜-space into another.

Equations

• One or more equations did not get rendered due to their size.

def AffineIsometry.linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : V →ₗᵢ[𝕜] V₂

The underlying linear map of an affine isometry is in fact a linear isometry.

Equations

• AffineIsometry.linearIsometry f = let src := f.linear; { toLinearMap := src, norm_map' := (_ : ∀ (x : V), ‖f.linear x‖ = ‖x‖) }

@[simp]

theorem AffineIsometry.linear_eq_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : f.linear = (AffineIsometry.linearIsometry f).toLinearMap

instance AffineIsometry.instFunLikeAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] : FunLike (P →ᵃⁱ[𝕜] P₂) P P₂

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometry.coe_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : ⇑f.toAffineMap = ⇑f

theorem AffineIsometry.toAffineMap_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] : Function.Injective AffineIsometry.toAffineMap

theorem AffineIsometry.coeFn_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] : Function.Injective DFunLike.coe

theorem AffineIsometry.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] {f : P →ᵃⁱ[𝕜] P₂} {g : P →ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), f x = g x) : f = g

def LinearIsometry.toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : V →ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry as an affine isometry.

Equations

• LinearIsometry.toAffineIsometry f = let src := LinearMap.toAffineMap f.toLinearMap; { toAffineMap := src, norm_map := (_ : ∀ (x : V), ‖f x‖ = ‖x‖) }

@[simp]

theorem LinearIsometry.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : ⇑(LinearIsometry.toAffineIsometry f) = ⇑f

@[simp]

theorem LinearIsometry.toAffineIsometry_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : AffineIsometry.linearIsometry (LinearIsometry.toAffineIsometry f) = f

@[simp]

theorem LinearIsometry.toAffineIsometry_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : (LinearIsometry.toAffineIsometry f).toAffineMap = LinearMap.toAffineMap f.toLinearMap

@[simp]

theorem AffineIsometry.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p : P) (v : V) : f (v +ᵥ p) = (AffineIsometry.linearIsometry f) v +ᵥ f p

@[simp]

theorem AffineIsometry.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) : (AffineIsometry.linearIsometry f) (p1 -ᵥ p2) = f p1 -ᵥ f p2

@[simp]

theorem AffineIsometry.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) : dist (f x) (f y) = dist x y

@[simp]

theorem AffineIsometry.nndist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) : nndist (f x) (f y) = nndist x y

@[simp]

theorem AffineIsometry.edist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) : edist (f x) (f y) = edist x y

theorem AffineIsometry.isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : Isometry ⇑f

theorem AffineIsometry.injective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (f₁ : P₁ →ᵃⁱ[𝕜] P₂) : Function.Injective ⇑f₁

@[simp]

theorem AffineIsometry.map_eq_iff {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (f₁ : P₁ →ᵃⁱ[𝕜] P₂) {x : P₁} {y : P₁} : f₁ x = f₁ y ↔ x = y

theorem AffineIsometry.map_ne {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (f₁ : P₁ →ᵃⁱ[𝕜] P₂) {x : P₁} {y : P₁} (h : x ≠ y) : f₁ x ≠ f₁ y

theorem AffineIsometry.lipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : LipschitzWith 1 ⇑f

theorem AffineIsometry.antilipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : AntilipschitzWith 1 ⇑f

theorem AffineIsometry.continuous {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : Continuous ⇑f

theorem AffineIsometry.ediam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P) : EMetric.diam (⇑f '' s) = EMetric.diam s

theorem AffineIsometry.ediam_range {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ

theorem AffineIsometry.diam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P) : Metric.diam (⇑f '' s) = Metric.diam s

theorem AffineIsometry.diam_range {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : Metric.diam (Set.range ⇑f) = Metric.diam Set.univ

@[simp]

theorem AffineIsometry.comp_continuous_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) {α : Type u_12} [TopologicalSpace α] {g : α → P} : Continuous (⇑f ∘ g) ↔ Continuous g

def AffineIsometry.id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : P →ᵃⁱ[𝕜] P

The identity affine isometry.

