statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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uniform_inducing (hf : isometry f) : uniform_inducing f | hf.antilipschitz.uniform_inducing hf.uniform_continuous | theorem | isometry.uniform_inducing | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"uniform_inducing"
] | An isometry from a metric space is a uniform inducing map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_iff {ι : Type*} {f : α → β} {g : ι → α} {a : filter ι} {b : α}
(hf : isometry f) :
filter.tendsto g a (𝓝 b) ↔ filter.tendsto (f ∘ g) a (𝓝 (f b)) | hf.uniform_inducing.inducing.tendsto_nhds_iff | lemma | isometry.tendsto_nhds_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"filter",
"filter.tendsto",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : isometry f) : continuous f | hf.lipschitz.continuous | lemma | isometry.continuous | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous",
"isometry"
] | An isometry is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_inv {f : α → β} {g : β → α} (h : isometry f) (hg : right_inverse g f) :
isometry g | λ x y, by rw [← h, hg _, hg _] | lemma | isometry.right_inv | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | The right inverse of an isometry is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_emetric_closed_ball (h : isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' (emetric.closed_ball (f x) r) = emetric.closed_ball x r | by { ext y, simp [h.edist_eq] } | lemma | isometry.preimage_emetric_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.closed_ball",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_emetric_ball (h : isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' (emetric.ball (f x) r) = emetric.ball x r | by { ext y, simp [h.edist_eq] } | lemma | isometry.preimage_emetric_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.ball",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image (hf : isometry f) (s : set α) : emetric.diam (f '' s) = emetric.diam s | eq_of_forall_ge_iff $ λ d,
by simp only [emetric.diam_le_iff, ball_image_iff, hf.edist_eq] | lemma | isometry.ediam_image | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.diam",
"emetric.diam_le_iff",
"eq_of_forall_ge_iff",
"isometry"
] | Isometries preserve the diameter in pseudoemetric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ediam_range (hf : isometry f) : emetric.diam (range f) = emetric.diam (univ : set α) | by { rw ← image_univ, exact hf.ediam_image univ } | lemma | isometry.ediam_range | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.diam",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_emetric_ball (hf : isometry f) (x : α) (r : ℝ≥0∞) :
maps_to f (emetric.ball x r) (emetric.ball (f x) r) | (hf.preimage_emetric_ball x r).ge | lemma | isometry.maps_to_emetric_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.ball",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_emetric_closed_ball (hf : isometry f) (x : α) (r : ℝ≥0∞) :
maps_to f (emetric.closed_ball x r) (emetric.closed_ball (f x) r) | (hf.preimage_emetric_closed_ball x r).ge | lemma | isometry.maps_to_emetric_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.closed_ball",
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.isometry_subtype_coe {s : set α} : isometry (coe : s → α) | λx y, rfl | lemma | isometry_subtype_coe | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | The injection from a subtype is an isometry | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_on_iff {γ} [topological_space γ] (hf : isometry f) {g : γ → α} {s : set γ} :
continuous_on (f ∘ g) s ↔ continuous_on g s | hf.uniform_inducing.inducing.continuous_on_iff.symm | lemma | isometry.comp_continuous_on_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous_on",
"isometry",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {γ} [topological_space γ] (hf : isometry f) {g : γ → α} :
continuous (f ∘ g) ↔ continuous g | hf.uniform_inducing.inducing.continuous_iff.symm | lemma | isometry.comp_continuous_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous",
"isometry",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (h : isometry f) : injective f | h.antilipschitz.injective | lemma | isometry.injective | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | An isometry from an emetric space is injective | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding (hf : isometry f) : uniform_embedding f | hf.antilipschitz.uniform_embedding hf.lipschitz.uniform_continuous | theorem | isometry.uniform_embedding | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"uniform_embedding"
