fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
protected lipschitz : LipschitzWith 1 f :=
(IsometryClass.isometry f).lipschitz | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | lipschitz | null |
protected antilipschitz : AntilipschitzWith 1 f :=
(IsometryClass.isometry f).antilipschitz | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | antilipschitz | null |
ediam_image (s : Set α) : EMetric.diam (f '' s) = EMetric.diam s :=
(IsometryClass.isometry f).ediam_image s | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_image | null |
ediam_range : EMetric.diam (range f) = EMetric.diam (univ : Set α) :=
(IsometryClass.isometry f).ediam_range | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_range | null |
toContinuousMapClass : ContinuousMapClass F α β where
map_continuous := IsometryClass.continuous | instance | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toContinuousMapClass | null |
toHomeomorphClass [EquivLike F α β] [IsometryClass F α β] : HomeomorphClass F α β where
map_continuous := IsometryClass.continuous
inv_continuous f := ((IsometryClass.isometry f).right_inv (EquivLike.right_inv f)).continuous | instance | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toHomeomorphClass | null |
protected dist_eq (x y : α) : dist (f x) (f y) = dist x y :=
(IsometryClass.isometry f).dist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | dist_eq | null |
protected nndist_eq (x y : α) : nndist (f x) (f y) = nndist x y :=
(IsometryClass.isometry f).nndist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | nndist_eq | null |
diam_image (s : Set α) : Metric.diam (f '' s) = Metric.diam s :=
(IsometryClass.isometry f).diam_image s | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_image | null |
diam_range : Metric.diam (range f) = Metric.diam (univ : Set α) :=
(IsometryClass.isometry f).diam_range | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_range | null |
IsometryEquiv (α : Type u) (β : Type v) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
extends α ≃ β where
isometry_toFun : Isometry toFun
@[inherit_doc]
infixl:25 " ≃ᵢ " => IsometryEquiv | structure | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | IsometryEquiv | `α` and `β` are isometric if there is an isometric bijection between them. |
toEquiv_injective : Injective (toEquiv : (α ≃ᵢ β) → (α ≃ β))
| ⟨_, _⟩, ⟨_, _⟩, rfl => rfl
@[simp] theorem toEquiv_inj {e₁ e₂ : α ≃ᵢ β} : e₁.toEquiv = e₂.toEquiv ↔ e₁ = e₂ :=
toEquiv_injective.eq_iff | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toEquiv_injective | null |
coe_eq_toEquiv (h : α ≃ᵢ β) (a : α) : h a = h.toEquiv a := rfl
@[simp] theorem coe_toEquiv (h : α ≃ᵢ β) : ⇑h.toEquiv = h := rfl
@[simp] theorem coe_mk (e : α ≃ β) (h) : ⇑(mk e h) = e := rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | coe_eq_toEquiv | null |
protected isometry (h : α ≃ᵢ β) : Isometry h :=
h.isometry_toFun | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | isometry | null |
protected bijective (h : α ≃ᵢ β) : Bijective h :=
h.toEquiv.bijective | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | bijective | null |
protected injective (h : α ≃ᵢ β) : Injective h :=
h.toEquiv.injective | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | injective | null |
protected surjective (h : α ≃ᵢ β) : Surjective h :=
h.toEquiv.surjective | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | surjective | null |
protected edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y :=
h.isometry.edist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | edist_eq | null |
protected dist_eq {α β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
(x y : α) : dist (h x) (h y) = dist x y :=
h.isometry.dist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | dist_eq | null |
protected nndist_eq {α β : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β] (h : α ≃ᵢ β)
(x y : α) : nndist (h x) (h y) = nndist x y :=
h.isometry.nndist_eq x y | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | nndist_eq | null |
protected continuous (h : α ≃ᵢ β) : Continuous h :=
h.isometry.continuous
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | continuous | null |
ediam_image (h : α ≃ᵢ β) (s : Set α) : EMetric.diam (h '' s) = EMetric.diam s :=