Equations

• AffineIsometry.id = { toAffineMap := AffineMap.id 𝕜 P, norm_map := (_ : ∀ (x : V), ‖(AffineMap.id 𝕜 P).linear x‖ = ‖(AffineMap.id 𝕜 P).linear x‖) }

@[simp]

theorem AffineIsometry.coe_id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : ⇑AffineIsometry.id = id

@[simp]

theorem AffineIsometry.id_apply {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : AffineIsometry.id x = x

@[simp]

theorem AffineIsometry.id_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : AffineIsometry.id.toAffineMap = AffineMap.id 𝕜 P

instance AffineIsometry.instInhabitedAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Inhabited (P →ᵃⁱ[𝕜] P)

Equations

• AffineIsometry.instInhabitedAffineIsometry = { default := AffineIsometry.id }

def AffineIsometry.comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : P →ᵃⁱ[𝕜] P₃

Composition of affine isometries.

Equations

• AffineIsometry.comp g f = { toAffineMap := AffineMap.comp g.toAffineMap f.toAffineMap, norm_map := (_ : ∀ (x : V), ‖g.linear (f.linear x)‖ = ‖x‖) }

@[simp]

theorem AffineIsometry.coe_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : ⇑(AffineIsometry.comp g f) = ⇑g ∘ ⇑f

@[simp]

theorem AffineIsometry.id_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : AffineIsometry.comp AffineIsometry.id f = f

@[simp]

theorem AffineIsometry.comp_id {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : AffineIsometry.comp f AffineIsometry.id = f

theorem AffineIsometry.comp_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {V₄ : Type u_6} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} {P₄ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [PseudoMetricSpace P₄] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] [NormedAddTorsor V₄ P₄] (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂) : AffineIsometry.comp (AffineIsometry.comp f g) h = AffineIsometry.comp f (AffineIsometry.comp g h)

instance AffineIsometry.instMonoidAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Monoid (P →ᵃⁱ[𝕜] P)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometry.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : ⇑1 = id

@[simp]

theorem AffineIsometry.coe_mul {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (f : P →ᵃⁱ[𝕜] P) (g : P →ᵃⁱ[𝕜] P) : ⇑(f * g) = ⇑f ∘ ⇑g

def AffineSubspace.subtypeₐᵢ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : ↥s →ᵃⁱ[𝕜] P

AffineSubspace.subtype as an AffineIsometry.

Equations

• One or more equations did not get rendered due to their size.

theorem AffineSubspace.subtypeₐᵢ_linear {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : (AffineSubspace.subtypeₐᵢ s).toAffineMap.linear = Submodule.subtype (AffineSubspace.direction s)

@[simp]

theorem AffineSubspace.subtypeₐᵢ_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : AffineIsometry.linearIsometry (AffineSubspace.subtypeₐᵢ s) = Submodule.subtypeₗᵢ (AffineSubspace.direction s)

@[simp]

theorem AffineSubspace.coe_subtypeₐᵢ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : ⇑(AffineSubspace.subtypeₐᵢ s) = ⇑(AffineSubspace.subtype s)

@[simp]

theorem AffineSubspace.subtypeₐᵢ_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : (AffineSubspace.subtypeₐᵢ s).toAffineMap = AffineSubspace.subtype s

structure AffineIsometryEquiv (𝕜 : Type u_1) {V : Type u_2} {V₂ : Type u_4} (P : Type u_8) (P₂ : Type u_9) [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] extends AffineEquiv : Type (max (max (max u_2 u_4) u_8) u_9)

An affine isometric equivalence between two normed vector spaces.

