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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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comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (𝓤 α)) = 𝓤 α | by
refine le_antisymm ?_ le_comap_map
refine ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU ↦ ?_
refine ⟨U, hU.1, fun (x₁, x₂) ⟨(y₁, y₂), hyU, hxy⟩ ↦ ?_⟩
simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy
exact ((hxy.1.prod hxy.2).mem_open_iff hU.2).1 hyU | theorem | SeparationQuotient.comap_map_mk_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"comap",
"le_antisymm",
"uniformity_hasBasis_open"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instUniformSpace : UniformSpace (SeparationQuotient α) | where
uniformity := map (Prod.map mk mk) (𝓤 α)
symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
comp := fun t ht ↦ by
rcases comp_open_symm_mem_uniformity_sets ht with ⟨U, hU, hUo, -, hUt⟩
refine mem_of_superset (mem_lift' <| image_mem_map hU) ?_
simp only [subset_def, Prod.forall, Se... | instance | SeparationQuotient.instUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Filter.comap_comap",
"SeparationQuotient",
"SetRel.mem_comp",
"UniformSpace",
"UniformSpace.isOpen_ball",
"comp_open_symm_mem_uniformity_sets",
"nhds_eq_comap_uniformity",
"symm",
"tendsto_swap_uniformity",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) | rfl | theorem | SeparationQuotient.uniformity_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_mk : UniformContinuous (mk : α → SeparationQuotient α) | le_rfl | theorem | SeparationQuotient.uniformContinuous_mk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient",
"UniformContinuous",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_dom {f : SeparationQuotient α → β} :
UniformContinuous f ↔ UniformContinuous (f ∘ mk) | .rfl | theorem | SeparationQuotient.uniformContinuous_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_dom₂ {f : SeparationQuotient α × SeparationQuotient β → γ} :
UniformContinuous f ↔ UniformContinuous fun p : α × β ↦ f (mk p.1, mk p.2) | by
simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq, prod_map_map_eq,
tendsto_map'_iff]
rfl | theorem | SeparationQuotient.uniformContinuous_dom₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient",
"UniformContinuous",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_lift {f : α → β} (h : ∀ a b, Inseparable a b → f a = f b) :
UniformContinuous (lift f h) ↔ UniformContinuous f | .rfl | theorem | SeparationQuotient.uniformContinuous_lift | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Inseparable",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_uncurry_lift₂ {f : α → β → γ}
(h : ∀ a c b d, Inseparable a b → Inseparable c d → f a c = f b d) :
UniformContinuous (uncurry <| lift₂ f h) ↔ UniformContinuous (uncurry f) | uniformContinuous_dom₂ | theorem | SeparationQuotient.uniformContinuous_uncurry_lift₂ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Inseparable",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_mk_uniformity : (𝓤 (SeparationQuotient α)).comap (Prod.map mk mk) = 𝓤 α | comap_map_mk_uniformity | theorem | SeparationQuotient.comap_mk_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift' [T0Space β] (f : α → β) : SeparationQuotient α → β | if hc : UniformContinuous f then lift f fun _ _ h => (h.map hc.continuous).eq
else fun x => f (Nonempty.some ⟨x.out⟩) | def | SeparationQuotient.lift' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Nonempty.some",
"SeparationQuotient",
"T0Space",
"UniformContinuous"
] | Factoring functions to a separated space through the separation quotient.
TODO: unify with `SeparationQuotient.lift`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
lift'_mk [T0Space β] {f : α → β} (h : UniformContinuous f) (a : α) :
lift' f (mk a) = f a | by rw [lift', dif_pos h, lift_mk] | theorem | SeparationQuotient.lift'_mk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"T0Space",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_lift' [T0Space β] (f : α → β) : UniformContinuous (lift' f) | by
by_cases hf : UniformContinuous f
· rwa [lift', dif_pos hf, uniformContinuous_lift]
· rw [lift', dif_neg hf]
exact uniformContinuous_of_const fun a _ => rfl | theorem | SeparationQuotient.uniformContinuous_lift' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"T0Space",
"UniformContinuous",
"by_cases",
"uniformContinuous_of_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map (f : α → β) : SeparationQuotient α → SeparationQuotient β | lift' (mk ∘ f) | def | SeparationQuotient.map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient"
] | The separation quotient functor acting on functions. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
map_mk {f : α → β} (h : UniformContinuous f) (a : α) : map f (mk a) = mk (f a) | by
rw [map, lift'_mk (uniformContinuous_mk.comp h)]; rfl | theorem | SeparationQuotient.map_mk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_map (f : α → β) : UniformContinuous (map f) | uniformContinuous_lift' _ | theorem | SeparationQuotient.uniformContinuous_map | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_unique {f : α → β} (hf : UniformContinuous f)
{g : SeparationQuotient α → SeparationQuotient β} (comm : mk ∘ f = g ∘ mk) : map f = g | by
ext ⟨a⟩
calc
map f ⟦a⟧ = ⟦f a⟧ := map_mk hf a
_ = g ⟦a⟧ := congr_fun comm a | theorem | SeparationQuotient.map_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SeparationQuotient",
"UniformContinuous",
"comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_id : map (@id α) = id | map_unique uniformContinuous_id rfl | theorem | SeparationQuotient.map_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) :
map g ∘ map f = map (g ∘ f) | (map_unique (hg.comp hf) <| by simp only [Function.comp_def, map_mk, hf, hg]).symm | theorem | SeparationQuotient.map_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"UniformContinuous",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of_uniformity_eq_top (h : uniformity α = ⊤) : IndiscreteTopology α | ⟨(UniformSpace.ext h.symm : ⊤ = u) ▸ rfl⟩ | theorem | IndiscreteTopology.of_uniformity_eq_top | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"IndiscreteTopology",
"UniformSpace.ext",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_top_uniformSpace [IndiscreteTopology α] : u = ⊤ | by
refine UniformSpace.ext ?_
rw [top_uniformity, ← Filter.ker_eq_univ]
ext x
rw [← inseparable_iff_ker_uniformity]
simp | lemma | IndiscreteTopology.eq_top_uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Filter.ker_eq_univ",
"IndiscreteTopology",
"UniformSpace.ext",
"inseparable_iff_ker_uniformity",
"top_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_top_iff_indiscrete : u = ⊤ ↔ IndiscreteTopology α | ⟨fun h ↦ IndiscreteTopology.mk <| h ▸ UniformSpace.toTopologicalSpace_top (α := α),
fun _ ↦ eq_top_uniformSpace⟩ | lemma | IndiscreteTopology.eq_top_iff_indiscrete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"IndiscreteTopology",
"UniformSpace.toTopologicalSpace_top"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous [IndiscreteTopology β] {f : α → β} : UniformContinuous f | by
rw [UniformContinuous, eq_top_uniformSpace (α := β), top_uniformity]
exact Filter.tendsto_top | lemma | IndiscreteTopology.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Filter.tendsto_top",
"IndiscreteTopology",
"UniformContinuous",
"top_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt (hx : x ∈ s)
(L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F : α → β, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
ContinuousWithinAt f s x | by
refine Uniform.continuousWithinAt_iff'_left.2 fun u₀ hu₀ => ?_
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀
obtain ⟨u₂, h₂, hsymm, u₂₁⟩ : ∃ u ∈ 𝓤 β, (∀ {a b}, (a, b) ∈ u → (b, a) ∈ u) ∧ u ○ u ⊆ u₁ :=
comp_symm_of_uniformity h₁
rcases L u₂ h₂ with ