statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
(hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
Equicontinuous (u ∘ (↑) : closure A → X → α) | fun x ↦ (hA x).closure' hu | theorem | Equicontinuous.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Set",
"closure"
] | If a set of functions is equicontinuous, the same is true for its closure in *any*
topology for which evaluation at any point is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `X → α` satisfying the right
continuity conditions. See also `Set.Equicontinuou... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.closure' {A : Set Y} {u : Y → X → α} {S : Set X}
(hA : EquicontinuousOn (u ∘ (↑) : A → X → α) S) (hu : Continuous (S.restrict ∘ u)) :
EquicontinuousOn (u ∘ (↑) : closure A → X → α) S | fun x hx ↦ (hA x hx).closure' hu <| by exact continuous_apply ⟨x, hx⟩ |>.comp hu | theorem | EquicontinuousOn.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"EquicontinuousOn",
"Set",
"closure",
"continuous_apply"
] | If a set of functions is equicontinuous on a set `S`, the same is true for its closure in *any*
topology for which evaluation at any `x ∈ S` is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `X → α` satisfying the right
continuity conditions. See also `Se... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
(closure A).Equicontinuous | fun x ↦ Set.EquicontinuousAt.closure (hA x) | theorem | Set.Equicontinuous.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"Set",
"Set.EquicontinuousAt.closure",
"closure"
] | If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousOn.closure {A : Set <| X → α} {S : Set X}
(hA : A.EquicontinuousOn S) : (closure A).EquicontinuousOn S | fun x hx ↦ Set.EquicontinuousWithinAt.closure (hA x hx) | theorem | Set.EquicontinuousOn.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set",
"Set.EquicontinuousWithinAt.closure",
"closure"
] | If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. This would also be true for the coarser topology of pointwise convergence on `S`,
see `EquicontinuousOn.closure'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuousOn.closure' {A : Set Y} {u : Y → β → α} {S : Set β}
(hA : UniformEquicontinuousOn (u ∘ (↑) : A → β → α) S) (hu : Continuous (S.restrict ∘ u)) :
UniformEquicontinuousOn (u ∘ (↑) : closure A → β → α) S | by
intro U hU
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [hA V hV, mem_inf_of_right (mem_principal_self _)]
rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
rw [SetCoe.forall] at *
change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
refine (closure_minimal hxy <| hVclosed.preimage <| .prodMk ?... | theorem | UniformEquicontinuousOn.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Set",
"SetCoe.forall",
"UniformEquicontinuousOn",
"closure",
"closure_minimal",
"continuous_apply",
"mem_uniformity_isClosed",
"trans"
] | If a set of functions is uniformly equicontinuous on a set `S`, the same is true for its
closure in *any* topology for which evaluation at any `x ∈ S` i continuous. Since this will be
applied to `DFunLike` types, we state it for any topological space with a map to `β → α` satisfying
the right continuity conditions. See... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
(hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
UniformEquicontinuous (u ∘ (↑) : closure A → β → α) | by
rw [← uniformEquicontinuousOn_univ] at hA ⊢
exact hA.closure' (Pi.continuous_restrict _ |>.comp hu) | theorem | UniformEquicontinuous.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Pi.continuous_restrict",
"Set",
"UniformEquicontinuous",
"closure",
"uniformEquicontinuousOn_univ"
] | If a set of functions is uniformly equicontinuous, the same is true for its closure in *any*
topology for which evaluation at any point is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `β → α` satisfying the right
continuity conditions. See also `Set.Uni... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuous.closure {A : Set <| β → α}
(hA : A.UniformEquicontinuous) : (closure A).UniformEquicontinuous | UniformEquicontinuous.closure' (u := id) hA continuous_id | theorem | Set.UniformEquicontinuous.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformEquicontinuous",
"UniformEquicontinuous.closure'",
"closure",
"continuous_id"
] | If a set of functions is uniformly equicontinuous, its closure for the product topology is also
uniformly equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuousOn.closure {A : Set <| β → α} {S : Set β}
(hA : A.UniformEquicontinuousOn S) : (closure A).UniformEquicontinuousOn S | UniformEquicontinuousOn.closure' (u := id) hA (Pi.continuous_restrict _) | theorem | Set.UniformEquicontinuousOn.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Pi.continuous_restrict",
"Set",
"UniformEquicontinuousOn",
"UniformEquicontinuousOn.closure'",
"closure"
] | If a set of functions is uniformly equicontinuous on a set `S`, its closure for the product
topology is also uniformly equicontinuous. This would also be true for the coarser topology of