• toFun : P → P₂
• invFun : P₂ → P
• left_inv : Function.LeftInverse self.invFun self.toFun
• right_inv : Function.RightInverse self.invFun self.toFun
• linear : V ≃ₗ[𝕜] V₂
• map_vadd' : ∀ (p : P) (v : V), self.toEquiv (v +ᵥ p) = self.linear v +ᵥ self.toEquiv p
• norm_map : ∀ (x : V), ‖self.linear x‖ = ‖x‖

def «term_≃ᵃⁱ[_]_» : Lean.TrailingParserDescr

Equations

• One or more equations did not get rendered due to their size.

def AffineIsometryEquiv.linearIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : V ≃ₗᵢ[𝕜] V₂

The underlying linear equiv of an affine isometry equiv is in fact a linear isometry equiv.

Equations

• AffineIsometryEquiv.linearIsometryEquiv e = let src := e.linear; { toLinearEquiv := src, norm_map' := (_ : ∀ (x : V), ‖e.linear x‖ = ‖x‖) }

@[simp]

theorem AffineIsometryEquiv.linear_eq_linear_isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.linear = (AffineIsometryEquiv.linearIsometryEquiv e).toLinearEquiv

instance AffineIsometryEquiv.instEquivLikeAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] : EquivLike (P ≃ᵃⁱ[𝕜] P₂) P P₂

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometryEquiv.coe_mk {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃ[𝕜] P₂) (he : ∀ (x : V), ‖e.linear x‖ = ‖x‖) : ⇑{ toAffineEquiv := e, norm_map := he } = ⇑e

@[simp]

theorem AffineIsometryEquiv.coe_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toAffineEquiv = ⇑e

theorem AffineIsometryEquiv.toAffineEquiv_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] : Function.Injective AffineIsometryEquiv.toAffineEquiv

theorem AffineIsometryEquiv.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] {e : P ≃ᵃⁱ[𝕜] P₂} {e' : P ≃ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), e x = e' x) : e = e'

def AffineIsometryEquiv.toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P →ᵃⁱ[𝕜] P₂

Reinterpret an AffineIsometryEquiv as an AffineIsometry.

Equations

• AffineIsometryEquiv.toAffineIsometry e = { toAffineMap := ↑e.toAffineEquiv, norm_map := (_ : ∀ (x : V), ‖e.linear x‖ = ‖x‖) }

@[simp]

theorem AffineIsometryEquiv.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑(AffineIsometryEquiv.toAffineIsometry e) = ⇑e

def AffineIsometryEquiv.mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) : P₁ ≃ᵃⁱ[𝕜] P₂

Construct an affine isometry equivalence by verifying the relation between the map and its linear part at one base point. Namely, this function takes a map e : P₁ → P₂, a linear isometry equivalence e' : V₁ ≃ᵢₗ[k] V₂, and a point p such that for any other point p' we have e p' = e' (p' -ᵥ p) +ᵥ e p.

Equations

• AffineIsometryEquiv.mk' e e' p h = let src := AffineEquiv.mk' e e'.toLinearEquiv p h; { toAffineEquiv := src, norm_map := (_ : ∀ (x : V₁), ‖e' x‖ = ‖x‖) }

@[simp]

theorem AffineIsometryEquiv.coe_mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) : ⇑(AffineIsometryEquiv.mk' e e' p h) = e

@[simp]

theorem AffineIsometryEquiv.linearIsometryEquiv_mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) : AffineIsometryEquiv.linearIsometryEquiv (AffineIsometryEquiv.mk' e e' p h) = e'

def LinearIsometryEquiv.toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : V ≃ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry equiv as an affine isometry equiv.

