# mathlibdocumentation

topology.metric_space.isometry

# 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 pseudo_metric_space and we specialize to metric_space when needed.

def isometry {α : Type u} {β : Type v} (f : α → β) :
Prop

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

Equations
theorem isometry_iff_nndist_eq {α : Type u} {β : Type v} {f : α → β} :
∀ (x y : α), has_nndist.nndist (f x) (f 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 : α → β} :
∀ (x y : α), has_dist.dist (f x) (f 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 : α → β} :
∀ (x y : α), has_dist.dist (f x) (f y) =

An isometry preserves distances.

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

A map that preserves distances is an isometry

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

An isometry preserves non-negative distances.

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

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

theorem isometry.edist_eq {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x y : α) :
has_edist.edist (f x) (f y) =

An isometry preserves edistances.

theorem isometry.lipschitz {α : Type u} {β : Type v} {f : α → β} (h : isometry f) :
theorem isometry.antilipschitz {α : Type u} {β : Type v} {f : α → β} (h : isometry f) :
theorem isometry_subsingleton {α : Type u} {β : Type v} {f : α → β} [subsingleton α] :

Any map on a subsingleton is an isometry

theorem isometry_id {α : Type u}  :

The identity is an isometry

theorem isometry.comp {α : Type u} {β : Type v} {γ : Type w} {g : β → γ} {f : α → β} (hg : isometry g) (hf : isometry f) :

The composition of isometries is an isometry

@[protected]
theorem isometry.uniform_continuous {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :

An isometry from a metric space is a uniform continuous map

@[protected]
theorem isometry.uniform_inducing {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :

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_1} {f : α → β} {g : ι → α} {a : filter ι} {b : α} (hf : isometry f) :
(nhds b) filter.tendsto (f g) a (nhds (f b))
@[protected]
theorem isometry.continuous {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :

An isometry is continuous.

theorem isometry.right_inv {α : Type u} {β : Type v} {f : α → β} {g : β → α} (h : isometry f) (hg : f) :

The right inverse of an isometry is an isometry.

theorem isometry.preimage_emetric_closed_ball {α : Type u} {β : Type v} {f : α → β} (h : isometry f) (x : α) (r : ennreal) :
theorem isometry.preimage_emetric_ball {α : Type u} {β : Type v} {f : α → β} (h : isometry f) (x : α) (r : ennreal) :
f ⁻¹' emetric.ball (f x) r = r
theorem isometry.ediam_image {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (s : set α) :

Isometries preserve the diameter in pseudoemetric spaces.

theorem isometry.ediam_range {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :
theorem isometry.maps_to_emetric_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ennreal) :
r) (emetric.ball (f x) r)
theorem isometry.maps_to_emetric_closed_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ennreal) :
r) (emetric.closed_ball (f x) r)
theorem isometry_subtype_coe {α : Type u} {s : set α} :

The injection from a subtype is an isometry

theorem isometry.comp_continuous_on_iff {α : Type u} {β : Type v} {f : α → β} {γ : Type u_1} (hf : isometry f) {g : γ → α} {s : set γ} :
theorem isometry.comp_continuous_iff {α : Type u} {β : Type v} {f : α → β} {γ : Type u_1} (hf : isometry f) {g : γ → α} :
@[protected]
theorem isometry.injective {α : Type u} {β : Type v} {f : α → β} (h : isometry f) :

An isometry from an emetric space is injective

@[protected]
theorem isometry.uniform_embedding {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :

An isometry from an emetric space is a uniform embedding

@[protected]
theorem isometry.embedding {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :

An isometry from an emetric space is an embedding

theorem isometry.closed_embedding {α : Type u} {γ : Type w} {f : α → γ} (hf : isometry f) :

An isometry from a complete emetric space is a closed embedding

theorem isometry.diam_image {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (s : set α) :

An isometry preserves the diameter in pseudometric spaces.

theorem isometry.diam_range {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) :
theorem isometry.preimage_set_of_dist {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (p : → Prop) :
f ⁻¹' {y : β | p (f x))} = {y : α | p x)}
theorem isometry.preimage_closed_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
theorem isometry.preimage_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
f ⁻¹' metric.ball (f x) r = r
theorem isometry.preimage_sphere {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
theorem isometry.maps_to_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
r) (metric.ball (f x) r)
theorem isometry.maps_to_sphere {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
r) (metric.sphere (f x) r)
theorem isometry.maps_to_closed_ball {α : Type u} {β : Type v} {f : α → β} (hf : isometry f) (x : α) (r : ) :
r) (metric.closed_ball (f x) r)
theorem uniform_embedding.to_isometry {α : Type u_1} {β : Type u_2} [metric_space β] {f : α → β} (h : uniform_embedding f) :

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_1} {β : Type u_2} [metric_space β] {f : α → β} (h : embedding f) :

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.