pointwise convergence on `S`, see `UniformEquicontinuousOn.closure'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt {l : Filter ι} [l.NeBot]
{F : ι → X → α} {f : X → α} {S : Set X} {x₀ : X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : Tendsto (F · x₀) l (𝓝 (f x₀))) (h₃ : EquicontinuousWithinAt F S x₀) :
ContinuousWithinAt f S x₀ | by
intro U hU; rw [mem_map]
rcases UniformSpace.mem_nhds_iff.mp hU with ⟨V, hV, hVU⟩
rcases mem_uniformity_isClosed hV with ⟨W, hW, hWclosed, hWV⟩
filter_upwards [h₃ W hW, eventually_mem_nhdsWithin] with x hx hxS using
hVU <| ball_mono hWV (f x₀) <| hWclosed.mem_of_tendsto (h₂.prodMk_nhds (h₁ x hxS)) <|
... | theorem | Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Filter",
"Set",
"eventually_mem_nhdsWithin",
"mem_uniformity_isClosed"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S ∪ {x₀} : Set X`* along some nontrivial
filter, and if the family `𝓕` is equicontinuous at `x₀ : X` within `S`, then the limit is
continuous at `x₀` within `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
ContinuousAt f x₀ | by
rw [← continuousWithinAt_univ, ← equicontinuousWithinAt_univ, tendsto_pi_nhds] at *
exact continuousWithinAt_of_equicontinuousWithinAt (fun x _ ↦ h₁ x) (h₁ x₀) h₂ | theorem | Filter.Tendsto.continuousAt_of_equicontinuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Filter",
"continuousWithinAt_univ",
"equicontinuousWithinAt_univ",
"tendsto_pi_nhds"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.continuous_of_equicontinuous {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : Equicontinuous F) : Continuous f | continuous_iff_continuousAt.mpr fun x => h₁.continuousAt_of_equicontinuousAt (h₂ x) | theorem | Filter.Tendsto.continuous_of_equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Filter"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous, then the limit is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.continuousOn_of_equicontinuousOn {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {S : Set X} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : EquicontinuousOn F S) : ContinuousOn f S | fun x hx ↦ Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt h₁ (h₁ x hx) (h₂ x hx) | theorem | Filter.Tendsto.continuousOn_of_equicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"EquicontinuousOn",
"Filter",
"Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt",
"Set"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S : Set X`* along some nontrivial
filter, and if the family `𝓕` is equicontinuous, then the limit is continuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn {l : Filter ι} [l.NeBot]
{F : ι → β → α} {f : β → α} {S : Set β} (h₁ : ∀ x ∈ S, Tendsto (F · x) l (𝓝 (f x)))
(h₂ : UniformEquicontinuousOn F S) :
UniformContinuousOn f S | by
intro U hU; rw [mem_map]
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [h₂ V hV, mem_inf_of_right (mem_principal_self _)]
rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
exact hVU <| hVclosed.mem_of_tendsto ((h₁ x hxS).prodMk_nhds (h₁ y hyS)) <|
Eventually.of_forall hxy | theorem | Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Filter",
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"mem_uniformity_isClosed"
] | If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise on `S : Set β`* along some nontrivial
filter, and if the family `𝓕` is uniformly equicontinuous on `S`, then the limit is uniformly
continuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous {l : Filter ι} [l.NeBot]
{F : ι → β → α} {f : β → α} (h₁ : Tendsto F l (𝓝 f)) (h₂ : UniformEquicontinuous F) :
UniformContinuous f | by
rw [← uniformContinuousOn_univ, ← uniformEquicontinuousOn_univ, tendsto_pi_nhds] at *
exact uniformContinuousOn_of_uniformEquicontinuousOn (fun x _ ↦ h₁ x) h₂ | theorem | Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Filter",
"UniformContinuous",
"UniformEquicontinuous",
"tendsto_pi_nhds",
"uniformContinuousOn_univ",
"uniformEquicontinuousOn_univ"
] | If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
{s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
(hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
Tendsto (F · x) l (𝓝 z) | by
rw [(nhds_basis_uniformity (𝓤 α).basis_sets).tendsto_right_iff] at hf ⊢
intro U hU
rcases comp_comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVs, hVU⟩
rw [mem_closure_iff_nhdsWithin_neBot] at hx
have : ∀ᶠ y in 𝓝[s] x, y ∈ s ∧ (∀ i, (F i x, F i y) ∈ V) ∧ (f y, z) ∈ V :=
eventually_mem_nhdsWithin.and <... | theorem | EquicontinuousAt.tendsto_of_mem_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"Filter",
"Set",
"closure",
"comp_comp_symm_mem_uniformity_sets",
"mem_closure_iff_nhdsWithin_neBot",
"nhdsWithin_le_nhds",
"nhds_basis_uniformity"
] | If `F : ι → X → α` is a family of functions equicontinuous at `x`,
it tends to `f y` along a filter `l` for any `y ∈ s`,
the limit function `f` tends to `z` along `𝓝[s] x`, and `x ∈ closure s`,
then `(F · x)` tends to `z` along `l`.