h.isometry.ediam_image s
@[ext] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_image | null |
ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ :=
DFunLike.ext _ _ H | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ext | null |
mk' {α : Type u} [EMetricSpace α] (f : α → β) (g : β → α) (hfg : ∀ x, f (g x) = x)
(hf : Isometry f) : α ≃ᵢ β where
toFun := f
invFun := g
left_inv _ := hf.injective <| hfg _
right_inv := hfg
isometry_toFun := hf | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | mk' | Alternative constructor for isometric bijections,
taking as input an isometry, and a right inverse. |
protected refl (α : Type*) [PseudoEMetricSpace α] : α ≃ᵢ α :=
{ Equiv.refl α with isometry_toFun := isometry_id } | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | refl | The identity isometry of a space. |
protected trans (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) : α ≃ᵢ γ :=
{ Equiv.trans h₁.toEquiv h₂.toEquiv with
isometry_toFun := h₂.isometry_toFun.comp h₁.isometry_toFun }
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | trans | The composition of two isometric isomorphisms, as an isometric isomorphism. |
trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) :=
rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | trans_apply | null |
protected symm (h : α ≃ᵢ β) : β ≃ᵢ α where
isometry_toFun := h.isometry.right_inv h.right_inv
toEquiv := h.toEquiv.symm | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm | The inverse of an isometric isomorphism, as an isometric isomorphism. |
Simps.apply (h : α ≃ᵢ β) : α → β := h | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.symm_apply (h : α ≃ᵢ β) : β → α :=
h.symm
initialize_simps_projections IsometryEquiv (toFun → apply, invFun → symm_apply)
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Simps.symm_apply | See Note [custom simps projection] |
symm_symm (h : α ≃ᵢ β) : h.symm.symm = h := rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_symm | null |
symm_bijective : Bijective (IsometryEquiv.symm : (α ≃ᵢ β) → β ≃ᵢ α) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_bijective | null |
apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y :=
h.toEquiv.apply_symm_apply y
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | apply_symm_apply | null |
symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x :=
h.toEquiv.symm_apply_apply x | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_apply_apply | null |
symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} : h.symm y = x ↔ y = h x :=
h.toEquiv.symm_apply_eq | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_apply_eq | null |
eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} : x = h.symm y ↔ h x = y :=
h.toEquiv.eq_symm_apply | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | eq_symm_apply | null |
symm_comp_self (h : α ≃ᵢ β) : (h.symm : β → α) ∘ h = id := funext h.left_inv | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_comp_self | null |
self_comp_symm (h : α ≃ᵢ β) : (h : α → β) ∘ h.symm = id := funext h.right_inv | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | self_comp_symm | null |
range_eq_univ (h : α ≃ᵢ β) : range h = univ := by simp | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | range_eq_univ | null |
image_symm (h : α ≃ᵢ β) : image h.symm = preimage h :=
image_eq_preimage_of_inverse h.symm.toEquiv.left_inv h.symm.toEquiv.right_inv | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_symm | null |
preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h :=
(image_eq_preimage_of_inverse h.toEquiv.left_inv h.toEquiv.right_inv).symm
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_symm | null |
symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
(h₁.trans h₂).symm x = h₁.symm (h₂.symm x) :=
rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | symm_trans_apply | null |
ediam_univ (h : α ≃ᵢ β) : EMetric.diam (univ : Set α) = EMetric.diam (univ : Set β) := by
rw [← h.range_eq_univ, h.isometry.ediam_range]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_univ | null |
ediam_preimage (h : α ≃ᵢ β) (s : Set β) : EMetric.diam (h ⁻¹' s) = EMetric.diam s := by
