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Mathlib.Topology.MetricSpace.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 PseudoMetricSpace and we specialize to MetricSpace when needed.

def Isometry {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (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} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} :
    Isometry f ∀ (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} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} :
    Isometry f ∀ (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} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} :
    Isometry f∀ (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} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} :
    (∀ (x y : α), dist (f x) (f y) = dist x y)Isometry f

    A map that preserves distances is an isometry

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

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

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

    Any map on a subsingleton is an isometry

    theorem isometry_id {α : Type u} [PseudoEMetricSpace α] :

    The identity is an isometry

    theorem Isometry.prod_map {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {δ : Type u_2} [PseudoEMetricSpace δ] {f : αβ} {g : γδ} (hf : Isometry f) (hg : Isometry g) :
    theorem isometry_dcomp {ι : Type u_4} [Fintype ι] {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} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] {g : βγ} {f : αβ} (hg : Isometry g) (hf : Isometry f) :

    The composition of isometries is an isometry.

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

    An isometry from a metric space is a uniform continuous map

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

    An isometry is continuous.

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

    The right inverse of an isometry is an isometry.

    theorem Isometry.preimage_emetric_closedBall {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) (x : α) (r : ENNReal) :
    theorem Isometry.preimage_emetric_ball {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) (x : α) (r : ENNReal) :
    theorem Isometry.ediam_image {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (hf : Isometry f) (s : Set α) :

    Isometries preserve the diameter in pseudoemetric spaces.

    theorem Isometry.ediam_range {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (hf : Isometry f) :
    theorem Isometry.mapsTo_emetric_ball {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ENNReal) :
    theorem Isometry.mapsTo_emetric_closedBall {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ENNReal) :
    theorem isometry_subtype_coe {α : Type u} [PseudoEMetricSpace α] {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} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} {γ : Type u_2} [TopologicalSpace γ] (hf : Isometry f) {g : γα} {s : Set γ} :
    theorem Isometry.comp_continuous_iff {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {f : αβ} {γ : Type u_2} [TopologicalSpace γ] (hf : Isometry f) {g : γα} :
    theorem Isometry.injective {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) :

    An isometry from an emetric space is injective

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

    An isometry from an emetric space is a uniform embedding

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

    An isometry from an emetric space is an embedding

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

    An isometry from a complete emetric space is a closed embedding

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

    An isometry preserves the diameter in pseudometric spaces.

    theorem Isometry.diam_range {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) :
    theorem Isometry.preimage_setOf_dist {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (p : Prop) :
    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} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem Isometry.preimage_ball {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem Isometry.preimage_sphere {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem Isometry.mapsTo_ball {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem Isometry.mapsTo_sphere {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem Isometry.mapsTo_closedBall {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] {f : αβ} (hf : Isometry f) (x : α) (r : ) :
    theorem UniformEmbedding.to_isometry {α : Type u_2} {β : Type u_3} [UniformSpace α] [MetricSpace β] {f : αβ} (h : UniformEmbedding 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_2} {β : Type u_3} [TopologicalSpace α] [MetricSpace β] {f : αβ} (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.

    structure IsometryEquiv (α : Type u) (β : Type v) [PseudoEMetricSpace α] [PseudoEMetricSpace β] extends Equiv :
    Type (max u v)