Equations

• LinearIsometryEquiv.toAffineIsometryEquiv e = let src := LinearEquiv.toAffineEquiv e.toLinearEquiv; { toAffineEquiv := src, norm_map := (_ : ∀ (x : V), ‖e x‖ = ‖x‖) }

@[simp]

theorem LinearIsometryEquiv.coe_toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : ⇑(LinearIsometryEquiv.toAffineIsometryEquiv e) = ⇑e

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_linearIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : AffineIsometryEquiv.linearIsometryEquiv (LinearIsometryEquiv.toAffineIsometryEquiv e) = e

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : (LinearIsometryEquiv.toAffineIsometryEquiv e).toAffineEquiv = LinearEquiv.toAffineEquiv e.toLinearEquiv

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : AffineIsometryEquiv.toAffineIsometry (LinearIsometryEquiv.toAffineIsometryEquiv e) = LinearIsometry.toAffineIsometry (LinearIsometryEquiv.toLinearIsometry e)

theorem AffineIsometryEquiv.isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Isometry ⇑e

def AffineIsometryEquiv.toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P ≃ᵢ P₂

Reinterpret an AffineIsometryEquiv as an IsometryEquiv.

Equations

• AffineIsometryEquiv.toIsometryEquiv e = { toEquiv := e.toEquiv, isometry_toFun := (_ : Isometry ⇑e) }

@[simp]

theorem AffineIsometryEquiv.coe_toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑(AffineIsometryEquiv.toIsometryEquiv e) = ⇑e

theorem AffineIsometryEquiv.range_eq_univ {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Set.range ⇑e = Set.univ

def AffineIsometryEquiv.toHomeomorph {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P ≃ₜ P₂

Reinterpret an AffineIsometryEquiv as a Homeomorph.

Equations

• AffineIsometryEquiv.toHomeomorph e = IsometryEquiv.toHomeomorph (AffineIsometryEquiv.toIsometryEquiv e)

@[simp]

theorem AffineIsometryEquiv.coe_toHomeomorph {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑(AffineIsometryEquiv.toHomeomorph e) = ⇑e

theorem AffineIsometryEquiv.continuous {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Continuous ⇑e

theorem AffineIsometryEquiv.continuousAt {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} : ContinuousAt (⇑e) x

theorem AffineIsometryEquiv.continuousOn {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {s : Set P} : ContinuousOn (⇑e) s

theorem AffineIsometryEquiv.continuousWithinAt {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {s : Set P} {x : P} : ContinuousWithinAt (⇑e) s x

def AffineIsometryEquiv.refl (𝕜 : Type u_1) {V : Type u_2} (P : Type u_8) [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : P ≃ᵃⁱ[𝕜] P

Identity map as an AffineIsometryEquiv.

Equations

• AffineIsometryEquiv.refl 𝕜 P = { toAffineEquiv := AffineEquiv.refl 𝕜 P, norm_map := (_ : ∀ (x : V), ‖(AffineEquiv.refl 𝕜 P).linear x‖ = ‖(AffineEquiv.refl 𝕜 P).linear x‖) }

instance AffineIsometryEquiv.instInhabitedAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Inhabited (P ≃ᵃⁱ[𝕜] P)

Equations

• AffineIsometryEquiv.instInhabitedAffineIsometryEquiv = { default := AffineIsometryEquiv.refl 𝕜 P }

@[simp]

theorem AffineIsometryEquiv.coe_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : ⇑(AffineIsometryEquiv.refl 𝕜 P) = id

@[simp]

theorem AffineIsometryEquiv.toAffineEquiv_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : (AffineIsometryEquiv.refl 𝕜 P).toAffineEquiv = AffineEquiv.refl 𝕜 P

@[simp]

theorem AffineIsometryEquiv.toIsometryEquiv_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : AffineIsometryEquiv.toIsometryEquiv (AffineIsometryEquiv.refl 𝕜 P) = IsometryEquiv.refl P

@[simp]

theorem AffineIsometryEquiv.toHomeomorph_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : AffineIsometryEquiv.toHomeomorph (AffineIsometryEquiv.refl 𝕜 P) = Homeomorph.refl P

def AffineIsometryEquiv.symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P₂ ≃ᵃⁱ[𝕜] P

The inverse AffineIsometryEquiv.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometryEquiv.apply_symm_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P₂) : e ((AffineIsometryEquiv.symm e) x) = x