] | An isometry from an emetric space is a uniform embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding (hf : isometry f) : embedding f | hf.uniform_embedding.embedding | theorem | isometry.embedding | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"embedding",
"isometry"
] | An isometry from an emetric space is an embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding [complete_space α] [emetric_space γ]
{f : α → γ} (hf : isometry f) : closed_embedding f | hf.antilipschitz.closed_embedding hf.lipschitz.uniform_continuous | theorem | isometry.closed_embedding | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"closed_embedding",
"complete_space",
"emetric_space",
"isometry"
] | An isometry from a complete emetric space is a closed embedding | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_image (hf : isometry f) (s : set α) : metric.diam (f '' s) = metric.diam s | by rw [metric.diam, metric.diam, hf.ediam_image] | lemma | isometry.diam_image | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.diam"
] | An isometry preserves the diameter in pseudometric spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_range (hf : isometry f) : metric.diam (range f) = metric.diam (univ : set α) | by { rw ← image_univ, exact hf.diam_image univ } | lemma | isometry.diam_range | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_set_of_dist (hf : isometry f) (x : α) (p : ℝ → Prop) :
f ⁻¹' {y | p (dist y (f x))} = {y | p (dist y x)} | by { ext y, simp [hf.dist_eq] } | lemma | isometry.preimage_set_of_dist | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (hf : isometry f) (x : α) (r : ℝ) :
f ⁻¹' (metric.closed_ball (f x) r) = metric.closed_ball x r | hf.preimage_set_of_dist x (≤ r) | lemma | isometry.preimage_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ball (hf : isometry f) (x : α) (r : ℝ) :
f ⁻¹' (metric.ball (f x) r) = metric.ball x r | hf.preimage_set_of_dist x (< r) | lemma | isometry.preimage_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_sphere (hf : isometry f) (x : α) (r : ℝ) :
f ⁻¹' (metric.sphere (f x) r) = metric.sphere x r | hf.preimage_set_of_dist x (= r) | lemma | isometry.preimage_sphere | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_ball (hf : isometry f) (x : α) (r : ℝ) :
maps_to f (metric.ball x r) (metric.ball (f x) r) | (hf.preimage_ball x r).ge | lemma | isometry.maps_to_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_sphere (hf : isometry f) (x : α) (r : ℝ) :
maps_to f (metric.sphere x r) (metric.sphere (f x) r) | (hf.preimage_sphere x r).ge | lemma | isometry.maps_to_sphere | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_closed_ball (hf : isometry f) (x : α) (r : ℝ) :
maps_to f (metric.closed_ball x r) (metric.closed_ball (f x) r) | (hf.preimage_closed_ball x r).ge | lemma | isometry.maps_to_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding.to_isometry {α β} [uniform_space α] [metric_space β] {f : α → β}
(h : uniform_embedding f) :
@isometry α β
(@pseudo_metric_space.to_pseudo_emetric_space α
(@metric_space.to_pseudo_metric_space α (h.comap_metric_space f)))
(by apply_instance) f | begin
apply isometry.of_dist_eq,
assume x y,
refl
end | lemma | uniform_embedding.to_isometry | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"metric_space",
"pseudo_metric_space.to_pseudo_emetric_space",
"uniform_embedding",
"uniform_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.to_isometry {α β} [topological_space α] [metric_space β] {f : α → β}
(h : embedding f) :
@isometry α β
(@pseudo_metric_space.to_pseudo_emetric_space α
(@metric_space.to_pseudo_metric_space α (h.comap_metric_space f)))
(by apply_instance) f | begin
apply isometry.of_dist_eq,
assume x y,
refl
end | lemma | embedding.to_isometry | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"embedding",
"isometry",
"metric_space",
"pseudo_metric_space.to_pseudo_emetric_space",
"topological_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_equiv (α β : Type*) [pseudo_emetric_space α] [pseudo_emetric_space β]
extends α ≃ β | (isometry_to_fun : isometry to_fun) | structure | isometry_equiv | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry",
"pseudo_emetric_space"