rw [← image_symm, ediam_image]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | ediam_preimage | null |
preimage_emetric_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r := by
rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_emetric_ball | null |
preimage_emetric_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
h ⁻¹' EMetric.closedBall x r = EMetric.closedBall (h.symm x) r := by
rw [← h.isometry.preimage_emetric_closedBall (h.symm x) r, h.apply_symm_apply]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_emetric_closedBall | null |
image_emetric_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
h '' EMetric.ball x r = EMetric.ball (h x) r := by
rw [← h.preimage_symm, h.symm.preimage_emetric_ball, symm_symm]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_emetric_ball | null |
image_emetric_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
h '' EMetric.closedBall x r = EMetric.closedBall (h x) r := by
rw [← h.preimage_symm, h.symm.preimage_emetric_closedBall, symm_symm] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_emetric_closedBall | null |
@[simps toEquiv]
protected toHomeomorph (h : α ≃ᵢ β) : α ≃ₜ β where
continuous_toFun := h.continuous
continuous_invFun := h.symm.continuous
toEquiv := h.toEquiv
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toHomeomorph | The (bundled) homeomorphism associated to an isometric isomorphism. |
coe_toHomeomorph (h : α ≃ᵢ β) : ⇑h.toHomeomorph = h :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | coe_toHomeomorph | null |
coe_toHomeomorph_symm (h : α ≃ᵢ β) : ⇑h.toHomeomorph.symm = h.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | coe_toHomeomorph_symm | null |
comp_continuousOn_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} {s : Set γ} :
ContinuousOn (h ∘ f) s ↔ ContinuousOn f s :=
h.toHomeomorph.comp_continuousOn_iff _ _
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp_continuousOn_iff | null |
comp_continuous_iff {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : γ → α} :
Continuous (h ∘ f) ↔ Continuous f :=
h.toHomeomorph.comp_continuous_iff
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp_continuous_iff | null |
comp_continuous_iff' {γ} [TopologicalSpace γ] (h : α ≃ᵢ β) {f : β → γ} :
Continuous (f ∘ h) ↔ Continuous f :=
h.toHomeomorph.comp_continuous_iff' | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | comp_continuous_iff' | null |
@[simps!]
piCongrLeft' {ι' : Type*} [Fintype ι] [Fintype ι'] {Y : ι → Type*}
[∀ j, PseudoEMetricSpace (Y j)] (e : ι ≃ ι') : (∀ i, Y i) ≃ᵢ ∀ j, Y (e.symm j) where
toEquiv := Equiv.piCongrLeft' _ e
isometry_toFun x1 x2 := by
simp_rw [edist_pi_def, Finset.sup_univ_eq_iSup]
exact (Equiv.iSup_comp (g := fun b ↦ edist (x1 b) (x2 b)) e.symm) | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | piCongrLeft' | The group of isometries. -/
instance : Group (α ≃ᵢ α) where
one := IsometryEquiv.refl _
mul e₁ e₂ := e₂.trans e₁
inv := IsometryEquiv.symm
mul_assoc _ _ _ := rfl
one_mul _ := ext fun _ => rfl
mul_one _ := ext fun _ => rfl
inv_mul_cancel e := ext e.symm_apply_apply
@[simp] theorem coe_one : ⇑(1 : α ≃ᵢ α) = id := rfl
@[simp] theorem coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ := rfl
theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) := rfl
@[simp] theorem inv_apply_self (e : α ≃ᵢ α) (x : α) : e⁻¹ (e x) = x := e.symm_apply_apply x
@[simp] theorem apply_inv_self (e : α ≃ᵢ α) (x : α) : e (e⁻¹ x) = x := e.apply_symm_apply x
theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by
simp only [completeSpace_iff_isComplete_univ, ← e.range_eq_univ, ← image_univ,
isComplete_image_iff e.isometry.isUniformInducing]
protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α :=
e.completeSpace_iff.2 ‹_›
/-- The natural isometry `∀ i, Y i ≃ᵢ ∀ j, Y (e.symm j)` obtained from a bijection `ι ≃ ι'` of
fintypes. `Equiv.piCongrLeft'` as an `IsometryEquiv`. |
@[simps!]
piCongrLeft {ι' : Type*} [Fintype ι] [Fintype ι'] {Y : ι' → Type*}
[∀ j, PseudoEMetricSpace (Y j)] (e : ι ≃ ι') : (∀ i, Y (e i)) ≃ᵢ ∀ j, Y j :=
(piCongrLeft' e.symm).symm | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | piCongrLeft | The natural isometry `∀ i, Y (e i) ≃ᵢ ∀ j, Y j` obtained from a bijection `ι ≃ ι'` of fintypes.