@[simp]

theorem AffineIsometryEquiv.symm_apply_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) : (AffineIsometryEquiv.symm e) (e x) = x

@[simp]

theorem AffineIsometryEquiv.symm_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineIsometryEquiv.symm (AffineIsometryEquiv.symm e) = e

@[simp]

theorem AffineIsometryEquiv.toAffineEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineEquiv.symm e.toAffineEquiv = (AffineIsometryEquiv.symm e).toAffineEquiv

@[simp]

theorem AffineIsometryEquiv.toIsometryEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : IsometryEquiv.symm (AffineIsometryEquiv.toIsometryEquiv e) = AffineIsometryEquiv.toIsometryEquiv (AffineIsometryEquiv.symm e)

@[simp]

theorem AffineIsometryEquiv.toHomeomorph_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Homeomorph.symm (AffineIsometryEquiv.toHomeomorph e) = AffineIsometryEquiv.toHomeomorph (AffineIsometryEquiv.symm e)

def AffineIsometryEquiv.trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] (e : P ≃ᵃⁱ[𝕜] P₂) (e' : P₂ ≃ᵃⁱ[𝕜] P₃) : P ≃ᵃⁱ[𝕜] P₃

Composition of AffineIsometryEquivs as an AffineIsometryEquiv.

Equations

• AffineIsometryEquiv.trans e e' = { toAffineEquiv := AffineEquiv.trans e.toAffineEquiv e'.toAffineEquiv, norm_map := (_ : ∀ (x : V), ‖e'.linear (↑e.linear x)‖ = ‖x‖) }

@[simp]

theorem AffineIsometryEquiv.coe_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(AffineIsometryEquiv.trans e₁ e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem AffineIsometryEquiv.trans_refl {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineIsometryEquiv.trans e (AffineIsometryEquiv.refl 𝕜 P₂) = e

@[simp]

theorem AffineIsometryEquiv.refl_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineIsometryEquiv.trans (AffineIsometryEquiv.refl 𝕜 P) e = e

@[simp]

theorem AffineIsometryEquiv.self_trans_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineIsometryEquiv.trans e (AffineIsometryEquiv.symm e) = AffineIsometryEquiv.refl 𝕜 P

@[simp]

theorem AffineIsometryEquiv.symm_trans_self {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AffineIsometryEquiv.trans (AffineIsometryEquiv.symm e) e = AffineIsometryEquiv.refl 𝕜 P₂

@[simp]

theorem AffineIsometryEquiv.coe_symm_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(AffineIsometryEquiv.symm (AffineIsometryEquiv.trans e₁ e₂)) = ⇑(AffineIsometryEquiv.symm e₁) ∘ ⇑(AffineIsometryEquiv.symm e₂)

theorem AffineIsometryEquiv.trans_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {V₃ : Type u_5} {V₄ : Type u_6} {P : Type u_8} {P₂ : Type u_9} {P₃ : Type u_10} {P₄ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [NormedSpace 𝕜 V₃] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [PseudoMetricSpace P₃] [PseudoMetricSpace P₄] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] [NormedAddTorsor V₃ P₃] [NormedAddTorsor V₄ P₄] (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄) : AffineIsometryEquiv.trans ePP₂ (AffineIsometryEquiv.trans eP₂G eGG') = AffineIsometryEquiv.trans (AffineIsometryEquiv.trans ePP₂ eP₂G) eGG'

instance AffineIsometryEquiv.instGroupAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Group (P ≃ᵃⁱ[𝕜] P)

The group of affine isometries of a NormedAddTorsor, P.