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

    Instances For

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

      Equations
      Instances For
        @[simp]
        theorem IsometryEquiv.toEquiv_inj {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {e₁ : α ≃ᵢ β} {e₂ : α ≃ᵢ β} :
        e₁.toEquiv = e₂.toEquiv e₁ = e₂
        Equations
        • IsometryEquiv.instEquivLikeIsometryEquiv = { 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} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (a : α) :
        h a = h.toEquiv a
        @[simp]
        theorem IsometryEquiv.coe_toEquiv {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
        h.toEquiv = h
        @[simp]
        theorem IsometryEquiv.coe_mk {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (e : α β) (h : Isometry e.toFun) :
        { toEquiv := e, isometry_toFun := h } = e
        theorem IsometryEquiv.isometry {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
        theorem IsometryEquiv.edist_eq {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (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} [PseudoMetricSpace α] [PseudoMetricSpace β] (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} [PseudoMetricSpace α] [PseudoMetricSpace β] (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} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
        @[simp]
        theorem IsometryEquiv.ediam_image {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (s : Set α) :
        theorem IsometryEquiv.ext {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] ⦃h₁ : α ≃ᵢ β ⦃h₂ : α ≃ᵢ β (H : ∀ (x : α), h₁ x = h₂ x) :
        h₁ = h₂
        def IsometryEquiv.mk' {β : Type v} [PseudoEMetricSpace β] {α : Type u} [EMetricSpace α] (f : αβ) (g : βα) (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
        • IsometryEquiv.mk' f g hfg hf = { toEquiv := { toFun := f, invFun := g, left_inv := , right_inv := hfg }, isometry_toFun := hf }
        Instances For

          The identity isometry of a space.

          Equations
          Instances For
            def IsometryEquiv.trans {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) :
            α ≃ᵢ γ

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

            Equations
            • IsometryEquiv.trans h₁ 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} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) :
              (IsometryEquiv.trans h₁ h₂) x = h₂ (h₁ x)
              def IsometryEquiv.symm {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
              β ≃ᵢ α

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

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

                  See Note [custom simps projection]

                  Equations
                  Instances For
                    @[simp]
                    theorem IsometryEquiv.apply_symm_apply {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (y : β) :
                    h ((IsometryEquiv.symm h) y) = y
                    @[simp]
                    theorem IsometryEquiv.symm_apply_apply {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α) :
                    (IsometryEquiv.symm h) (h x) = x
                    theorem IsometryEquiv.symm_apply_eq {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) {x : α} {y : β} :
                    (IsometryEquiv.symm h) y = x y = h x
                    theorem IsometryEquiv.eq_symm_apply {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) {x : α} {y : β} :
                    x = (IsometryEquiv.symm h) y h x = y
                    theorem IsometryEquiv.symm_comp_self {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    (IsometryEquiv.symm h) h = id
                    theorem IsometryEquiv.self_comp_symm {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    h (IsometryEquiv.symm h) = id
                    @[simp]
                    theorem IsometryEquiv.range_eq_univ {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    Set.range h = Set.univ
                    @[simp]
                    theorem IsometryEquiv.symm_trans_apply {α : Type u} {β : Type v} {γ : Type w} [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ] (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
                    theorem IsometryEquiv.ediam_univ {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    EMetric.diam Set.univ = EMetric.diam Set.univ
                    @[simp]
                    theorem IsometryEquiv.ediam_preimage {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (s : Set β) :
                    @[simp]
                    theorem IsometryEquiv.preimage_emetric_ball {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ENNReal) :
                    @[simp]
                    theorem IsometryEquiv.image_emetric_ball {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ENNReal) :
                    h '' EMetric.ball x r = EMetric.ball (h x) r
                    @[simp]
                    theorem IsometryEquiv.image_emetric_closedBall {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ENNReal) :
                    @[simp]
                    theorem IsometryEquiv.toHomeomorph_toEquiv {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    (IsometryEquiv.toHomeomorph h).toEquiv = h.toEquiv
                    def IsometryEquiv.toHomeomorph {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (h : α ≃ᵢ β) :
                    α ≃ₜ β

                    The (bundled) homeomorphism associated to an isometric isomorphism.

                    Equations
                    Instances For
                      @[simp]
                      @[simp]
                      theorem IsometryEquiv.comp_continuousOn_iff {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γα} {s : Set γ} :
                      @[simp]
                      theorem IsometryEquiv.comp_continuous_iff {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γα} :
                      @[simp]
                      theorem IsometryEquiv.comp_continuous_iff' {α : Type u} {β : Type v} [PseudoEMetricSpace α] [PseudoEMetricSpace β] {γ : Type u_2} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : βγ} :

                      The group of isometries.