`Equiv.piCongrLeft` as an `IsometryEquiv`. |
@[simps!]
sumArrowIsometryEquivProdArrow [Fintype α] [Fintype β] : (α ⊕ β → γ) ≃ᵢ (α → γ) × (β → γ) where
toEquiv := Equiv.sumArrowEquivProdArrow _ _ _
isometry_toFun _ _ := by simp [Prod.edist_eq, edist_pi_def, Finset.sup_univ_eq_iSup, iSup_sum]
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | sumArrowIsometryEquivProdArrow | The natural isometry `(α ⊕ β → γ) ≃ᵢ (α → γ) × (β → γ)` between the type of maps on a sum of
fintypes `α ⊕ β` and the pairs of functions on the types `α` and `β`.
`Equiv.sumArrowEquivProdArrow` as an `IsometryEquiv`. |
sumArrowIsometryEquivProdArrow_toHomeomorph {α β : Type*} [Fintype α] [Fintype β] :
sumArrowIsometryEquivProdArrow.toHomeomorph
= Homeomorph.sumArrowHomeomorphProdArrow (ι := α) (ι' := β) (X := γ) :=
rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | sumArrowIsometryEquivProdArrow_toHomeomorph | null |
_root_.Fin.edist_append_eq_max_edist (m n : ℕ) {x x2 : Fin m → α} {y y2 : Fin n → α} :
edist (Fin.append x y) (Fin.append x2 y2) = max (edist x x2) (edist y y2) := by
simp [edist_pi_def, Finset.sup_univ_eq_iSup, ← Equiv.iSup_comp (e := finSumFinEquiv),
iSup_sum] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.Fin.edist_append_eq_max_edist | null |
@[simps!]
_root_.Fin.appendIsometry (m n : ℕ) : (Fin m → α) × (Fin n → α) ≃ᵢ (Fin (m + n) → α) where
toEquiv := Fin.appendEquiv _ _
isometry_toFun _ _ := by simp_rw [Fin.appendEquiv, Fin.edist_append_eq_max_edist, Prod.edist_eq]
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.Fin.appendIsometry | The natural `IsometryEquiv` between `(Fin m → α) × (Fin n → α)` and `Fin (m + n) → α`.
`Fin.appendEquiv` as an `IsometryEquiv`. |
_root_.Fin.appendIsometry_toHomeomorph (m n : ℕ) :
(Fin.appendIsometry m n).toHomeomorph = Fin.appendHomeomorph (X := α) m n :=
rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.Fin.appendIsometry_toHomeomorph | null |
@[simps!]
_root_.Fin.appendIsometryOfEq {n m l : ℕ} (hmln : m + l = n) :
(Fin m → α) × (Fin l → α) ≃ᵢ (Fin n → α) :=
(Fin.appendIsometry m l).trans (IsometryEquiv.piCongrLeft (Y := fun _ ↦ α) (finCongr hmln))
variable (ι α) | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | _root_.Fin.appendIsometryOfEq | The natural `IsometryEquiv` `(Fin m → ℝ) × (Fin l → ℝ) ≃ᵢ (Fin n → ℝ)` when `m + l = n`. |
@[simps!]
funUnique [Unique ι] [Fintype ι] : (ι → α) ≃ᵢ α where
toEquiv := Equiv.funUnique ι α
isometry_toFun x hx := by simp [edist_pi_def, Finset.univ_unique, Finset.sup_singleton] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | funUnique | `Equiv.funUnique` as an `IsometryEquiv`. |
@[simps!]
