# Isometries #

We define isometries, i.e., maps between emetric spaces that preserve the edistance (on metric spaces, these are exactly the maps that preserve distances), and prove their basic properties. We also introduce isometric bijections.

Since a lot of elementary properties don't require eq_of_dist_eq_zero we start setting up the theory for PseudoMetricSpace and we specialize to MetricSpace when needed.

def Isometry {α : Type u} {β : Type v} (f : αβ) :

An isometry (also known as isometric embedding) is a map preserving the edistance between pseudoemetric spaces, or equivalently the distance between pseudometric space.

Equations
Instances For
theorem isometry_iff_nndist_eq {α : Type u} {β : Type v} {f : αβ} :
∀ (x y : α), nndist (f x) (f y) = nndist x y

On pseudometric spaces, a map is an isometry if and only if it preserves nonnegative distances.

theorem isometry_iff_dist_eq {α : Type u} {β : Type v} {f : αβ} :
∀ (x y : α), dist (f x) (f y) = dist x y

On pseudometric spaces, a map is an isometry if and only if it preserves distances.

theorem Isometry.dist_eq {α : Type u} {β : Type v} {f : αβ} :
∀ (x y : α), dist (f x) (f y) = dist x y

An isometry preserves distances.

theorem Isometry.of_dist_eq {α : Type u} {β : Type v} {f : αβ} :
(∀ (x y : α), dist (f x) (f y) = dist x y)

A map that preserves distances is an isometry

theorem Isometry.nndist_eq {α : Type u} {β : Type v} {f : αβ} :
∀ (x y : α), nndist (f x) (f y) = nndist x y

An isometry preserves non-negative distances.

theorem Isometry.of_nndist_eq {α : Type u} {β : Type v} {f : αβ} :
(∀ (x y : α), nndist (f x) (f y) = nndist x y)

A map that preserves non-negative distances is an isometry.

theorem Isometry.edist_eq {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (y : α) :
edist (f x) (f y) = edist x y

An isometry preserves edistances.

theorem Isometry.lipschitz {α : Type u} {β : Type v} {f : αβ} (h : ) :
theorem Isometry.antilipschitz {α : Type u} {β : Type v} {f : αβ} (h : ) :
theorem isometry_subsingleton {α : Type u} {β : Type v} {f : αβ} [] :

Any map on a subsingleton is an isometry

theorem isometry_id {α : Type u} :

The identity is an isometry

theorem Isometry.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_2} {f : αβ} {g : γδ} (hf : ) (hg : ) :
theorem isometry_dcomp {ι : Type u_4} [] {α : ιType u_2} {β : ιType u_3} [(i : ι) → PseudoEMetricSpace (α i)] [(i : ι) → PseudoEMetricSpace (β i)] (f : (i : ι) → α iβ i) (hf : ∀ (i : ι), Isometry (f i)) :
Isometry fun (g : (i : ι) → α i) (i : ι) => f i (g i)
theorem Isometry.comp {α : Type u} {β : Type v} {γ : Type w} {g : βγ} {f : αβ} (hg : ) (hf : ) :

The composition of isometries is an isometry.

theorem Isometry.uniformContinuous {α : Type u} {β : Type v} {f : αβ} (hf : ) :

An isometry from a metric space is a uniform continuous map

theorem Isometry.uniformInducing {α : Type u} {β : Type v} {f : αβ} (hf : ) :

An isometry from a metric space is a uniform inducing map

theorem Isometry.tendsto_nhds_iff {α : Type u} {β : Type v} {ι : Type u_2} {f : αβ} {g : ια} {a : } {b : α} (hf : ) :
Filter.Tendsto g a (nhds b) Filter.Tendsto (f g) a (nhds (f b))
theorem Isometry.continuous {α : Type u} {β : Type v} {f : αβ} (hf : ) :

An isometry is continuous.

theorem Isometry.right_inv {α : Type u} {β : Type v} {f : αβ} {g : βα} (h : ) (hg : ) :

The right inverse of an isometry is an isometry.

