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_5} (P : Type u_10) (P₂ : Type u_11) [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] extends AffineMap : Type (max (max (max u_10 u_11) u_2) u_5)

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.toFun (v +ᵥ p) = self.linear v +ᵥ self.toFun p
• norm_map : ∀ (x : V), ‖self.linear x‖ = ‖x‖

theorem AffineIsometry.norm_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (self : P →ᵃⁱ[𝕜] P₂) (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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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

• f.linearIsometry = { toLinearMap := f.linear, norm_map' := ⋯ }

@[simp]

theorem AffineIsometry.linear_eq_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : f.linear = f.linearIsometry.toLinearMap

instance AffineIsometry.instFunLike {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] : FunLike (P →ᵃⁱ[𝕜] P₂) P P₂

Equations

• AffineIsometry.instFunLike = { coe := fun (f : P →ᵃⁱ[𝕜] P₂) => f.toFun, coe_injective' := ⋯ }

@[simp]

theorem AffineIsometry.coe_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] : Function.Injective AffineIsometry.toAffineMap

theorem AffineIsometry.coeFn_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] : Function.Injective DFunLike.coe

theorem AffineIsometry.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : V →ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry as an affine isometry.

Equations

• f.toAffineIsometry = { toAffineMap := f.toAffineMap, norm_map := ⋯ }

@[simp]

theorem LinearIsometry.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : ⇑f.toAffineIsometry = ⇑f

@[simp]

theorem LinearIsometry.toAffineIsometry_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : f.toAffineIsometry.linearIsometry = f

@[simp]

theorem LinearIsometry.toAffineIsometry_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) : f.toAffineIsometry.toAffineMap = f.toAffineMap

@[simp]

theorem AffineIsometry.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p : P) (v : V) : f (v +ᵥ p) = f.linearIsometry v +ᵥ f p

@[simp]

theorem AffineIsometry.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) : f.linearIsometry (p1 -ᵥ p2) = f p1 -ᵥ f p2

@[simp]

theorem AffineIsometry.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : Isometry ⇑f

theorem AffineIsometry.injective {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) : Function.Injective ⇑f₁

@[simp]

theorem AffineIsometry.map_eq_iff {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) {α : Type u_14} [TopologicalSpace α] {g : α → P} : Continuous (⇑f ∘ g) ↔ Continuous g

def AffineIsometry.id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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 := ⋯ }

@[simp]

theorem AffineIsometry.coe_id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : AffineIsometry.id.toAffineMap = AffineMap.id 𝕜 P

instance AffineIsometry.instInhabited {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Inhabited (P →ᵃⁱ[𝕜] P)

Equations

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

def AffineIsometry.comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : P →ᵃⁱ[𝕜] P₃

Composition of affine isometries.

Equations

• g.comp f = { toAffineMap := g.comp f.toAffineMap, norm_map := ⋯ }

@[simp]

theorem AffineIsometry.coe_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem AffineIsometry.id_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : AffineIsometry.id.comp f = f

@[simp]

theorem AffineIsometry.comp_id {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) : f.comp AffineIsometry.id = f

theorem AffineIsometry.comp_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P₄] [NormedAddTorsor V₄ P₄] (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂) : (f.comp g).comp h = f.comp (g.comp h)

instance AffineIsometry.instMonoid {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Monoid (P →ᵃⁱ[𝕜] P)

Equations

• AffineIsometry.instMonoid = Monoid.mk ⋯ ⋯ npowRecAuto ⋯ ⋯

@[simp]

theorem AffineIsometry.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [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

• s.subtypeₐᵢ = { toAffineMap := s.subtype, norm_map := ⋯ }

theorem AffineSubspace.subtypeₐᵢ_linear {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.linear = s.direction.subtype

@[simp]

theorem AffineSubspace.subtypeₐᵢ_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.linearIsometry = s.direction.subtypeₗᵢ

@[simp]

theorem AffineSubspace.coe_subtypeₐᵢ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : ⇑s.subtypeₐᵢ = ⇑s.subtype

@[simp]

theorem AffineSubspace.subtypeₐᵢ_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (s : AffineSubspace 𝕜 P) [Nonempty ↥s] : s.subtypeₐᵢ.toAffineMap = s.subtype

structure AffineIsometryEquiv (𝕜 : Type u_1) {V : Type u_2} {V₂ : Type u_5} (P : Type u_10) (P₂ : Type u_11) [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] extends AffineEquiv , Equiv : Type (max (max (max u_10 u_11) u_2) u_5)

An affine isometric equivalence between two normed vector spaces.

theorem AffineIsometryEquiv.norm_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (self : P ≃ᵃⁱ[𝕜] P₂) (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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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

• e.linearIsometryEquiv = { toLinearEquiv := e.linear, norm_map' := ⋯ }

@[simp]

theorem AffineIsometryEquiv.linear_eq_linear_isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.linear = e.linearIsometryEquiv.toLinearEquiv

instance AffineIsometryEquiv.instEquivLike {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] : EquivLike (P ≃ᵃⁱ[𝕜] P₂) P P₂