piFinTwo (α : Fin 2 → Type*) [∀ i, PseudoEMetricSpace (α i)] : (∀ i, α i) ≃ᵢ α 0 × α 1 where
toEquiv := piFinTwoEquiv α
isometry_toFun x hx := by simp [edist_pi_def, Fin.univ_succ, Prod.edist_eq] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | piFinTwo | `piFinTwoEquiv` as an `IsometryEquiv`. |
@[simp]
diam_image (s : Set α) : Metric.diam (h '' s) = Metric.diam s :=
h.isometry.diam_image s
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_image | null |
diam_preimage (s : Set β) : Metric.diam (h ⁻¹' s) = Metric.diam s := by
rw [← image_symm, diam_image]
include h in | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_preimage | null |
diam_univ : Metric.diam (univ : Set α) = Metric.diam (univ : Set β) :=
congr_arg ENNReal.toReal h.ediam_univ
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | diam_univ | null |
preimage_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
h ⁻¹' Metric.ball x r = Metric.ball (h.symm x) r := by
rw [← h.isometry.preimage_ball (h.symm x) r, h.apply_symm_apply]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_ball | null |
preimage_sphere (h : α ≃ᵢ β) (x : β) (r : ℝ) :
h ⁻¹' Metric.sphere x r = Metric.sphere (h.symm x) r := by
rw [← h.isometry.preimage_sphere (h.symm x) r, h.apply_symm_apply]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_sphere | null |
preimage_closedBall (h : α ≃ᵢ β) (x : β) (r : ℝ) :
h ⁻¹' Metric.closedBall x r = Metric.closedBall (h.symm x) r := by
rw [← h.isometry.preimage_closedBall (h.symm x) r, h.apply_symm_apply]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | preimage_closedBall | null |
image_ball (h : α ≃ᵢ β) (x : α) (r : ℝ) : h '' Metric.ball x r = Metric.ball (h x) r := by
rw [← h.preimage_symm, h.symm.preimage_ball, symm_symm]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_ball | null |
image_sphere (h : α ≃ᵢ β) (x : α) (r : ℝ) :
h '' Metric.sphere x r = Metric.sphere (h x) r := by
rw [← h.preimage_symm, h.symm.preimage_sphere, symm_symm]
@[simp] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_sphere | null |
image_closedBall (h : α ≃ᵢ β) (x : α) (r : ℝ) :
h '' Metric.closedBall x r = Metric.closedBall (h x) r := by
rw [← h.preimage_symm, h.symm.preimage_closedBall, symm_symm] | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | image_closedBall | null |
@[simps! +simpRhs toEquiv apply]
Isometry.isometryEquivOnRange [EMetricSpace α] [PseudoEMetricSpace β] {f : α → β}
(h : Isometry f) : α ≃ᵢ range f where
isometry_toFun := h
toEquiv := Equiv.ofInjective f h.injective
open NNReal in | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Isometry.isometryEquivOnRange | An isometry induces an isometric isomorphism between the source space and the
range of the isometry. |
Isometry.lipschitzWith_iff {α β γ : Type*} [PseudoEMetricSpace α] [PseudoEMetricSpace β]
[PseudoEMetricSpace γ] {f : α → β} {g : β → γ} (K : ℝ≥0) (h : Isometry g) :
LipschitzWith K (g ∘ f) ↔ LipschitzWith K f := by
simp [LipschitzWith, h.edist_eq] | lemma | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | Isometry.lipschitzWith_iff | Post-composition by an isometry does not change the Lipschitz-property of a function. |
@[coe]
toIsometryEquiv (f : F) : α ≃ᵢ β :=
{ (f : α ≃ β) with
isometry_toFun := IsometryClass.isometry f }
@[simp] | def | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toIsometryEquiv | Turn an element of a type `F` satisfying `EquivLike F α β` and `IsometryClass F α β` into
an actual `IsometryEquiv`. This is declared as the default coercion from `F` to `α ≃ᵢ β`. |
coe_coe (f : F) : ⇑(toIsometryEquiv f) = ⇑f := rfl | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | coe_coe | null |
toIsometryEquiv_injective : Function.Injective ((↑) : F → α ≃ᵢ β) :=
fun _ _ e ↦ DFunLike.ext _ _ fun a ↦ DFunLike.congr_fun e a | theorem | Topology | [