Equations

• AffineIsometryEquiv.instGroupAffineIsometryEquiv = Group.mk (_ : ∀ (e : P ≃ᵃⁱ[𝕜] P), AffineIsometryEquiv.trans e (AffineIsometryEquiv.symm e) = AffineIsometryEquiv.refl 𝕜 P)

@[simp]

theorem AffineIsometryEquiv.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : ⇑1 = id

@[simp]

theorem AffineIsometryEquiv.coe_mul {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (e : P ≃ᵃⁱ[𝕜] P) (e' : P ≃ᵃⁱ[𝕜] P) : ⇑(e * e') = ⇑e ∘ ⇑e'

@[simp]

theorem AffineIsometryEquiv.coe_inv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (e : P ≃ᵃⁱ[𝕜] P) : ⇑e⁻¹ = ⇑(AffineIsometryEquiv.symm e)

@[simp]

theorem AffineIsometryEquiv.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p : P) (v : V) : e (v +ᵥ p) = (AffineIsometryEquiv.linearIsometryEquiv e) v +ᵥ e p

@[simp]

theorem AffineIsometryEquiv.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) : (AffineIsometryEquiv.linearIsometryEquiv e) (p1 -ᵥ p2) = e p1 -ᵥ e p2

@[simp]

theorem AffineIsometryEquiv.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) (y : P) : dist (e x) (e y) = dist x y

@[simp]

theorem AffineIsometryEquiv.edist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) (y : P) : edist (e x) (e y) = edist x y

theorem AffineIsometryEquiv.bijective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Function.Bijective ⇑e

theorem AffineIsometryEquiv.injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Function.Injective ⇑e

theorem AffineIsometryEquiv.surjective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Function.Surjective ⇑e

theorem AffineIsometryEquiv.map_eq_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} {y : P} : e x = e y ↔ x = y

theorem AffineIsometryEquiv.map_ne {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} {y : P} (h : x ≠ y) : e x ≠ e y

theorem AffineIsometryEquiv.lipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : LipschitzWith 1 ⇑e

theorem AffineIsometryEquiv.antilipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : AntilipschitzWith 1 ⇑e

@[simp]

theorem AffineIsometryEquiv.ediam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (s : Set P) : EMetric.diam (⇑e '' s) = EMetric.diam s

@[simp]

theorem AffineIsometryEquiv.diam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (s : Set P) : Metric.diam (⇑e '' s) = Metric.diam s

@[simp]

theorem AffineIsometryEquiv.comp_continuousOn_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_12} [TopologicalSpace α] {f : α → P} {s : Set α} : ContinuousOn (⇑e ∘ f) s ↔ ContinuousOn f s

@[simp]

theorem AffineIsometryEquiv.comp_continuous_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_12} [TopologicalSpace α] {f : α → P} : Continuous (⇑e ∘ f) ↔ Continuous f

def AffineIsometryEquiv.vaddConst (𝕜 : Type u_1) {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : V ≃ᵃⁱ[𝕜] P

The map v ↦ v +ᵥ p as an affine isometric equivalence between V and P.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometryEquiv.coe_vaddConst {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : ⇑(AffineIsometryEquiv.vaddConst 𝕜 p) = fun (v : V) => v +ᵥ p

@[simp]

theorem AffineIsometryEquiv.coe_vaddConst' {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : ⇑(AffineEquiv.vaddConst 𝕜 p) = fun (v : V) => v +ᵥ p

@[simp]

theorem AffineIsometryEquiv.coe_vaddConst_symm {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : ⇑(AffineIsometryEquiv.symm (AffineIsometryEquiv.vaddConst 𝕜 p)) = fun (p' : P) => p' -ᵥ p

@[simp]

theorem AffineIsometryEquiv.vaddConst_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : (AffineIsometryEquiv.vaddConst 𝕜 p).toAffineEquiv = AffineEquiv.vaddConst 𝕜 p

def AffineIsometryEquiv.constVSub (𝕜 : Type u_1) {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : P ≃ᵃⁱ[𝕜] V

p' ↦ p -ᵥ p' as an affine isometric equivalence.