Equations

• AffineIsometryEquiv.instEquivLike = { coe := fun (f : P ≃ᵃⁱ[𝕜] P₂) => f.toFun, inv := fun (f : P ≃ᵃⁱ[𝕜] P₂) => f.invFun, left_inv := ⋯, right_inv := ⋯, coe_injective' := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_mk {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] : Function.Injective AffineIsometryEquiv.toAffineEquiv

theorem AffineIsometryEquiv.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P →ᵃⁱ[𝕜] P₂

Reinterpret an AffineIsometryEquiv as an AffineIsometry.

Equations

• e.toAffineIsometry = { toAffineMap := ↑e.toAffineEquiv, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toAffineIsometry = ⇑e

def AffineIsometryEquiv.mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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 = { toAffineEquiv := AffineEquiv.mk' e e'.toLinearEquiv p h, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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).linearIsometryEquiv = e'

def LinearIsometryEquiv.toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : V ≃ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry equiv as an affine isometry equiv.

Equations

• e.toAffineIsometryEquiv = { toAffineEquiv := e.toAffineEquiv, norm_map := ⋯ }

@[simp]

theorem LinearIsometryEquiv.coe_toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : ⇑e.toAffineIsometryEquiv = ⇑e

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_linearIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : e.toAffineIsometryEquiv.linearIsometryEquiv = e

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : e.toAffineIsometryEquiv.toAffineEquiv = e.toAffineEquiv

@[simp]

theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) : e.toAffineIsometryEquiv.toAffineIsometry = e.toLinearIsometry.toAffineIsometry

theorem AffineIsometryEquiv.isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Isometry ⇑e

def AffineIsometryEquiv.toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P ≃ᵢ P₂

Reinterpret an AffineIsometryEquiv as an IsometryEquiv.

Equations

• e.toIsometryEquiv = { toEquiv := e.toEquiv, isometry_toFun := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toIsometryEquiv = ⇑e

theorem AffineIsometryEquiv.range_eq_univ {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P ≃ₜ P₂

Reinterpret an AffineIsometryEquiv as a Homeomorph.

Equations

• e.toHomeomorph = e.toIsometryEquiv.toHomeomorph

@[simp]

theorem AffineIsometryEquiv.coe_toHomeomorph {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : ⇑e.toHomeomorph = ⇑e

theorem AffineIsometryEquiv.continuous {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : Continuous ⇑e

theorem AffineIsometryEquiv.continuousAt {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_10) [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 := ⋯ }

instance AffineIsometryEquiv.instInhabited {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : Inhabited (P ≃ᵃⁱ[𝕜] P)

Equations

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

@[simp]

theorem AffineIsometryEquiv.coe_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : (AffineIsometryEquiv.refl 𝕜 P).toIsometryEquiv = IsometryEquiv.refl P

@[simp]

theorem AffineIsometryEquiv.toHomeomorph_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] : (AffineIsometryEquiv.refl 𝕜 P).toHomeomorph = Homeomorph.refl P

def AffineIsometryEquiv.symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : P₂ ≃ᵃⁱ[𝕜] P

The inverse AffineIsometryEquiv.

Equations

• e.symm = { toAffineEquiv := e.symm, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.apply_symm_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P₂) : e (e.symm x) = x

@[simp]

theorem AffineIsometryEquiv.symm_apply_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) : e.symm (e x) = x

@[simp]

theorem AffineIsometryEquiv.symm_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.symm.symm = e

@[simp]

theorem AffineIsometryEquiv.toAffineEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.symm = e.symm.toAffineEquiv

@[simp]

theorem AffineIsometryEquiv.toIsometryEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.toIsometryEquiv.symm = e.symm.toIsometryEquiv

@[simp]

theorem AffineIsometryEquiv.toHomeomorph_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.toHomeomorph.symm = e.symm.toHomeomorph

def AffineIsometryEquiv.trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (e : P ≃ᵃⁱ[𝕜] P₂) (e' : P₂ ≃ᵃⁱ[𝕜] P₃) : P ≃ᵃⁱ[𝕜] P₃

Composition of AffineIsometryEquivs as an AffineIsometryEquiv.