"Mathlib.Data.Fintype.Lattice",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.MetricSpace.Antilipschitz"
] | Mathlib/Topology/MetricSpace/Isometry.lean | toIsometryEquiv_injective | null |
embeddingOfSubset : ℓ^∞(ℕ) :=
⟨fun n => dist a (x n) - dist (x 0) (x n), by
apply memℓp_infty
use dist a (x 0)
rintro - ⟨n, rfl⟩
exact abs_dist_sub_le _ _ _⟩ | def | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | embeddingOfSubset | A metric space can be embedded in `l^∞(ℝ)` via the distances to points in
a fixed countable set, if this set is dense. This map is given in `kuratowskiEmbedding`,
without density assumptions. |
embeddingOfSubset_coe : embeddingOfSubset x a n = dist a (x n) - dist (x 0) (x n) :=
rfl | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | embeddingOfSubset_coe | null |
embeddingOfSubset_dist_le (a b : α) :
dist (embeddingOfSubset x a) (embeddingOfSubset x b) ≤ dist a b := by
refine lp.norm_le_of_forall_le dist_nonneg fun n => ?_
simp only [lp.coeFn_sub, Pi.sub_apply, embeddingOfSubset_coe]
convert abs_dist_sub_le a b (x n) using 2
ring | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | embeddingOfSubset_dist_le | The embedding map is always a semi-contraction. |
embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) := by
refine Isometry.of_dist_eq fun a b => ?_
refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
have C : dist b (x n) - dist a (x n) = embeddingOfSubset x b n - embeddingOfSubset x a n := by
simp only [embeddingOfSubset_coe, sub_sub_sub_cancel_right]
have :=
calc
dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
_ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [dist_comm]; ring
_ ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| := by grw [← le_abs_self]
_ ≤ 2 * (e / 2) + |embeddingOfSubset x b n - embeddingOfSubset x a n| := by
rw [C]
gcongr
_ ≤ 2 * (e / 2) + dist (embeddingOfSubset x b) (embeddingOfSubset x a) := by
gcongr
simp only [dist_eq_norm]
exact lp.norm_apply_le_norm ENNReal.top_ne_zero
(embeddingOfSubset x b - embeddingOfSubset x a) n
_ = dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e := by ring
simpa [dist_comm] using this | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | embeddingOfSubset_isometry | When the reference set is dense, the embedding map is an isometry on its image. |
exists_isometric_embedding (α : Type u) [MetricSpace α] [SeparableSpace α] :
∃ f : α → ℓ^∞(ℕ), Isometry f := by
rcases (univ : Set α).eq_empty_or_nonempty with h | h
· use fun _ => 0; intro x; exact absurd h (Nonempty.ne_empty ⟨x, mem_univ x⟩)
· -- We construct a map x : ℕ → α with dense image
rcases h with ⟨basepoint⟩
haveI : Inhabited α := ⟨basepoint⟩
have : ∃ s : Set α, s.Countable ∧ Dense s := exists_countable_dense α
rcases this with ⟨S, ⟨S_countable, S_dense⟩⟩
rcases Set.countable_iff_exists_subset_range.1 S_countable with ⟨x, x_range⟩
exact ⟨embeddingOfSubset x, embeddingOfSubset_isometry x (S_dense.mono x_range)⟩ | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | exists_isometric_embedding | Every separable metric space embeds isometrically in `ℓ^∞(ℕ)`. |
kuratowskiEmbedding (α : Type u) [MetricSpace α] [SeparableSpace α] : α → ℓ^∞(ℕ) :=
Classical.choose (KuratowskiEmbedding.exists_isometric_embedding α) | def | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | kuratowskiEmbedding | The Kuratowski embedding is an isometric embedding of a separable metric space in `ℓ^∞(ℕ, ℝ)`. |
protected kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [SeparableSpace α] :
Isometry (kuratowskiEmbedding α) :=
Classical.choose_spec (exists_isometric_embedding α) | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | kuratowskiEmbedding.isometry | The Kuratowski embedding is an isometry.