                      Equations
                      • IsometryEquiv.instGroupIsometryEquiv = Group.mk
                      @[simp]
                      theorem IsometryEquiv.coe_one {α : Type u} [PseudoEMetricSpace α] :
                      1 = id
                      @[simp]
                      theorem IsometryEquiv.coe_mul {α : Type u} [PseudoEMetricSpace α] (e₁ : α ≃ᵢ α) (e₂ : α ≃ᵢ α) :
                      (e₁ * e₂) = e₁ e₂
                      theorem IsometryEquiv.mul_apply {α : Type u} [PseudoEMetricSpace α] (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} [PseudoEMetricSpace α] (e : α ≃ᵢ α) (x : α) :
                      e⁻¹ (e x) = x
                      @[simp]
                      theorem IsometryEquiv.apply_inv_self {α : Type u} [PseudoEMetricSpace α] (e : α ≃ᵢ α) (x : α) :
                      e (e⁻¹ x) = x
                      @[simp]
                      theorem IsometryEquiv.funUnique_symm_apply (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (x : α) (i : ι) :
                      @[simp]
                      theorem IsometryEquiv.funUnique_apply (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (f : (i : ι) → (fun (a : ι) => α) i) :
                      (IsometryEquiv.funUnique ι α) f = f default
                      @[simp]
                      theorem IsometryEquiv.funUnique_invFun (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (x : α) (i : ι) :
                      (IsometryEquiv.funUnique ι α).invFun x i = x
                      @[simp]
                      theorem IsometryEquiv.funUnique_toFun (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] (f : (i : ι) → (fun (a : ι) => α) i) :
                      (IsometryEquiv.funUnique ι α) f = f default
                      def IsometryEquiv.funUnique (ι : Type u_1) (α : Type u) [PseudoEMetricSpace α] [Unique ι] [Fintype ι] :
                      (ια) ≃ᵢ α

                      Equiv.funUnique as an IsometryEquiv.

                      Equations
                      Instances For
                        @[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)) :
                        (IsometryEquiv.piFinTwo α).invFun p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
                        @[simp]
                        theorem IsometryEquiv.piFinTwo_toFun (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i) :
                        (IsometryEquiv.piFinTwo α) f = (f 0, f 1)
                        @[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)) :
                        (IsometryEquiv.symm (IsometryEquiv.piFinTwo α)) p i = Fin.cons p.1 (Fin.cons p.2 finZeroElim) i
                        @[simp]
                        theorem IsometryEquiv.piFinTwo_apply (α : Fin 2Type u_2) [(i : Fin 2) → PseudoEMetricSpace (α i)] (f : (i : Fin 2) → α i) :
                        (IsometryEquiv.piFinTwo α) f = (f 0, f 1)
                        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
                        Instances For
                          @[simp]
                          theorem IsometryEquiv.diam_image {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (s : Set α) :
                          @[simp]
                          theorem IsometryEquiv.diam_preimage {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (s : Set β) :
                          theorem IsometryEquiv.diam_univ {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) :
                          Metric.diam Set.univ = Metric.diam Set.univ
                          @[simp]
                          theorem IsometryEquiv.preimage_ball {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ) :
                          @[simp]
                          theorem IsometryEquiv.preimage_sphere {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : β) (r : ) :
                          @[simp]
                          @[simp]
                          theorem IsometryEquiv.image_ball {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ) :
                          h '' Metric.ball x r = Metric.ball (h x) r
                          @[simp]
                          theorem IsometryEquiv.image_sphere {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ) :
                          h '' Metric.sphere x r = Metric.sphere (h x) r
                          @[simp]
                          theorem IsometryEquiv.image_closedBall {α : Type u} {β : Type v} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β) (x : α) (r : ) :
                          @[simp]
                          theorem Isometry.isometryEquivOnRange_apply {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) (a : α) :
                          (Isometry.isometryEquivOnRange h) a = { val := f a, property := }
                          @[simp]
                          theorem Isometry.isometryEquivOnRange_toEquiv {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) :
                          def Isometry.isometryEquivOnRange {α : Type u} {β : Type v} [EMetricSpace α] [PseudoEMetricSpace β] {f : αβ} (h : Isometry f) :
                          α ≃ᵢ (Set.range f)

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

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