In some sense, this is a converse of `EquicontinuousAt.closure`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Equicontinuous.isClosed_setOf_tendsto {l : Filter ι} {F : ι → X → α} {f : X → α}
(hF : Equicontinuous F) (hf : Continuous f) :
IsClosed {x | Tendsto (F · x) l (𝓝 (f x))} | closure_subset_iff_isClosed.mp fun x hx ↦
(hF x).tendsto_of_mem_closure (hf.continuousAt.mono_left inf_le_left) (fun _ ↦ id) hx | theorem | Equicontinuous.isClosed_setOf_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Filter",
"IsClosed"
] | If `F : ι → X → α` is an equicontinuous family of functions,
`f : X → α` is a continuous function, and `l` is a filter on `ι`,
then `{x | Filter.Tendsto (F · x) l (𝓝 (f x))}` is a closed set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquiv (α : Type*) (β : Type*) [UniformSpace α] [UniformSpace β] extends
α ≃ β where
/-- Uniform continuity of the function -/
uniformContinuous_toFun : UniformContinuous toFun
/-- Uniform continuity of the inverse -/
uniformContinuous_invFun : UniformContinuous invFun | structure | UniformEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | Uniform isomorphism between `α` and `β` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toEquiv_injective : Function.Injective (toEquiv : α ≃ᵤ β → α ≃ β) | | ⟨e, h₁, h₂⟩, ⟨e', h₁', h₂'⟩, h => by simpa only [mk.injEq] | theorem | UniformEquiv.toEquiv_injective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquiv_mk_coe (a : Equiv α β) (b c) : (UniformEquiv.mk a b c : α → β) = a | rfl | theorem | UniformEquiv.uniformEquiv_mk_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm (h : α ≃ᵤ β) : β ≃ᵤ α | where
uniformContinuous_toFun := h.uniformContinuous_invFun
uniformContinuous_invFun := h.uniformContinuous_toFun
toEquiv := h.toEquiv.symm | def | UniformEquiv.symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"symm"
] | Inverse of a uniform isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Simps.apply (h : α ≃ᵤ β) : α → β | h | def | UniformEquiv.Simps.apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | 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/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Simps.symm_apply (h : α ≃ᵤ β) : β → α | h.symm
initialize_simps_projections UniformEquiv (toFun → apply, invFun → symm_apply) | def | UniformEquiv.Simps.symm_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformEquiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv = h | rfl | theorem | UniformEquiv.coe_toEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_symm_toEquiv (h : α ≃ᵤ β) : ⇑h.toEquiv.symm = h.symm | rfl | theorem | UniformEquiv.coe_symm_toEquiv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {h h' : α ≃ᵤ β} (H : ∀ x, h x = h' x) : h = h' | toEquiv_injective <| Equiv.ext H | theorem | UniformEquiv.ext | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.ext"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl (α : Type*) [UniformSpace α] : α ≃ᵤ α | where
uniformContinuous_toFun := uniformContinuous_id
uniformContinuous_invFun := uniformContinuous_id
toEquiv := Equiv.refl α | def | UniformEquiv.refl | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.refl",
"UniformSpace",
"refl",
"uniformContinuous_id"
] | Identity map as a uniform isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trans (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) : α ≃ᵤ γ | where
uniformContinuous_toFun := h₂.uniformContinuous_toFun.comp h₁.uniformContinuous_toFun
uniformContinuous_invFun := h₁.uniformContinuous_invFun.comp h₂.uniformContinuous_invFun
toEquiv := Equiv.trans h₁.toEquiv h₂.toEquiv | def | UniformEquiv.trans | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.trans",
"trans"
] | Composition of two uniform isomorphisms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
trans_apply (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α) : h₁.trans h₂ a = h₂ (h₁ a) | rfl | theorem | UniformEquiv.trans_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquiv_mk_coe_symm (a : Equiv α β) (b c) :
((UniformEquiv.mk a b c).symm : β → α) = a.symm | rfl | theorem | UniformEquiv.uniformEquiv_mk_coe_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
refl_symm : (UniformEquiv.refl α).symm = UniformEquiv.refl α | rfl | theorem | UniformEquiv.refl_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformEquiv.refl",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous (h : α ≃ᵤ β) : UniformContinuous h | h.uniformContinuous_toFun | theorem | UniformEquiv.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous (h : α ≃ᵤ β) : Continuous h | h.uniformContinuous.continuous | theorem | UniformEquiv.continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_symm (h : α ≃ᵤ β) : UniformContinuous h.symm | h.uniformContinuous_invFun | theorem | UniformEquiv.uniformContinuous_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_symm (h : α ≃ᵤ β) : Continuous h.symm | h.uniformContinuous_symm.continuous | theorem | UniformEquiv.continuous_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toHomeomorph (e : α ≃ᵤ β) : α ≃ₜ β | { e.toEquiv with
continuous_toFun := e.continuous
continuous_invFun := e.continuous_symm } | def | UniformEquiv.toHomeomorph | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | A uniform isomorphism as a homeomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toHomeomorph_apply (e : α ≃ᵤ β) : (e.toHomeomorph : α → β) = e | rfl | lemma | UniformEquiv.toHomeomorph_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toHomeomorph_symm_apply (e : α ≃ᵤ β) : (e.toHomeomorph.symm : β → α) = e.symm | rfl | lemma | UniformEquiv.toHomeomorph_symm_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