Equations

• e.trans e' = { toAffineEquiv := e.trans e'.toAffineEquiv, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁

@[simp]

theorem AffineIsometryEquiv.trans_refl {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.trans (AffineIsometryEquiv.refl 𝕜 P₂) = e

@[simp]

theorem AffineIsometryEquiv.refl_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : (AffineIsometryEquiv.refl 𝕜 P).trans e = e

@[simp]

theorem AffineIsometryEquiv.self_trans_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.trans e.symm = AffineIsometryEquiv.refl 𝕜 P

@[simp]

theorem AffineIsometryEquiv.symm_trans_self {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) : e.symm.trans e = AffineIsometryEquiv.refl 𝕜 P₂

@[simp]

theorem AffineIsometryEquiv.coe_symm_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) : ⇑(e₁.trans e₂).symm = ⇑e₁.symm ∘ ⇑e₂.symm

theorem AffineIsometryEquiv.trans_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] [SeminormedAddCommGroup V₄] [NormedSpace 𝕜 V₄] [PseudoMetricSpace P₄] [NormedAddTorsor V₄ P₄] (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄) : ePP₂.trans (eP₂G.trans eGG') = (ePP₂.trans eP₂G).trans eGG'

instance AffineIsometryEquiv.instGroup {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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.instGroup = Group.mk ⋯

@[simp]

theorem AffineIsometryEquiv.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (e : P ≃ᵃⁱ[𝕜] P) : ⇑e⁻¹ = ⇑e.symm

@[simp]

theorem AffineIsometryEquiv.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p : P) (v : V) : e (v +ᵥ p) = e.linearIsometryEquiv v +ᵥ e p

@[simp]

theorem AffineIsometryEquiv.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) : e.linearIsometryEquiv (p1 -ᵥ p2) = e p1 -ᵥ e p2

@[simp]

theorem AffineIsometryEquiv.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_14} [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_14} [TopologicalSpace α] {f : α → P} : Continuous (⇑e ∘ f) ↔ Continuous f

def AffineIsometryEquiv.vaddConst (𝕜 : Type u_1) {V : Type u_2} {P : Type u_10} [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

• AffineIsometryEquiv.vaddConst 𝕜 p = { toAffineEquiv := AffineEquiv.vaddConst 𝕜 p, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_vaddConst {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : ⇑(AffineIsometryEquiv.vaddConst 𝕜 p).symm = fun (p' : P) => p' -ᵥ p

@[simp]

theorem AffineIsometryEquiv.vaddConst_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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 = { toAffineEquiv := AffineEquiv.constVSub 𝕜 p, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_constVSub {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p : P) : (AffineIsometryEquiv.constVSub 𝕜 p).symm = (LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (AffineIsometryEquiv.vaddConst 𝕜 p)

def AffineIsometryEquiv.constVAdd (𝕜 : Type u_1) {V : Type u_2} (P : Type u_10) [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

• AffineIsometryEquiv.constVAdd 𝕜 P v = { toAffineEquiv := AffineEquiv.constVAdd 𝕜 P v, norm_map := ⋯ }

@[simp]

theorem AffineIsometryEquiv.coe_constVAdd {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_10} [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.constVSub 𝕜 x).trans (AffineIsometryEquiv.vaddConst 𝕜 x)

theorem AffineIsometryEquiv.pointReflection_apply {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [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_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) : (AffineIsometryEquiv.pointReflection 𝕜 x).symm = AffineIsometryEquiv.pointReflection 𝕜 x

@[simp]

theorem AffineIsometryEquiv.dist_pointReflection_fixed {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (y : P) : dist ((AffineIsometryEquiv.pointReflection 𝕜 x) y) y = ‖2 • (x -ᵥ y)‖

theorem AffineIsometryEquiv.dist_pointReflection_self {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [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_10} [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_10} [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_10} [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_10} [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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.

noncomputable def AffineSubspace.equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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.

@[simp]

theorem AffineSubspace.linear_equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) : (E.equivMapOfInjective φ hφ).linear = (Submodule.equivMapOfInjective φ.linear ⋯ E.direction).trans (LinearEquiv.ofEq (Submodule.map φ.linear E.direction) (AffineSubspace.map φ E).direction ⋯)

@[simp]

theorem AffineSubspace.equivMapOfInjective_toFun {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁] [NormedSpace 𝕜 V₁] [PseudoMetricSpace P₁] [NormedAddTorsor V₁ P₁] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [Nonempty ↥E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ) (p : ↑↑E) : (E.equivMapOfInjective φ hφ) p = ⟨φ ↑p, ⋯⟩

noncomputable def AffineSubspace.isometryEquivMap {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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

• AffineSubspace.isometryEquivMap φ E = { toAffineEquiv := E.equivMapOfInjective φ.toAffineMap ⋯, norm_map := ⋯ }

@[simp]

theorem AffineSubspace.isometryEquivMap.apply_symm_apply {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] {E : AffineSubspace 𝕜 P₁'} [Nonempty ↥E] {φ : P₁' →ᵃⁱ[𝕜] P₂} (x : ↥(AffineSubspace.map φ.toAffineMap E)) : φ ↑((AffineSubspace.isometryEquivMap φ E).symm x) = ↑x

@[simp]

theorem AffineSubspace.isometryEquivMap.coe_apply {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [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_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty ↥E] : ↑(AffineSubspace.isometryEquivMap φ E).toAffineEquiv = ↑(E.equivMapOfInjective φ.toAffineMap ⋯)