Equations

• AffineIsometryEquiv.constVSub 𝕜 p = let src := AffineEquiv.constVSub 𝕜 p; { toAffineEquiv := src, norm_map := (_ : ∀ (a : V), ‖-a‖ = ‖a‖) }

@[simp]

theorem AffineIsometryEquiv.coe_constVSub {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : ⇑(AffineIsometryEquiv.constVSub 𝕜 p) = fun (x : P) => p -ᵥ x

@[simp]

theorem AffineIsometryEquiv.symm_constVSub {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : AffineIsometryEquiv.symm (AffineIsometryEquiv.constVSub 𝕜 p) = AffineIsometryEquiv.trans (LinearIsometryEquiv.toAffineIsometryEquiv (LinearIsometryEquiv.neg 𝕜)) (AffineIsometryEquiv.vaddConst 𝕜 p)

def AffineIsometryEquiv.constVAdd (𝕜 : Type u_1) {V : Type u_2} (P : Type u_8) [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) : P ≃ᵃⁱ[𝕜] P

Translation by v (that is, the map p ↦ v +ᵥ p) as an affine isometric automorphism of P.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineIsometryEquiv.coe_constVAdd {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) : ⇑(AffineIsometryEquiv.constVAdd 𝕜 P v) = fun (x : P) => v +ᵥ x

@[simp]

theorem AffineIsometryEquiv.constVAdd_zero {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : AffineIsometryEquiv.constVAdd 𝕜 P 0 = AffineIsometryEquiv.refl 𝕜 P

theorem AffineIsometryEquiv.vadd_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] {f : P → P₂} (hf : Isometry f) {p : P} {g : V → V₂} (hg : ∀ (v : V), g v = f (v +ᵥ p) -ᵥ f p) : Isometry g

The map g from V to V₂ corresponding to a map f from P to P₂, at a base point p, is an isometry if f is one.

def AffineIsometryEquiv.pointReflection (𝕜 : Type u_1) {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : P ≃ᵃⁱ[𝕜] P

Point reflection in x as an affine isometric automorphism.

Equations

• AffineIsometryEquiv.pointReflection 𝕜 x = AffineIsometryEquiv.trans (AffineIsometryEquiv.constVSub 𝕜 x) (AffineIsometryEquiv.vaddConst 𝕜 x)

theorem AffineIsometryEquiv.pointReflection_apply {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (y : P) : (AffineIsometryEquiv.pointReflection 𝕜 x) y = x -ᵥ y +ᵥ x

@[simp]

theorem AffineIsometryEquiv.pointReflection_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : (AffineIsometryEquiv.pointReflection 𝕜 x).toAffineEquiv = AffineEquiv.pointReflection 𝕜 x

@[simp]

theorem AffineIsometryEquiv.pointReflection_self {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : (AffineIsometryEquiv.pointReflection 𝕜 x) x = x

theorem AffineIsometryEquiv.pointReflection_involutive {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : Function.Involutive ⇑(AffineIsometryEquiv.pointReflection 𝕜 x)

@[simp]

theorem AffineIsometryEquiv.pointReflection_symm {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : AffineIsometryEquiv.symm (AffineIsometryEquiv.pointReflection 𝕜 x) = AffineIsometryEquiv.pointReflection 𝕜 x

@[simp]

theorem AffineIsometryEquiv.dist_pointReflection_fixed {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (y : P) : dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) x = dist y x

theorem AffineIsometryEquiv.dist_pointReflection_self' {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (y : P) : dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) y = ‖bit0 (x -ᵥ y)‖

theorem AffineIsometryEquiv.dist_pointReflection_self {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (y : P) : dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) y = ‖2‖ * dist x y

theorem AffineIsometryEquiv.pointReflection_fixed_iff {𝕜 : Type u_1} {V : Type u_2} {P : Type u_8} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [Invertible 2] {x : P} {y : P} : (AffineIsometryEquiv.pointReflection 𝕜 x) y = y ↔ y = x

theorem AffineIsometryEquiv.dist_pointReflection_self_real {V : Type u_2} {P : Type u_8} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x : P) (y : P) : dist ((AffineIsometryEquiv.pointReflection ℝ x) y) y = 2 * dist x y