apply_symm_apply (h : α ≃ᵤ β) (x : β) : h (h.symm x) = x | h.toEquiv.apply_symm_apply x | theorem | UniformEquiv.apply_symm_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symm_apply_apply (h : α ≃ᵤ β) (x : α) : h.symm (h x) = x | h.toEquiv.symm_apply_apply x | theorem | UniformEquiv.symm_apply_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
bijective (h : α ≃ᵤ β) : Function.Bijective h | h.toEquiv.bijective | theorem | UniformEquiv.bijective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Function.Bijective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
injective (h : α ≃ᵤ β) : Function.Injective h | h.toEquiv.injective | theorem | UniformEquiv.injective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
surjective (h : α ≃ᵤ β) : Function.Surjective h | h.toEquiv.surjective | theorem | UniformEquiv.surjective | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
changeInv (f : α ≃ᵤ β) (g : β → α) (hg : Function.RightInverse g f) : α ≃ᵤ β | have : g = f.symm :=
funext fun x => calc
g x = f.symm (f (g x)) := (f.left_inv (g x)).symm
_ = f.symm x := by rw [hg x]
{ toFun := f
invFun := g
left_inv := by convert! f.left_inv
right_inv := by convert! f.right_inv using 1
uniformContinuous_toFun := f.uniformContinuous
uniformCo... | def | UniformEquiv.changeInv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"symm"
] | Change the uniform equiv `f` to make the inverse function definitionally equal to `g`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
symm_comp_self (h : α ≃ᵤ β) : (h.symm : β → α) ∘ h = id | funext h.symm_apply_apply | theorem | UniformEquiv.symm_comp_self | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_comp_symm (h : α ≃ᵤ β) : (h : α → β) ∘ h.symm = id | funext h.apply_symm_apply | theorem | UniformEquiv.self_comp_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
range_coe (h : α ≃ᵤ β) : range h = univ | by simp | theorem | UniformEquiv.range_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_symm (h : α ≃ᵤ β) : image h.symm = preimage h | funext h.symm.toEquiv.image_eq_preimage_symm | theorem | UniformEquiv.image_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_symm (h : α ≃ᵤ β) : preimage h.symm = image h | (funext h.toEquiv.image_eq_preimage_symm).symm | theorem | UniformEquiv.preimage_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
image_preimage (h : α ≃ᵤ β) (s : Set β) : h '' h ⁻¹' s = s | h.toEquiv.image_preimage s | theorem | UniformEquiv.image_preimage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
preimage_image (h : α ≃ᵤ β) (s : Set α) : h ⁻¹' h '' s = s | h.toEquiv.preimage_image s | theorem | UniformEquiv.preimage_image | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformInducing (h : α ≃ᵤ β) : IsUniformInducing h | IsUniformInducing.of_comp h.uniformContinuous h.symm.uniformContinuous <| by
simp only [symm_comp_self, IsUniformInducing.id] | theorem | UniformEquiv.isUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"IsUniformInducing",
"IsUniformInducing.id",
"IsUniformInducing.of_comp"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_eq (h : α ≃ᵤ β) : UniformSpace.comap h ‹_› = ‹_› | h.isUniformInducing.comap_uniformSpace | theorem | UniformEquiv.comap_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding (h : α ≃ᵤ β) : IsUniformEmbedding h | ⟨h.isUniformInducing, h.injective⟩ | lemma | UniformEquiv.isUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"IsUniformEmbedding"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
completeSpace_iff (h : α ≃ᵤ β) : CompleteSpace α ↔ CompleteSpace β | completeSpace_congr h.isUniformEmbedding | theorem | UniformEquiv.completeSpace_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"CompleteSpace",
"completeSpace_congr"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofIsUniformEmbedding (f : α → β) (hf : IsUniformEmbedding f) :
α ≃ᵤ Set.range f | where
uniformContinuous_toFun := hf.isUniformInducing.uniformContinuous.subtype_mk _
uniformContinuous_invFun := by
rw [hf.isUniformInducing.uniformContinuous_iff, Equiv.invFun_as_coe,
Equiv.self_comp_ofInjective_symm]
exact uniformContinuous_subtype_val
toEquiv := Equiv.ofInjective f hf.injective | def | UniformEquiv.ofIsUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.invFun_as_coe",
"Equiv.ofInjective",
"Equiv.self_comp_ofInjective_symm",
"IsUniformEmbedding",
"Set.range",
"uniformContinuous_subtype_val"
] | Uniform equiv given a uniform embedding. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
setCongr {s t : Set α} (h : s = t) : s ≃ᵤ t | where
uniformContinuous_toFun := uniformContinuous_subtype_val.subtype_mk _
uniformContinuous_invFun := uniformContinuous_subtype_val.subtype_mk _
toEquiv := Equiv.setCongr h | def | UniformEquiv.setCongr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.setCongr",
"Set"
] | If two sets are equal, then they are uniformly equivalent. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodCongr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : α × γ ≃ᵤ β × δ | where
uniformContinuous_toFun :=
(h₁.uniformContinuous.comp uniformContinuous_fst).prodMk
(h₂.uniformContinuous.comp uniformContinuous_snd)
uniformContinuous_invFun :=