theorem Isometry.preimage_emetric_closedBall {α : Type u} {β : Type v} {f : αβ} (h : ) (x : α) (r : ENNReal) :
theorem Isometry.preimage_emetric_ball {α : Type u} {β : Type v} {f : αβ} (h : ) (x : α) (r : ENNReal) :
f ⁻¹' EMetric.ball (f x) r =
theorem Isometry.ediam_image {α : Type u} {β : Type v} {f : αβ} (hf : ) (s : Set α) :

Isometries preserve the diameter in pseudoemetric spaces.

theorem Isometry.ediam_range {α : Type u} {β : Type v} {f : αβ} (hf : ) :
= EMetric.diam Set.univ
theorem Isometry.mapsTo_emetric_ball {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ENNReal) :
Set.MapsTo f () (EMetric.ball (f x) r)
theorem Isometry.mapsTo_emetric_closedBall {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ENNReal) :
theorem isometry_subtype_coe {α : Type u} {s : Set α} :
Isometry Subtype.val

The injection from a subtype is an isometry

theorem Isometry.comp_continuousOn_iff {α : Type u} {β : Type v} {f : αβ} {γ : Type u_2} [] (hf : ) {g : γα} {s : Set γ} :
theorem Isometry.comp_continuous_iff {α : Type u} {β : Type v} {f : αβ} {γ : Type u_2} [] (hf : ) {g : γα} :
theorem Isometry.injective {α : Type u} {β : Type v} [] {f : αβ} (h : ) :

An isometry from an emetric space is injective

theorem Isometry.uniformEmbedding {α : Type u} {β : Type v} [] {f : αβ} (hf : ) :

An isometry from an emetric space is a uniform embedding

theorem Isometry.embedding {α : Type u} {β : Type v} [] {f : αβ} (hf : ) :

An isometry from an emetric space is an embedding

theorem Isometry.closedEmbedding {α : Type u} {γ : Type w} [] [] [] {f : αγ} (hf : ) :

An isometry from a complete emetric space is a closed embedding

theorem Isometry.diam_image {α : Type u} {β : Type v} {f : αβ} (hf : ) (s : Set α) :

An isometry preserves the diameter in pseudometric spaces.

theorem Isometry.diam_range {α : Type u} {β : Type v} {f : αβ} (hf : ) :
= Metric.diam Set.univ
theorem Isometry.preimage_setOf_dist {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (p : ) :
f ⁻¹' {y : β | p (dist y (f x))} = {y : α | p (dist y x)}
theorem Isometry.preimage_closedBall {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
theorem Isometry.preimage_ball {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
f ⁻¹' Metric.ball (f x) r =
theorem Isometry.preimage_sphere {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
theorem Isometry.mapsTo_ball {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
Set.MapsTo f () (Metric.ball (f x) r)
theorem Isometry.mapsTo_sphere {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
Set.MapsTo f () (Metric.sphere (f x) r)
theorem Isometry.mapsTo_closedBall {α : Type u} {β : Type v} {f : αβ} (hf : ) (x : α) (r : ) :
theorem UniformEmbedding.to_isometry {α : Type u_2} {β : Type u_3} [] [] {f : αβ} (h : ) :

A uniform embedding from a uniform space to a metric space is an isometry with respect to the induced metric space structure on the source space.

theorem Embedding.to_isometry {α : Type u_2} {β : Type u_3} [] [] {f : αβ} (h : ) :

An embedding from a topological space to a metric space is an isometry with respect to the induced metric space structure on the source space.

structure IsometryEquiv (α : Type u) (β : Type v) extends :
Type (max u v)

α and β are isometric if there is an isometric bijection between them.

Instances For
theorem IsometryEquiv.isometry_toFun {α : Type u} {β : Type v} (self : α ≃ᵢ β) :
Isometry self.toFun

α and β are isometric if there is an isometric bijection between them.