Theorem 2.1 of [Assaf Naor, *Metric Embeddings and Lipschitz Extensions*][Naor-2015]. |
LipschitzOnWith.extend_lp_infty [PseudoMetricSpace α] {s : Set α} {ι : Type*}
{f : α → ℓ^∞(ι)} {K : ℝ≥0} (hfl : LipschitzOnWith K f s) :
∃ g : α → ℓ^∞(ι), LipschitzWith K g ∧ EqOn f g s := by
rw [LipschitzOnWith.coordinate] at hfl
have (i : ι) : ∃ g : α → ℝ, LipschitzWith K g ∧ EqOn (fun x => f x i) g s :=
LipschitzOnWith.extend_real (hfl i) -- use the nonlinear Hahn-Banach theorem here!
choose g hgl hgeq using this
rcases s.eq_empty_or_nonempty with rfl | ⟨a₀, ha₀_in_s⟩
· exact ⟨0, LipschitzWith.const' 0, by simp⟩
· -- Show that the extensions are uniformly bounded
have hf_extb : ∀ a : α, Memℓp (swap g a) ∞ := by
apply LipschitzWith.uniformly_bounded (swap g) hgl a₀
use ‖f a₀‖
rintro - ⟨i, rfl⟩
simp_rw [← hgeq i ha₀_in_s]
exact lp.norm_apply_le_norm top_ne_zero (f a₀) i
let f_ext' : α → ℓ^∞(ι) := fun i ↦ ⟨swap g i, hf_extb i⟩
refine ⟨f_ext', ?_, ?_⟩
· rw [LipschitzWith.coordinate]
exact hgl
· intro a hyp
ext i
exact (hgeq i) hyp | theorem | Topology | [
"Mathlib.Analysis.Normed.Lp.lpSpace",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/MetricSpace/Kuratowski.lean | LipschitzOnWith.extend_lp_infty | Version of the Kuratowski embedding for nonempty compacts -/
nonrec def NonemptyCompacts.kuratowskiEmbedding (α : Type u) [MetricSpace α] [CompactSpace α]
[Nonempty α] : NonemptyCompacts ℓ^∞(ℕ) where
carrier := range (kuratowskiEmbedding α)
isCompact' := isCompact_range (kuratowskiEmbedding.isometry α).continuous
nonempty' := range_nonempty _
/--
A function `f : α → ℓ^∞(ι, ℝ)` which is `K`-Lipschitz on a subset `s` admits a `K`-Lipschitz
extension to the whole space.