(h₁.symm.uniformContinuous.comp uniformContinuous_fst).prodMk
(h₂.symm.uniformContinuous.comp uniformContinuous_snd)
toEquiv :... | def | UniformEquiv.prodCongr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"uniformContinuous_fst",
"uniformContinuous_snd"
] | Product of two uniform isomorphisms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodCongr_symm (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
(h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm | rfl | theorem | UniformEquiv.prodCongr_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_prodCongr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : ⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ | rfl | theorem | UniformEquiv.coe_prodCongr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prodComm : α × β ≃ᵤ β × α | where
uniformContinuous_toFun := uniformContinuous_snd.prodMk uniformContinuous_fst
uniformContinuous_invFun := uniformContinuous_snd.prodMk uniformContinuous_fst
toEquiv := Equiv.prodComm α β | def | UniformEquiv.prodComm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.prodComm",
"uniformContinuous_fst"
] | `α × β` is uniformly isomorphic to `β × α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodComm_symm : (prodComm α β).symm = prodComm β α | rfl | theorem | UniformEquiv.prodComm_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_prodComm : ⇑(prodComm α β) = Prod.swap | rfl | theorem | UniformEquiv.coe_prodComm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prodAssoc : (α × β) × γ ≃ᵤ α × β × γ | where
uniformContinuous_toFun :=
(uniformContinuous_fst.comp uniformContinuous_fst).prodMk
((uniformContinuous_snd.comp uniformContinuous_fst).prodMk uniformContinuous_snd)
uniformContinuous_invFun :=
(uniformContinuous_fst.prodMk (uniformContinuous_fst.comp
uniformContinuous_snd)).prodMk (unifo... | def | UniformEquiv.prodAssoc | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.prodAssoc",
"uniformContinuous_fst",
"uniformContinuous_snd"
] | `(α × β) × γ` is uniformly isomorphic to `α × (β × γ)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodPUnit : α × PUnit ≃ᵤ α | where
toEquiv := Equiv.prodPUnit α
uniformContinuous_toFun := uniformContinuous_fst
uniformContinuous_invFun := uniformContinuous_id.prodMk uniformContinuous_const
@[deprecated (since := "2026-02-08")] alias prodPunit := prodPUnit | def | UniformEquiv.prodPUnit | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.prodPUnit",
"uniformContinuous_const",
"uniformContinuous_fst"
] | `α × {*}` is uniformly isomorphic to `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
punitProd : PUnit × α ≃ᵤ α | (prodComm _ _).trans (prodPUnit _) | def | UniformEquiv.punitProd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"trans"
] | `{*} × α` is uniformly isomorphic to `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_punitProd : ⇑(punitProd α) = Prod.snd | rfl | theorem | UniformEquiv.coe_punitProd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongrLeft {ι ι' : Type*} {β : ι' → Type*} [∀ j, UniformSpace (β j)]
(e : ι ≃ ι') : (∀ i, β (e i)) ≃ᵤ ∀ j, β j | where
uniformContinuous_toFun := uniformContinuous_pi.mpr <| e.forall_congr_right.mp fun i ↦ by
simpa only [Equiv.toFun_as_coe, Equiv.piCongrLeft_apply_apply] using
Pi.uniformContinuous_proj _ i
uniformContinuous_invFun := Pi.uniformContinuous_precomp' _ e
toEquiv := Equiv.piCongrLeft _ e | def | UniformEquiv.piCongrLeft | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.piCongrLeft",
"Equiv.piCongrLeft_apply_apply",
"Equiv.toFun_as_coe",
"Pi.uniformContinuous_precomp'",
"Pi.uniformContinuous_proj",
"UniformSpace"
] | `Equiv.piCongrLeft` as a uniform isomorphism: this is the natural isomorphism
`Π i, β (e i) ≃ᵤ Π j, β j` obtained from a bijection `ι ≃ ι'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
piCongrLeft_refl {ι : Type*} {X : ι → Type*} [∀ i, UniformSpace (X i)] :
piCongrLeft (.refl ι) = .refl (∀ i, X i) | rfl | lemma | UniformEquiv.piCongrLeft_refl | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongrLeft_symm_apply {ι ι' : Type*} {X : ι' → Type*} [∀ j, UniformSpace (X j)]
(e : ι ≃ ι') : ⇑(piCongrLeft (β := X) e).symm = (· <| e ·) | rfl | lemma | UniformEquiv.piCongrLeft_symm_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongrLeft_apply_apply {ι ι' : Type*} {X : ι' → Type*} [∀ j, UniformSpace (X j)]
(e : ι ≃ ι') (x : ∀ i, X (e i)) i : piCongrLeft e x (e i) = x i | Equiv.piCongrLeft_apply_apply .. | lemma | UniformEquiv.piCongrLeft_apply_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.piCongrLeft_apply_apply",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongrRight {ι : Type*} {β₁ β₂ : ι → Type*} [∀ i, UniformSpace (β₁ i)]
[∀ i, UniformSpace (β₂ i)] (F : ∀ i, β₁ i ≃ᵤ β₂ i) : (∀ i, β₁ i) ≃ᵤ ∀ i, β₂ i | where
uniformContinuous_toFun := Pi.uniformContinuous_postcomp' _ fun i ↦ (F i).uniformContinuous
uniformContinuous_invFun := Pi.uniformContinuous_postcomp' _ fun i ↦ (F i).symm.uniformContinuous
toEquiv := Equiv.piCongrRight fun i => (F i).toEquiv | def | UniformEquiv.piCongrRight | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.piCongrRight",
"Pi.uniformContinuous_postcomp'",
"UniformSpace"
] | `Equiv.piCongrRight` as a uniform isomorphism: this is the natural isomorphism
`Π i, β₁ i ≃ᵤ Π j, β₂ i` obtained from uniform isomorphisms `β₁ i ≃ᵤ β₂ i` for each `i`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