] | `α` and `β` are isometric if there is an isometric bijection between them. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_eq_to_equiv (h : α ≃ᵢ β) (a : α) : h a = h.to_equiv a | rfl | lemma | isometry_equiv.coe_eq_to_equiv | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_equiv (h : α ≃ᵢ β) : ⇑h.to_equiv = h | rfl | lemma | isometry_equiv.coe_to_equiv | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry (h : α ≃ᵢ β) : isometry h | h.isometry_to_fun | lemma | isometry_equiv.isometry | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective (h : α ≃ᵢ β) : bijective h | h.to_equiv.bijective | lemma | isometry_equiv.bijective | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (h : α ≃ᵢ β) : injective h | h.to_equiv.injective | lemma | isometry_equiv.injective | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (h : α ≃ᵢ β) : surjective h | h.to_equiv.surjective | lemma | isometry_equiv.surjective | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
edist_eq (h : α ≃ᵢ β) (x y : α) : edist (h x) (h y) = edist x y | h.isometry.edist_eq x y | lemma | isometry_equiv.edist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq {α β : Type*} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β)
(x y : α) : dist (h x) (h y) = dist x y | h.isometry.dist_eq x y | lemma | isometry_equiv.dist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nndist_eq {α β : Type*} [pseudo_metric_space α] [pseudo_metric_space β] (h : α ≃ᵢ β)
(x y : α) : nndist (h x) (h y) = nndist x y | h.isometry.nndist_eq x y | lemma | isometry_equiv.nndist_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (h : α ≃ᵢ β) : continuous h | h.isometry.continuous | lemma | isometry_equiv.continuous | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_image (h : α ≃ᵢ β) (s : set α) : emetric.diam (h '' s) = emetric.diam s | h.isometry.ediam_image s | lemma | isometry_equiv.ediam_image | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_inj : ∀ ⦃h₁ h₂ : α ≃ᵢ β⦄, (h₁.to_equiv = h₂.to_equiv) → h₁ = h₂ | | ⟨e₁, h₁⟩ ⟨e₂, h₂⟩ H := by { dsimp at H, subst e₁ } | lemma | isometry_equiv.to_equiv_inj | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext ⦃h₁ h₂ : α ≃ᵢ β⦄ (H : ∀ x, h₁ x = h₂ x) : h₁ = h₂ | to_equiv_inj $ equiv.ext H | lemma | isometry_equiv.ext | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' {α : Type u} [emetric_space α] (f : α → β) (g : β → α) (hfg : ∀ x, f (g x) = x)
(hf : isometry f) : α ≃ᵢ β | { to_fun := f,
inv_fun := g,
left_inv := λ x, hf.injective $ hfg _,
right_inv := hfg,
isometry_to_fun := hf } | def | isometry_equiv.mk' | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric_space",
"inv_fun",
"isometry",
"mk'"
] | Alternative constructor for isometric bijections,
taking as input an isometry, and a right inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (α : Type*) [pseudo_emetric_space α] : α ≃ᵢ α | { isometry_to_fun := isometry_id, .. equiv.refl α } | def | isometry_equiv.refl | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"equiv.refl",
"isometry_id",
"pseudo_emetric_space"
] | The identity isometry of a space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) : α ≃ᵢ γ | { isometry_to_fun := h₂.isometry_to_fun.comp h₁.isometry_to_fun,
.. equiv.trans h₁.to_equiv h₂.to_equiv } | def | isometry_equiv.trans | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"equiv.trans"
] | The composition of two isometric isomorphisms, as an isometric isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : α) : h₁.trans h₂ x = h₂ (h₁ x) | rfl | lemma | isometry_equiv.trans_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : α ≃ᵢ β) : β ≃ᵢ α | { isometry_to_fun := h.isometry.right_inv h.right_inv,
to_equiv := h.to_equiv.symm } | def | isometry_equiv.symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | The inverse of an isometric isomorphism, as an isometric isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : α ≃ᵢ β) : α → β | h | def | isometry_equiv.simps.apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (h : α ≃ᵢ β) : β → α | h.symm
initialize_simps_projections isometry_equiv
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply) | def | isometry_equiv.simps.symm_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"isometry_equiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (h : α ≃ᵢ β) : h.symm.symm = h | to_equiv_inj h.to_equiv.symm_symm | lemma | isometry_equiv.symm_symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply (h : α ≃ᵢ β) (y : β) : h (h.symm y) = y | h.to_equiv.apply_symm_apply y | lemma | isometry_equiv.apply_symm_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (h : α ≃ᵢ β) (x : α) : h.symm (h x) = x | h.to_equiv.symm_apply_apply x | lemma | isometry_equiv.symm_apply_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_eq (h : α ≃ᵢ β) {x : α} {y : β} :