Theorem 2.2 of [Assaf Naor, *Metric Embeddings and Lipschitz Extensions*][Naor-2015]
The same result for the case of a finite type `ι` is implemented in
`LipschitzOnWith.extend_pi`. |
lipschitzWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{f : α → β} : LipschitzWith K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y := by
simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast
alias ⟨LipschitzWith.dist_le_mul, LipschitzWith.of_dist_le_mul⟩ := lipschitzWith_iff_dist_le_mul | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | lipschitzWith_iff_dist_le_mul | null |
lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{s : Set α} {f : α → β} :
LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y := by
simp only [LipschitzOnWith, edist_nndist, dist_nndist]
norm_cast
alias ⟨LipschitzOnWith.dist_le_mul, LipschitzOnWith.of_dist_le_mul⟩ :=
lipschitzOnWith_iff_dist_le_mul | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | lipschitzOnWith_iff_dist_le_mul | null |
protected of_dist_le' {K : ℝ} (h : ∀ x y, dist (f x) (f y) ≤ K * dist x y) :
LipschitzWith (Real.toNNReal K) f :=
of_dist_le_mul fun x y =>
le_trans (h x y) <| by gcongr; apply Real.le_coe_toNNReal | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_dist_le' | null |
protected mk_one (h : ∀ x y, dist (f x) (f y) ≤ dist x y) : LipschitzWith 1 f :=
of_dist_le_mul <| by simpa only [NNReal.coe_one, one_mul] using h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | mk_one | null |
protected of_le_add_mul' {f : α → ℝ} (K : ℝ) (h : ∀ x y, f x ≤ f y + K * dist x y) :
LipschitzWith (Real.toNNReal K) f :=
have I : ∀ x y, f x - f y ≤ K * dist x y := fun x y => sub_le_iff_le_add'.2 (h x y)
LipschitzWith.of_dist_le' fun x y => abs_sub_le_iff.2 ⟨I x y, dist_comm y x ▸ I y x⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add_mul' | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
doesn't assume `0≤K`. |
protected of_le_add_mul {f : α → ℝ} (K : ℝ≥0) (h : ∀ x y, f x ≤ f y + K * dist x y) :
LipschitzWith K f := by simpa only [Real.toNNReal_coe] using LipschitzWith.of_le_add_mul' K h | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add_mul | For functions to `ℝ`, it suffices to prove `f x ≤ f y + K * dist x y`; this version
assumes `0≤K`. |
protected of_le_add {f : α → ℝ} (h : ∀ x y, f x ≤ f y + dist x y) : LipschitzWith 1 f :=
LipschitzWith.of_le_add_mul 1 <| by simpa only [NNReal.coe_one, one_mul] | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | of_le_add | null |
protected le_add_mul {f : α → ℝ} {K : ℝ≥0} (h : LipschitzWith K f) (x y) :
f x ≤ f y + K * dist x y :=
sub_le_iff_le_add'.1 <| le_trans (le_abs_self _) <| h.dist_le_mul x y | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | le_add_mul | null |
protected iff_le_add_mul {f : α → ℝ} {K : ℝ≥0} :
LipschitzWith K f ↔ ∀ x y, f x ≤ f y + K * dist x y :=
⟨LipschitzWith.le_add_mul, LipschitzWith.of_le_add_mul K⟩ | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | iff_le_add_mul | null |
nndist_le (hf : LipschitzWith K f) (x y : α) : nndist (f x) (f y) ≤ K * nndist x y :=
hf.dist_le_mul x y | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | nndist_le | null |
dist_le_mul_of_le (hf : LipschitzWith K f) (hr : dist x y ≤ r) : dist (f x) (f y) ≤ K * r :=
(hf.dist_le_mul x y).trans <| by gcongr | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist_le_mul_of_le | null |
mapsTo_closedBall (hf : LipschitzWith K f) (x : α) (r : ℝ) :
MapsTo f (Metric.closedBall x r) (Metric.closedBall (f x) (K * r)) := fun _y hy =>
hf.dist_le_mul_of_le hy | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | mapsTo_closedBall | null |
dist_lt_mul_of_lt (hf : LipschitzWith K f) (hK : K ≠ 0) (hr : dist x y < r) :
dist (f x) (f y) < K * r :=
(hf.dist_le_mul x y).trans_lt <| by gcongr | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | dist_lt_mul_of_lt | null |
mapsTo_ball (hf : LipschitzWith K f) (hK : K ≠ 0) (x : α) (r : ℝ) :
MapsTo f (Metric.ball x r) (Metric.ball (f x) (K * r)) := fun _y hy =>
hf.dist_lt_mul_of_lt hK hy | theorem | Topology | [
"Mathlib.Order.Interval.Set.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.Maps.Proper.Basic",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Lipschitz.lean | mapsTo_ball | null |
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