piCongrRight_symm {ι : Type*} {β₁ β₂ : ι → Type*} [∀ i, UniformSpace (β₁ i)]
[∀ i, UniformSpace (β₂ i)] (F : ∀ i, β₁ i ≃ᵤ β₂ i) :
(piCongrRight F).symm = piCongrRight fun i => (F i).symm | rfl | theorem | UniformEquiv.piCongrRight_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongrRight_refl {ι : Type*} {X : ι → Type*} [∀ i, UniformSpace (X i)] :
piCongrRight (fun i ↦ .refl (X i)) = .refl (∀ i, X i) | rfl | theorem | UniformEquiv.piCongrRight_refl | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace",
"refl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
piCongr {ι₁ ι₂ : Type*} {β₁ : ι₁ → Type*} {β₂ : ι₂ → Type*}
[∀ i₁, UniformSpace (β₁ i₁)] [∀ i₂, UniformSpace (β₂ i₂)]
(e : ι₁ ≃ ι₂) (F : ∀ i₁, β₁ i₁ ≃ᵤ β₂ (e i₁)) : (∀ i₁, β₁ i₁) ≃ᵤ ∀ i₂, β₂ i₂ | (UniformEquiv.piCongrRight F).trans (UniformEquiv.piCongrLeft e) | def | UniformEquiv.piCongr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformEquiv.piCongrLeft",
"UniformEquiv.piCongrRight",
"UniformSpace",
"trans"
] | `Equiv.piCongr` as a uniform isomorphism: this is the natural isomorphism
`Π i₁, β₁ i ≃ᵤ Π i₂, β₂ i₂` obtained from a bijection `ι₁ ≃ ι₂` and isomorphisms
`β₁ i₁ ≃ᵤ β₂ (e i₁)` for each `i₁ : ι₁`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ulift : ULift.{v, u} α ≃ᵤ α | { Equiv.ulift with
uniformContinuous_toFun := uniformContinuous_comap
uniformContinuous_invFun := by
have hf : IsUniformInducing (@Equiv.ulift.{v, u} α).toFun := ⟨rfl⟩
simp_rw [hf.uniformContinuous_iff]
exact uniformContinuous_id } | def | UniformEquiv.ulift | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.ulift",
"IsUniformInducing",
"uniformContinuous_comap",
"uniformContinuous_id"
] | Uniform equivalence between `ULift α` and `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
funUnique (ι α : Type*) [Unique ι] [UniformSpace α] : (ι → α) ≃ᵤ α | where
toEquiv := Equiv.funUnique ι α
uniformContinuous_toFun := Pi.uniformContinuous_proj _ _
uniformContinuous_invFun := uniformContinuous_pi.mpr fun _ => uniformContinuous_id | def | UniformEquiv.funUnique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.funUnique",
"Pi.uniformContinuous_proj",
"UniformSpace",
"Unique",
"uniformContinuous_id"
] | If `ι` has a unique element, then `ι → α` is uniformly isomorphic to `α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
piFinTwo (α : Fin 2 → Type u) [∀ i, UniformSpace (α i)] : (∀ i, α i) ≃ᵤ α 0 × α 1 | where
toEquiv := piFinTwoEquiv α
uniformContinuous_toFun := (Pi.uniformContinuous_proj _ 0).prodMk (Pi.uniformContinuous_proj _ 1)
uniformContinuous_invFun :=
uniformContinuous_pi.mpr <| Fin.forall_fin_two.2 ⟨uniformContinuous_fst, uniformContinuous_snd⟩ | def | UniformEquiv.piFinTwo | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Pi.uniformContinuous_proj",
"UniformSpace",
"piFinTwoEquiv"
] | Uniform isomorphism between dependent functions `Π i : Fin 2, α i` and `α 0 × α 1`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
finTwoArrow (α : Type*) [UniformSpace α] : (Fin 2 → α) ≃ᵤ α × α | { piFinTwo fun _ => α with toEquiv := finTwoArrowEquiv α } | def | UniformEquiv.finTwoArrow | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"UniformSpace",
"finTwoArrowEquiv"
] | Uniform isomorphism between `α² = Fin 2 → α` and `α × α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
image (e : α ≃ᵤ β) (s : Set α) : s ≃ᵤ e '' s | where
uniformContinuous_toFun := (e.uniformContinuous.comp uniformContinuous_subtype_val).subtype_mk _
uniformContinuous_invFun :=
(e.symm.uniformContinuous.comp uniformContinuous_subtype_val).subtype_mk _
toEquiv := e.toEquiv.image s | def | UniformEquiv.image | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Set",
"uniformContinuous_subtype_val"
] | A subset of a uniform space is uniformly isomorphic to its image under a uniform isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subtype {p : α → Prop} {q : β → Prop} (e : α ≃ᵤ β) (h : ∀ a, p a ↔ q (e a)) :
{ a : α // p a } ≃ᵤ { b : β // q b } | where
uniformContinuous_toFun := by
simpa [Equiv.coe_subtypeEquiv_eq_map] using e.uniformContinuous.subtype_map _
uniformContinuous_invFun := by
simpa [Equiv.coe_subtypeEquiv_eq_map] using e.symm.uniformContinuous.subtype_map _
__ := e.subtypeEquiv h | def | UniformEquiv.subtype | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"Equiv.coe_subtypeEquiv_eq_map"
] | A uniform isomorphism `e : α ≃ᵤ β` lifts to subtypes `{ a : α // p a } ≃ᵤ { b : β // q b }`
provided `p = q ∘ e`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Equiv.toUniformEquivOfIsUniformInducing [UniformSpace α] [UniformSpace β] (f : α ≃ β)
(hf : IsUniformInducing f) : α ≃ᵤ β | { f with
uniformContinuous_toFun := hf.uniformContinuous
uniformContinuous_invFun := hf.uniformContinuous_iff.2 <| by simpa using uniformContinuous_id } | def | Equiv.toUniformEquivOfIsUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equiv.lean | [] | [
"IsUniformInducing",
"UniformSpace",
"uniformContinuous_id"
] | A uniform inducing equiv between uniform spaces is a uniform isomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α → β}
(h : Continuous f) : UniformContinuous f | calc map (Prod.map f f) (𝓤 α)
= map (Prod.map f f) (𝓝ˢ (diagonal α)) := by rw [nhdsSet_diagonal_eq_uniformity]
_ ≤ 𝓝ˢ (diagonal β) := (h.prodMap h).tendsto_nhdsSet mapsTo_prodMap_diagonal
_ ≤ 𝓤 β := nhdsSet_diagonal_le_uniformity | theorem | CompactSpace.uniformContinuous_of_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"CompactSpace",