h.symm y = x ↔ y = h x | h.to_equiv.symm_apply_eq | lemma | isometry_equiv.symm_apply_eq | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_apply (h : α ≃ᵢ β) {x : α} {y : β} :
x = h.symm y ↔ h x = y | h.to_equiv.eq_symm_apply | lemma | isometry_equiv.eq_symm_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_self (h : α ≃ᵢ β) : ⇑h.symm ∘ ⇑h = id | funext $ assume a, h.to_equiv.left_inv a | lemma | isometry_equiv.symm_comp_self | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_comp_symm (h : α ≃ᵢ β) : ⇑h ∘ ⇑h.symm = id | funext $ assume a, h.to_equiv.right_inv a | lemma | isometry_equiv.self_comp_symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq_univ (h : α ≃ᵢ β) : range h = univ | h.to_equiv.range_eq_univ | lemma | isometry_equiv.range_eq_univ | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_symm (h : α ≃ᵢ β) : image h.symm = preimage h | image_eq_preimage_of_inverse h.symm.to_equiv.left_inv h.symm.to_equiv.right_inv | lemma | isometry_equiv.image_symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_symm (h : α ≃ᵢ β) : preimage h.symm = image h | (image_eq_preimage_of_inverse h.to_equiv.left_inv h.to_equiv.right_inv).symm | lemma | isometry_equiv.preimage_symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
(h₁.trans h₂).symm x = h₁.symm (h₂.symm x) | rfl | lemma | isometry_equiv.symm_trans_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.ediam_univ | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_preimage (h : α ≃ᵢ β) (s : set β) : emetric.diam (h ⁻¹' s) = emetric.diam s | by rw [← image_symm, ediam_image] | lemma | isometry_equiv.ediam_preimage | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.preimage_emetric_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_emetric_closed_ball (h : α ≃ᵢ β) (x : β) (r : ℝ≥0∞) :
h ⁻¹' (emetric.closed_ball x r) = emetric.closed_ball (h.symm x) r | by rw [← h.isometry.preimage_emetric_closed_ball (h.symm x) r, h.apply_symm_apply] | lemma | isometry_equiv.preimage_emetric_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.image_emetric_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_emetric_closed_ball (h : α ≃ᵢ β) (x : α) (r : ℝ≥0∞) :
h '' (emetric.closed_ball x r) = emetric.closed_ball (h x) r | by rw [← h.preimage_symm, h.symm.preimage_emetric_closed_ball, symm_symm] | lemma | isometry_equiv.image_emetric_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph (h : α ≃ᵢ β) : α ≃ₜ β | { continuous_to_fun := h.continuous,
continuous_inv_fun := h.symm.continuous,
to_equiv := h.to_equiv } | def | isometry_equiv.to_homeomorph | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | The (bundled) homeomorphism associated to an isometric isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_homeomorph (h : α ≃ᵢ β) : ⇑(h.to_homeomorph) = h | rfl | lemma | isometry_equiv.coe_to_homeomorph | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_homeomorph_symm (h : α ≃ᵢ β) : ⇑(h.to_homeomorph.symm) = h.symm | rfl | lemma | isometry_equiv.coe_to_homeomorph_symm | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_on_iff {γ} [topological_space γ] (h : α ≃ᵢ β)
{f : γ → α} {s : set γ} :
continuous_on (h ∘ f) s ↔ continuous_on f s | h.to_homeomorph.comp_continuous_on_iff _ _ | lemma | isometry_equiv.comp_continuous_on_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff {γ} [topological_space γ] (h : α ≃ᵢ β) {f : γ → α} :
continuous (h ∘ f) ↔ continuous f | h.to_homeomorph.comp_continuous_iff | lemma | isometry_equiv.comp_continuous_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff' {γ} [topological_space γ] (h : α ≃ᵢ β) {f : β → γ} :
continuous (f ∘ h) ↔ continuous f | h.to_homeomorph.comp_continuous_iff' | lemma | isometry_equiv.comp_continuous_iff' | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : α ≃ᵢ α) = id | rfl | lemma | isometry_equiv.coe_one | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (e₁ e₂ : α ≃ᵢ α) : ⇑(e₁ * e₂) = e₁ ∘ e₂ | rfl | lemma | isometry_equiv.coe_mul | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) | rfl | lemma | isometry_equiv.mul_apply | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply_self (e : α ≃ᵢ α) (x: α) : e⁻¹ (e x) = x | e.symm_apply_apply x | lemma | isometry_equiv.inv_apply_self | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_inv_self (e : α ≃ᵢ α) (x: α) : e (e⁻¹ x) = x | e.apply_symm_apply x | lemma | isometry_equiv.apply_inv_self | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space [complete_space β] (e : α ≃ᵢ β) : complete_space α | complete_space_of_is_complete_univ $ is_complete_of_complete_image e.isometry.uniform_inducing $