Equations
Instances For
theorem IsometryEquiv.toEquiv_injective {α : Type u} {β : Type v} :
Function.Injective IsometryEquiv.toEquiv
@[simp]
theorem IsometryEquiv.toEquiv_inj {α : Type u} {β : Type v} {e₁ : α ≃ᵢ β} {e₂ : α ≃ᵢ β} :
e₁.toEquiv = e₂.toEquiv e₁ = e₂
instance IsometryEquiv.instEquivLike {α : Type u} {β : Type v} :
EquivLike (α ≃ᵢ β) α β
Equations
• IsometryEquiv.instEquivLike = { coe := fun (e : α ≃ᵢ β) => e.toEquiv, inv := fun (e : α ≃ᵢ β) => e.symm, left_inv := , right_inv := , coe_injective' := }
theorem IsometryEquiv.coe_eq_toEquiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) (a : α) :
h a = h.toEquiv a
@[simp]
theorem IsometryEquiv.coe_toEquiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.toEquiv = h
@[simp]
theorem IsometryEquiv.coe_mk {α : Type u} {β : Type v} (e : α β) (h : Isometry e.toFun) :
{ toEquiv := e, isometry_toFun := h } = e
theorem IsometryEquiv.isometry {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem IsometryEquiv.bijective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem IsometryEquiv.injective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem IsometryEquiv.surjective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem IsometryEquiv.edist_eq {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (y : α) :
edist (h x) (h y) = edist x y
theorem IsometryEquiv.dist_eq {α : Type u_2} {β : Type u_3} (h : α ≃ᵢ β) (x : α) (y : α) :
dist (h x) (h y) = dist x y
theorem IsometryEquiv.nndist_eq {α : Type u_2} {β : Type u_3} (h : α ≃ᵢ β) (x : α) (y : α) :
nndist (h x) (h y) = nndist x y
theorem IsometryEquiv.continuous {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem IsometryEquiv.ediam_image {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : Set α) :
EMetric.diam (h '' s) =
theorem IsometryEquiv.ext {α : Type u} {β : Type v} ⦃h₁ : α ≃ᵢ β ⦃h₂ : α ≃ᵢ β (H : ∀ (x : α), h₁ x = h₂ x) :
h₁ = h₂
def IsometryEquiv.mk' {β : Type v} {α : Type u} [] (f : αβ) (g : βα) (hfg : ∀ (x : β), f (g x) = x) (hf : ) :
α ≃ᵢ β

Alternative constructor for isometric bijections, taking as input an isometry, and a right inverse.

Equations
• IsometryEquiv.mk' f g hfg hf = { toFun := f, invFun := g, left_inv := , right_inv := hfg, isometry_toFun := hf }
Instances For
def IsometryEquiv.refl (α : Type u_2) :
α ≃ᵢ α

The identity isometry of a space.

Equations
• = let __src := ; { toEquiv := __src, isometry_toFun := }
Instances For
def IsometryEquiv.trans {α : Type u} {β : Type v} {γ : Type w} (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) :
α ≃ᵢ γ

The composition of two isometric isomorphisms, as an isometric isomorphism.

Equations
• h₁.trans h₂ = let __src := h₁.trans h₂.toEquiv; { toEquiv := __src, isometry_toFun := }
Instances For
@[simp]
theorem IsometryEquiv.trans_apply {α : Type u} {β : Type v} {γ : Type w} (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) :
(h₁.trans h₂) x = h₂ (h₁ x)
def IsometryEquiv.symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
β ≃ᵢ α

The inverse of an isometric isomorphism, as an isometric isomorphism.

Equations
• h.symm = { toEquiv := h.symm, isometry_toFun := }
Instances For
def IsometryEquiv.Simps.apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
αβ

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

Equations
Instances For
def IsometryEquiv.Simps.symm_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
βα

See Note [custom simps projection]