"Continuous",
"UniformContinuous",
"nhdsSet_diagonal_eq_uniformity",
"nhdsSet_diagonal_le_uniformity"
] | Heine-Cantor: a continuous function on a compact uniform space is uniformly
continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.uniformContinuousOn_of_continuous {s : Set α} {f : α → β} (hs : IsCompact s)
(hf : ContinuousOn f s) : UniformContinuousOn f s | by
rw [uniformContinuousOn_iff_restrict]
rw [isCompact_iff_compactSpace] at hs
rw [continuousOn_iff_continuous_restrict] at hf
exact CompactSpace.uniformContinuous_of_continuous hf | theorem | IsCompact.uniformContinuousOn_of_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"CompactSpace.uniformContinuous_of_continuous",
"ContinuousOn",
"IsCompact",
"Set",
"UniformContinuousOn",
"continuousOn_iff_continuous_restrict",
"isCompact_iff_compactSpace",
"uniformContinuousOn_iff_restrict"
] | Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly
continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.uniformContinuousAt_of_continuousAt {r : Set (β × β)} {s : Set α}
(hs : IsCompact s) (f : α → β) (hf : ∀ a ∈ s, ContinuousAt f a) (hr : r ∈ 𝓤 β) :
{ x : α × α | x.1 ∈ s → (f x.1, f x.2) ∈ r } ∈ 𝓤 α | by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
choose U hU T hT hb using fun a ha =>
exists_mem_nhds_ball_subset_of_mem_nhds ((hf a ha).preimage_mem_nhds <| mem_nhds_left _ ht)
obtain ⟨fs, hsU⟩ := hs.elim_nhds_subcover' U hU
apply mem_of_superset ((biInter_finset_mem fs).2 fun a _ => hT ... | theorem | IsCompact.uniformContinuousAt_of_continuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"ContinuousAt",
"IsCompact",
"Set",
"SetRel.symm",
"comp_symm_mem_uniformity_sets",
"exists_mem_nhds_ball_subset_of_mem_nhds",
"mem_nhds_left"
] | If `s` is compact and `f` is continuous at all points of `s`, then `f` is
"uniformly continuous at the set `s`", i.e. `f x` is close to `f y` whenever `x ∈ s` and `y` is
close to `x` (even if `y` is not itself in `s`, so this is a stronger assertion than
`UniformContinuousOn s`). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.uniformContinuous_of_tendsto_cocompact {f : α → β} {x : β}
(h_cont : Continuous f) (hx : Tendsto f (cocompact α) (𝓝 x)) : UniformContinuous f | uniformContinuous_def.2 fun r hr => by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx <| mem_nhds_left _ ht)
apply
mem_of_superset
(symmetrize_mem_uniformity <|
(hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.con... | theorem | Continuous.uniformContinuous_of_tendsto_cocompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"Continuous",
"SetRel.symm",
"UniformContinuous",
"by_cases",
"comp_symm_mem_uniformity_sets",
"mem_nhds_left",
"symmetrize_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
HasCompactMulSupport.uniformContinuous_of_continuous {f : α → β} [One β]
(h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f | h2.uniformContinuous_of_tendsto_cocompact h1.is_one_at_infty | theorem | HasCompactMulSupport.uniformContinuous_of_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"Continuous",
"HasCompactMulSupport",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOn.tendstoUniformly [LocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
{f : α → β → γ} {x : α} {U : Set α} (hxU : U ∈ 𝓝 x) (h : ContinuousOn ↿f (U ×ˢ univ)) :
TendstoUniformly f (f x) (𝓝 x) | by
rcases LocallyCompactSpace.local_compact_nhds _ _ hxU with ⟨K, hxK, hKU, hK⟩
have : UniformContinuousOn ↿f (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ)
(h.mono <| prod_mono hKU Subset.rfl)
exact this.tendstoUniformly hxK | theorem | ContinuousOn.tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"CompactSpace",
"ContinuousOn",
"IsCompact.uniformContinuousOn_of_continuous",
"LocallyCompactSpace",
"Set",
"TendstoUniformly",
"UniformContinuousOn",
"UniformSpace",
"isCompact_univ"
] | A family of functions `α → β → γ` tends uniformly to its value at `x` if `α` is locally compact,
`β` is compact and `f` is continuous on `U × (univ : Set β)` for some neighborhood `U` of `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Continuous.tendstoUniformly [WeaklyLocallyCompactSpace α] [CompactSpace β] [UniformSpace γ]
(f : α → β → γ) (h : Continuous ↿f) (x : α) : TendstoUniformly f (f x) (𝓝 x) | let ⟨K, hK, hxK⟩ := exists_compact_mem_nhds x
have : UniformContinuousOn ↿f (K ×ˢ univ) :=
IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ) h.continuousOn
this.tendstoUniformly hxK | theorem | Continuous.tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"CompactSpace",
"Continuous",
"IsCompact.uniformContinuousOn_of_continuous",
"TendstoUniformly",
"UniformContinuousOn",
"UniformSpace",
"WeaklyLocallyCompactSpace",
"isCompact_univ"
] | A continuous family of functions `α → β → γ` tends uniformly to its value at `x`
if `α` is weakly locally compact and `β` is compact. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsCompact.mem_uniformity_of_prod
{α β E : Type*} [TopologicalSpace α] [TopologicalSpace β] [UniformSpace E]
{f : α → β → E} {s : Set α} {k : Set β} {q : α} {u : Set (E × E)}
(hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ k)) (hq : q ∈ s) (hu : u ∈ 𝓤 E) :
∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, (f p x, ... | by
apply hk.induction_on (p := fun t ↦ ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ t, (f p x, f q x) ∈ u)
· exact ⟨univ, univ_mem, by simp⟩
· intro t' t ht't ⟨v, v_mem, hv⟩