by rwa [set.image_univ, isometry_equiv.range_eq_univ, ← complete_space_iff_is_complete_univ] | lemma | isometry_equiv.complete_space | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"complete_space",
"complete_space_iff_is_complete_univ",
"complete_space_of_is_complete_univ",
"is_complete_of_complete_image",
"isometry_equiv.range_eq_univ",
"set.image_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_iff (e : α ≃ᵢ β) : complete_space α ↔ complete_space β | by { split; introI H, exacts [e.symm.complete_space, e.complete_space] } | lemma | isometry_equiv.complete_space_iff | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"complete_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fun_unique [unique ι] [fintype ι] : (ι → α) ≃ᵢ α | { to_equiv := equiv.fun_unique ι α,
isometry_to_fun := λ x hx, by simp [edist_pi_def, finset.univ_unique, finset.sup_singleton] } | def | isometry_equiv.fun_unique | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"edist_pi_def",
"equiv.fun_unique",
"finset.sup_singleton",
"finset.univ_unique",
"fintype",
"unique"
] | `equiv.fun_unique` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_fin_two (α : fin 2 → Type*) [Π i, pseudo_emetric_space (α i)] :
(Π i, α i) ≃ᵢ α 0 × α 1 | { to_equiv := pi_fin_two_equiv α,
isometry_to_fun := λ x hx, by simp [edist_pi_def, fin.univ_succ, prod.edist_eq] } | def | isometry_equiv.pi_fin_two | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"edist_pi_def",
"fin.univ_succ",
"pi_fin_two_equiv",
"prod.edist_eq",
"pseudo_emetric_space"
] | `pi_fin_two_equiv` as an `isometry_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diam_image (s : set α) : metric.diam (h '' s) = metric.diam s | h.isometry.diam_image s | lemma | isometry_equiv.diam_image | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_preimage (s : set β) : metric.diam (h ⁻¹' s) = metric.diam s | by rw [← image_symm, diam_image] | lemma | isometry_equiv.diam_preimage | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diam_univ : metric.diam (univ : set α) = metric.diam (univ : set β) | congr_arg ennreal.to_real h.ediam_univ | lemma | isometry_equiv.diam_univ | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"ennreal.to_real",
"metric.diam"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.preimage_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.preimage_sphere | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (h : α ≃ᵢ β) (x : β) (r : ℝ) :
h ⁻¹' (metric.closed_ball x r) = metric.closed_ball (h.symm x) r | by rw [← h.isometry.preimage_closed_ball (h.symm x) r, h.apply_symm_apply] | lemma | isometry_equiv.preimage_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.image_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | isometry_equiv.image_sphere | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_closed_ball (h : α ≃ᵢ β) (x : α) (r : ℝ) :
h '' (metric.closed_ball x r) = metric.closed_ball (h x) r | by rw [← h.preimage_symm, h.symm.preimage_closed_ball, symm_symm] | lemma | isometry_equiv.image_closed_ball | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"metric.closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
isometry.isometry_equiv_on_range [emetric_space α] [pseudo_emetric_space β] {f : α → β}
(h : isometry f) : α ≃ᵢ range f | { isometry_to_fun := λx y, by simpa [subtype.edist_eq] using h x y,
to_equiv := equiv.of_injective f h.injective } | def | isometry.isometry_equiv_on_range | topology.metric_space | src/topology/metric_space/isometry.lean | [
"topology.metric_space.antilipschitz"
] | [
"emetric_space",
"equiv.of_injective",
"isometry",
"pseudo_emetric_space",
"subtype.edist_eq"
] | An isometry induces an isometric isomorphism between the source space and the
range of the isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_of_subset : ℓ_infty_ℝ | ⟨ λ n, dist a (x n) - dist (x 0) (x n),
begin
apply mem_ℓp_infty,
use dist a (x 0),
rintros - ⟨n, rfl⟩,
exact abs_dist_sub_le _ _ _
end ⟩ | def | Kuratowski_embedding.embedding_of_subset | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"abs_dist_sub_le",
"mem_ℓp_infty"