Equations
• = h.symm
Instances For
@[simp]
theorem IsometryEquiv.symm_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.symm.symm = h
theorem IsometryEquiv.symm_bijective {α : Type u} {β : Type v} :
Function.Bijective IsometryEquiv.symm
@[simp]
theorem IsometryEquiv.apply_symm_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) (y : β) :
h (h.symm y) = y
@[simp]
theorem IsometryEquiv.symm_apply_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) :
h.symm (h x) = x
theorem IsometryEquiv.symm_apply_eq {α : Type u} {β : Type v} (h : α ≃ᵢ β) {x : α} {y : β} :
h.symm y = x y = h x
theorem IsometryEquiv.eq_symm_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) {x : α} {y : β} :
x = h.symm y h x = y
theorem IsometryEquiv.symm_comp_self {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.symm h = id
theorem IsometryEquiv.self_comp_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h h.symm = id
@[simp]
theorem IsometryEquiv.range_eq_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
= Set.univ
theorem IsometryEquiv.image_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
Set.image h.symm =
theorem IsometryEquiv.preimage_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
Set.preimage h.symm =
@[simp]
theorem IsometryEquiv.symm_trans_apply {α : Type u} {β : Type v} {γ : Type w} (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
(h₁.trans h₂).symm x = h₁.symm (h₂.symm x)
theorem IsometryEquiv.ediam_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
EMetric.diam Set.univ = EMetric.diam Set.univ
@[simp]
theorem IsometryEquiv.ediam_preimage {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : Set β) :
@[simp]
theorem IsometryEquiv.preimage_emetric_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ENNReal) :
h ⁻¹' = EMetric.ball (h.symm x) r
@[simp]
theorem IsometryEquiv.preimage_emetric_closedBall {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ENNReal) :
h ⁻¹' = EMetric.closedBall (h.symm x) r
@[simp]
theorem IsometryEquiv.image_emetric_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ENNReal) :
h '' = EMetric.ball (h x) r
@[simp]
theorem IsometryEquiv.image_emetric_closedBall {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ENNReal) :
h '' = EMetric.closedBall (h x) r
@[simp]
theorem IsometryEquiv.toHomeomorph_toEquiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.toHomeomorph.toEquiv = h.toEquiv
def IsometryEquiv.toHomeomorph {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
α ≃ₜ β

The (bundled) homeomorphism associated to an isometric isomorphism.

Equations
• h.toHomeomorph = { toEquiv := h.toEquiv, continuous_toFun := , continuous_invFun := }
Instances For
@[simp]
theorem IsometryEquiv.coe_toHomeomorph {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.toHomeomorph = h
@[simp]
theorem IsometryEquiv.coe_toHomeomorph_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.toHomeomorph.symm = h.symm
@[simp]
theorem IsometryEquiv.comp_continuousOn_iff {α : Type u} {β : Type v} {γ : Type u_2} [] (h : α ≃ᵢ β) {f : γα} {s : Set γ} :
ContinuousOn (h f) s
@[simp]
theorem IsometryEquiv.comp_continuous_iff {α : Type u} {β : Type v} {γ : Type u_2} [] (h : α ≃ᵢ β) {f : γα} :
Continuous (h f)
@[simp]
theorem IsometryEquiv.comp_continuous_iff' {α : Type u} {β : Type v} {γ : Type u_2} [] (h : α ≃ᵢ β) {f : βγ} :
Continuous (f h)
instance IsometryEquiv.instGroup {α : Type u} :
Group (α ≃ᵢ α)

The group of isometries.

Equations
• IsometryEquiv.instGroup =
@[simp]
theorem IsometryEquiv.coe_one {α : Type u} :
1 = id
@[simp]
theorem IsometryEquiv.coe_mul {α : Type u} (e₁ : α ≃ᵢ α) (e₂ : α ≃ᵢ α) :
(e₁ * e₂) = e₁ e₂
theorem IsometryEquiv.mul_apply {α : Type u} (e₁ : α ≃ᵢ α) (e₂ : α ≃ᵢ α) (x : α) :
(e₁ * e₂) x = e₁ (e₂ x)
@[simp]
theorem IsometryEquiv.inv_apply_self {α : Type u} (e : α ≃ᵢ α) (x : α) :
e⁻¹ (e x) = x
@[simp]
theorem IsometryEquiv.apply_inv_self {α : Type u} (e : α ≃ᵢ α) (x : α) :
e (e⁻¹ x) = x
theorem IsometryEquiv.completeSpace_iff {α : Type u} {β : Type v} (e : α ≃ᵢ β) :
theorem IsometryEquiv.completeSpace {α : Type u} {β : Type v} [] (e : α ≃ᵢ β) :
@[simp]
theorem IsometryEquiv.funUnique_apply (ι : Type u_1) (α : Type u) [] [] (f : (i : ι) → (fun (a : ι) => α) i) :
() f = f default
@[simp]
theorem IsometryEquiv.funUnique_invFun (ι : Type u_1) (α : Type u) [] [] (x : α) (i : ι) :
().invFun x i = x
@[simp]
theorem IsometryEquiv.funUnique_toFun (ι : Type u_1) (α : Type u) [] [] (f : (i : ι) → (fun (a : ι) => α) i) :
() f = f default
@[simp]
theorem IsometryEquiv.funUnique_symm_apply (ι : Type u_1) (α : Type u) [] [] (x : α) (i : ι) :
().symm x i = x
def IsometryEquiv.funUnique (ι : Type u_1) (α : Type u) [] [] :
(ια) ≃ᵢ α

Equiv.funUnique as an IsometryEquiv.

Equations
• = { toEquiv := , isometry_toFun := }
Instances For
@[simp]
theorem IsometryEquiv.piFinTwo_toFun (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i) :
= (f 0, f 1)
@[simp]
theorem IsometryEquiv.piFinTwo_apply (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i) :
= (f 0, f 1)
@[simp]
theorem IsometryEquiv.piFinTwo_invFun (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (p : α 0 × α 1) (i : Fin (1 + 1)) :
.invFun p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
@[simp]
theorem IsometryEquiv.piFinTwo_symm_apply (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (p : α 0 × α 1) (i : Fin (1 + 1)) :
.symm p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
def IsometryEquiv.piFinTwo (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] :
((i : Fin 2) → α i) ≃ᵢ α 0 × α 1

piFinTwoEquiv as an IsometryEquiv.

Equations
• = { toEquiv := , isometry_toFun := }
Instances For
@[simp]
theorem IsometryEquiv.diam_image {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : Set α) :
Metric.diam (h '' s) =
@[simp]
theorem IsometryEquiv.diam_preimage {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : Set β) :
Metric.diam (h ⁻¹' s) =
theorem IsometryEquiv.diam_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
Metric.diam Set.univ = Metric.diam Set.univ
@[simp]
theorem IsometryEquiv.preimage_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
h ⁻¹' = Metric.ball (h.symm x) r
@[simp]
theorem IsometryEquiv.preimage_sphere {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
h ⁻¹' = Metric.sphere (h.symm x) r
@[simp]
theorem IsometryEquiv.preimage_closedBall {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
h ⁻¹' = Metric.closedBall (h.symm x) r
@[simp]
theorem IsometryEquiv.image_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
h '' = Metric.ball (h x) r
@[simp]
theorem IsometryEquiv.image_sphere {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
h '' = Metric.sphere (h x) r
@[simp]
theorem IsometryEquiv.image_closedBall {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
h '' = Metric.closedBall (h x) r
@[simp]
theorem Isometry.isometryEquivOnRange_apply {α : Type u} {β : Type v} [] {f : αβ} (h : ) (a : α) :
h.isometryEquivOnRange a = f a,
@[simp]
theorem Isometry.isometryEquivOnRange_toEquiv {α : Type u} {β : Type v} [] {f : αβ} (h : ) :
h.isometryEquivOnRange.toEquiv =
def Isometry.isometryEquivOnRange {α : Type u} {β : Type v} [] {f : αβ} (h : ) :
α ≃ᵢ ()

An isometry induces an isometric isomorphism between the source space and the range of the isometry.

Equations
• h.isometryEquivOnRange = { toEquiv := , isometry_toFun := h }
Instances For
theorem Isometry.lipschitzWith_iff {α : Type u_2} {β : Type u_3} {γ : Type u_4} {f : αβ} {g : βγ} (K : NNReal) (h : ) :

Post-composition by an isometry does not change the Lipschitz-property of a function.