@[nolint]
structure isometric (α : Type u_1) (β : Type u_2)  :
Type (max u_1 u_2)

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

Instances for isometric
@[protected, instance]
def isometric.has_coe_to_fun {α : Type u} {β : Type v}  :
has_coe_to_fun ≃ᵢ β) (λ (_x : α ≃ᵢ β), α → β)
Equations
theorem isometric.coe_eq_to_equiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) (a : α) :
h a = (h.to_equiv) a
@[simp]
theorem isometric.coe_to_equiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
theorem isometric.isometry {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
theorem isometric.bijective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
theorem isometric.injective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
theorem isometric.surjective {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
theorem isometric.edist_eq {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x y : α) :
@[protected]
theorem isometric.dist_eq {α : Type u_1} {β : Type u_2} (h : α ≃ᵢ β) (x y : α) :
has_dist.dist (h x) (h y) =
@[protected]
theorem isometric.nndist_eq {α : Type u_1} {β : Type u_2} (h : α ≃ᵢ β) (x y : α) :
@[protected]
theorem isometric.continuous {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.ediam_image {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : set α) :
theorem isometric.to_equiv_inj {α : Type u} {β : Type v} ⦃h₁ h₂ : α ≃ᵢ β⦄ :
h₁.to_equiv = h₂.to_equivh₁ = h₂
@[ext]
theorem isometric.ext {α : Type u} {β : Type v} ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ (x : α), h₁ x = h₂ x) :
h₁ = h₂
def isometric.mk' {β : Type v} {α : Type u} (f : α → β) (g : β → α) (hfg : ∀ (x : β), f (g x) = x) (hf : isometry f) :
α ≃ᵢ β

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

Equations
@[protected]
def isometric.refl (α : Type u_1)  :
α ≃ᵢ α

The identity isometry of a space.

Equations
@[protected]
def isometric.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
@[simp]
theorem isometric.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)
@[protected]
def isometric.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
def isometric.simps.apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
α → β

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

Equations
def isometric.simps.symm_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
β → α
Equations
@[simp]
theorem isometric.symm_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
h.symm.symm = h
@[simp]
theorem isometric.apply_symm_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) (y : β) :
h ((h.symm) y) = y
@[simp]
theorem isometric.symm_apply_apply {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) :
(h.symm) (h x) = x
theorem isometric.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 isometric.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 isometric.symm_comp_self {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem isometric.self_comp_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.range_eq_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem isometric.image_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
theorem isometric.preimage_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
=
@[simp]
theorem isometric.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 isometric.ediam_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.ediam_preimage {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : set β) :
@[simp]
theorem isometric.preimage_emetric_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ennreal) :
@[simp]
theorem isometric.preimage_emetric_closed_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ennreal) :
@[simp]
theorem isometric.image_emetric_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ennreal) :
h '' r = emetric.ball (h x) r
@[simp]
theorem isometric.image_emetric_closed_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ennreal) :
@[simp]
theorem isometric.to_homeomorph_to_equiv {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[protected]
def isometric.to_homeomorph {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
α ≃ₜ β

The (bundled) homeomorphism associated to an isometric isomorphism.

Equations
@[simp]
theorem isometric.coe_to_homeomorph {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.coe_to_homeomorph_symm {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.comp_continuous_on_iff {α : Type u} {β : Type v} {γ : Type u_1} (h : α ≃ᵢ β) {f : γ → α} {s : set γ} :
@[simp]
theorem isometric.comp_continuous_iff {α : Type u} {β : Type v} {γ : Type u_1} (h : α ≃ᵢ β) {f : γ → α} :
@[simp]
theorem isometric.comp_continuous_iff' {α : Type u} {β : Type v} {γ : Type u_1} (h : α ≃ᵢ β) {f : β → γ} :
@[protected, instance]
def isometric.group {α : Type u}  :
group ≃ᵢ α)

The group of isometries.

Equations
@[simp]
theorem isometric.coe_one {α : Type u}  :
@[simp]
theorem isometric.coe_mul {α : Type u} (e₁ e₂ : α ≃ᵢ α) :
(e₁ * e₂) = e₁ e₂
theorem isometric.mul_apply {α : Type u} (e₁ e₂ : α ≃ᵢ α) (x : α) :
(e₁ * e₂) x = e₁ (e₂ x)
@[simp]
theorem isometric.inv_apply_self {α : Type u} (e : α ≃ᵢ α) (x : α) :
e⁻¹ (e x) = x
@[simp]
theorem isometric.apply_inv_self {α : Type u} (e : α ≃ᵢ α) (x : α) :
e (e⁻¹ x) = x
@[protected]
theorem isometric.complete_space {α : Type u} {β : Type v} (e : α ≃ᵢ β) :
theorem isometric.complete_space_iff {α : Type u} {β : Type v} (e : α ≃ᵢ β) :
@[simp]
theorem isometric.diam_image {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : set α) :
@[simp]
theorem isometric.diam_preimage {α : Type u} {β : Type v} (h : α ≃ᵢ β) (s : set β) :
theorem isometric.diam_univ {α : Type u} {β : Type v} (h : α ≃ᵢ β) :
@[simp]
theorem isometric.preimage_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
@[simp]
theorem isometric.preimage_sphere {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
@[simp]
theorem isometric.preimage_closed_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : β) (r : ) :
@[simp]
theorem isometric.image_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
h '' r = metric.ball (h x) r
@[simp]
theorem isometric.image_sphere {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
@[simp]
theorem isometric.image_closed_ball {α : Type u} {β : Type v} (h : α ≃ᵢ β) (x : α) (r : ) :
noncomputable def isometry.isometric_on_range {α : Type u} {β : Type v} {f : α → β} (h : isometry f) :

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

Equations
@[simp]
theorem isometry.isometric_on_range_apply {α : Type u} {β : Type v} {f : α → β} (h : isometry f) (a : α) :
(h.isometric_on_range) a = f a, _⟩
@[simp]
theorem isometry.isometric_on_range_to_equiv {α : Type u} {β : Type v} {f : α → β} (h : isometry f) :