] | 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 `Kuratowski_embedding`,
without density assumptions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_of_subset_coe : embedding_of_subset x a n = dist a (x n) - dist (x 0) (x n) | rfl | lemma | Kuratowski_embedding.embedding_of_subset_coe | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_of_subset_dist_le (a b : α) :
dist (embedding_of_subset x a) (embedding_of_subset x b) ≤ dist a b | begin
refine lp.norm_le_of_forall_le dist_nonneg (λn, _),
simp only [lp.coe_fn_sub, pi.sub_apply, embedding_of_subset_coe, real.dist_eq],
convert abs_dist_sub_le a b (x n) using 2,
ring
end | lemma | Kuratowski_embedding.embedding_of_subset_dist_le | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"abs_dist_sub_le",
"dist_nonneg",
"lp.coe_fn_sub",
"lp.norm_le_of_forall_le",
"real.dist_eq",
"ring"
] | The embedding map is always a semi-contraction. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_of_subset_isometry (H : dense_range x) : isometry (embedding_of_subset x) | begin
refine isometry.of_dist_eq (λa b, _),
refine (embedding_of_subset_dist_le x a b).antisymm (le_of_forall_pos_le_add (λe epos, _)),
/- First step: find n with dist a (x n) < e -/
rcases metric.mem_closure_range_iff.1 (H a) (e/2) (half_pos epos) with ⟨n, hn⟩,
/- Second step: use the norm control at index n... | lemma | Kuratowski_embedding.embedding_of_subset_isometry | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"dense_range",
"dist_comm",
"dist_triangle",
"ennreal.top_ne_zero",
"half_pos",
"isometry",
"le_abs_self",
"lp.norm_apply_le_norm",
"mul_le_mul_of_nonneg_left",
"ring"
] | When the reference set is dense, the embedding map is an isometry on its image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_isometric_embedding (α : Type u) [metric_space α] [separable_space α] :
∃(f : α → ℓ_infty_ℝ), isometry f | begin
cases (univ : set α).eq_empty_or_nonempty with h h,
{ use (λ_, 0), assume 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... | theorem | Kuratowski_embedding.exists_isometric_embedding | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"dense",
"isometry",
"metric_space"
] | Every separable metric space embeds isometrically in `ℓ_infty_ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Kuratowski_embedding (α : Type u) [metric_space α] [separable_space α] : α → ℓ_infty_ℝ | classical.some (Kuratowski_embedding.exists_isometric_embedding α) | def | Kuratowski_embedding | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"Kuratowski_embedding.exists_isometric_embedding",
"metric_space"
] | The Kuratowski embedding is an isometric embedding of a separable metric space in `ℓ^∞(ℝ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Kuratowski_embedding.isometry (α : Type u) [metric_space α] [separable_space α] :
isometry (Kuratowski_embedding α) | classical.some_spec (exists_isometric_embedding α) | lemma | Kuratowski_embedding.isometry | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"Kuratowski_embedding",
"isometry",
"metric_space"
] | The Kuratowski embedding is an isometry. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_compacts.Kuratowski_embedding (α : Type u) [metric_space α] [compact_space α]
[nonempty α] :
nonempty_compacts ℓ_infty_ℝ | { carrier := range (Kuratowski_embedding α),
is_compact' := is_compact_range (Kuratowski_embedding.isometry α).continuous,
nonempty' := range_nonempty _ } | def | nonempty_compacts.Kuratowski_embedding | topology.metric_space | src/topology/metric_space/kuratowski.lean | [
"analysis.normed_space.lp_space",
"topology.sets.compacts"
] | [
"Kuratowski_embedding",
"Kuratowski_embedding.isometry",
"compact_space",
"continuous",
"is_compact_range",
"metric_space"
] | Version of the Kuratowski embedding for nonempty compacts | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lipschitz_with [pseudo_emetric_space α] [pseudo_emetric_space β] (K : ℝ≥0) (f : α → β) | ∀x y, edist (f x) (f y) ≤ K * edist x y | def | lipschitz_with | topology.metric_space | src/topology/metric_space/lipschitz.lean | [
"logic.function.iterate",
"data.set.intervals.proj_Icc",
"topology.algebra.order.field",
"topology.metric_space.basic",
"topology.bornology.hom"
] | [
"pseudo_emetric_space"
] | A function `f` is Lipschitz continuous with constant `K ≥ 0` if for all `x, y`
we have `dist (f x) (f y) ≤ K * dist x y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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