@[simp]

theorem AffineIsometryEquiv.pointReflection_midpoint_left {V : Type u_2} {P : Type u_8} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x : P) (y : P) : (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) x = y

@[simp]

theorem AffineIsometryEquiv.pointReflection_midpoint_right {V : Type u_2} {P : Type u_8} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x : P) (y : P) : (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) y = x

theorem AffineMap.continuous_linear_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] {f : P →ᵃ[𝕜] P₂} : Continuous ⇑f.linear ↔ Continuous ⇑f

If f is an affine map, then its linear part is continuous iff f is continuous.

theorem AffineMap.isOpenMap_linear_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_4} {P : Type u_8} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P] [PseudoMetricSpace P₂] [NormedAddTorsor V P] [NormedAddTorsor V₂ P₂] {f : P →ᵃ[𝕜] P₂} : IsOpenMap ⇑f.linear ↔ IsOpenMap ⇑f

If f is an affine map, then its linear part is an open map iff f is an open map.

@[simp]

theorem AffineSubspace.linear_equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) : (AffineSubspace.equivMapOfInjective E φ hφ).linear = LinearEquiv.trans (Submodule.equivMapOfInjective φ.linear (_ : Function.Injective ⇑φ.linear) (AffineSubspace.direction E)) (LinearEquiv.ofEq (Submodule.map φ.linear (AffineSubspace.direction E)) (AffineSubspace.direction (AffineSubspace.map φ E)) (_ : Submodule.map φ.linear (AffineSubspace.direction E) = AffineSubspace.direction (AffineSubspace.map φ E)))

@[simp]

theorem AffineSubspace.equivMapOfInjective_toFun {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) (p : ↑↑E) : (AffineSubspace.equivMapOfInjective E φ hφ) p = { val := φ ↑p, property := (_ : φ ↑p ∈ ⇑φ '' ↑E) }

noncomputable def AffineSubspace.equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) : ↥E ≃ᵃ[𝕜] ↥(AffineSubspace.map φ E)

An affine subspace is isomorphic to its image under an injective affine map. This is the affine version of Submodule.equivMapOfInjective.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def AffineSubspace.isometryEquivMap {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (φ : P₁ →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] : ↥E ≃ᵃⁱ[𝕜] ↥(AffineSubspace.map φ.toAffineMap E)

Restricts an affine isometry to an affine isometry equivalence between a nonempty affine subspace E and its image.

This is an isometry version of AffineSubspace.equivMap, having a stronger premise and a stronger conclusion.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineSubspace.isometryEquivMap.apply_symm_apply {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] {E : AffineSubspace 𝕜 P₁} [Nonempty ↥E] {φ : P₁ →ᵃⁱ[𝕜] P₂} (x : ↥(AffineSubspace.map φ.toAffineMap E)) : φ ↑((AffineIsometryEquiv.symm (AffineSubspace.isometryEquivMap φ E)) x) = ↑x

@[simp]

theorem AffineSubspace.isometryEquivMap.coe_apply {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (φ : P₁ →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (g : ↥E) : ↑((AffineSubspace.isometryEquivMap φ E) g) = φ ↑g

@[simp]

theorem AffineSubspace.isometryEquivMap.toAffineMap_eq {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_4} {P₁ : Type u_7} {P₂ : Type u_9} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₁] [NormedSpace 𝕜 V₂] [MetricSpace P₁] [PseudoMetricSpace P₂] [NormedAddTorsor V₁ P₁] [NormedAddTorsor V₂ P₂] (φ : P₁ →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] : ↑(AffineSubspace.isometryEquivMap φ E).toAffineEquiv = ↑(AffineSubspace.equivMapOfInjective E φ.toAffineMap (_ : Function.Injective ⇑φ))