exact ⟨v, v_mem, fun p hp x hx ↦ hv p hp x (ht't hx)⟩
· intro t t' ⟨v, v_mem, hv⟩ ⟨v', v'_mem, hv'⟩
refine ⟨v ∩ v', inter_mem v_mem v'_mem, fun p... | lemma | IsCompact.mem_uniformity_of_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"ContinuousOn",
"IsCompact",
"Set",
"SetRel.prodMk_mem_comp",
"TopologicalSpace",
"UniformSpace",
"comp_symm_of_uniformity",
"mem_nhds_left"
] | In a product space `α × β`, assume that a function `f` is continuous on `s × k` where `k` is
compact. Then, along the fiber above any `q ∈ s`, `f` is transversely uniformly continuous, i.e.,
if `p ∈ s` is close enough to `q`, then `f p x` is uniformly close to `f q x` for all `x ∈ k`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
CompactSpace.uniformEquicontinuous_of_equicontinuous {ι : Type*} {F : ι → β → α}
[CompactSpace β] (h : Equicontinuous F) : UniformEquicontinuous F | by
rw [equicontinuous_iff_continuous] at h
rw [uniformEquicontinuous_iff_uniformContinuous]
exact CompactSpace.uniformContinuous_of_continuous h | theorem | CompactSpace.uniformEquicontinuous_of_equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/HeineCantor.lean | [] | [
"CompactSpace",
"CompactSpace.uniformContinuous_of_continuous",
"Equicontinuous",
"UniformEquicontinuous",
"equicontinuous_iff_continuous",
"uniformEquicontinuous_iff_uniformContinuous"
] | An equicontinuous family of functions defined on a compact uniform space is automatically
uniformly equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformlyOn (F : ι → α → β) (f : α → β) (p : Filter ι) (s : Set α) | ∀ u ∈ 𝓤 β, ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | TendstoLocallyUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Filter",
"Set"
] | A sequence of functions `Fₙ` converges locally uniformly on a set `s` to a limiting function
`f` with respect to a filter `p` if, for any entourage of the diagonal `u`, for any `x ∈ s`, one
has `p`-eventually `(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x` in `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformly (F : ι → α → β) (f : α → β) (p : Filter ι) | ∀ u ∈ 𝓤 β, ∀ x : α, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u | def | TendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Filter"
] | A sequence of functions `Fₙ` converges locally uniformly to a limiting function `f` with respect
to a filter `p` if, for any entourage of the diagonal `u`, for any `x`, one has `p`-eventually
`(f y, Fₙ y) ∈ u` for all `y` in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformlyOn_univ :
TendstoLocallyUniformlyOn F f p univ ↔ TendstoLocallyUniformly F f p | by
simp [TendstoLocallyUniformlyOn, TendstoLocallyUniformly, nhdsWithin_univ] | theorem | tendstoLocallyUniformlyOn_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"TendstoLocallyUniformlyOn",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_iff_forall_tendsto :
TendstoLocallyUniformlyOn F f p s ↔
∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β) | forall₂_comm.trans <| forall₄_congr fun _ _ _ _ => by
simp_rw [mem_map, mem_prod_iff_right, mem_preimage] | theorem | tendstoLocallyUniformlyOn_iff_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto (hs : IsOpen s) :
TendstoLocallyUniformlyOn F f p s ↔
∀ x ∈ s, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) | tendstoLocallyUniformlyOn_iff_forall_tendsto.trans <| forall₂_congr fun x hx => by
rw [hs.nhdsWithin_eq hx] | theorem | IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsOpen",
"TendstoLocallyUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformly_iff_forall_tendsto :
TendstoLocallyUniformly F f p ↔
∀ x, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) | by
simp [← tendstoLocallyUniformlyOn_univ, isOpen_univ.tendstoLocallyUniformlyOn_iff_forall_tendsto] | theorem | tendstoLocallyUniformly_iff_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe :
TendstoLocallyUniformlyOn F f p s ↔
TendstoLocallyUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p | by
simp only [tendstoLocallyUniformly_iff_forall_tendsto, Subtype.forall', tendsto_map'_iff,
tendstoLocallyUniformlyOn_iff_forall_tendsto, ← map_nhds_subtype_val, prod_map_right]; rfl | theorem | tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Subtype.forall'",
"TendstoLocallyUniformly",
"TendstoLocallyUniformlyOn",
"map_nhds_subtype_val",
"tendstoLocallyUniformlyOn_iff_forall_tendsto",
"tendstoLocallyUniformly_iff_forall_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.tendstoLocallyUniformlyOn (h : TendstoUniformlyOn F f p s) :
TendstoLocallyUniformlyOn F f p s | fun u hu _ _ =>
⟨s, self_mem_nhdsWithin, by simpa using h u hu⟩ | theorem | TendstoUniformlyOn.tendstoLocallyUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOn",
"self_mem_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformly.tendstoLocallyUniformly (h : TendstoUniformly F f p) :
TendstoLocallyUniformly F f p | fun u hu _ => ⟨univ, univ_mem, by simpa using h u hu⟩ | theorem | TendstoUniformly.tendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"TendstoUniformly"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.mono (h : TendstoLocallyUniformlyOn F f p s) (h' : s' ⊆ s) :
TendstoLocallyUniformlyOn F f p s' | by
intro u hu x hx
rcases h u hu x (h' hx) with ⟨t, ht, H⟩
exact ⟨t, nhdsWithin_mono x h' ht, H.mono fun n => id⟩ | theorem | TendstoLocallyUniformlyOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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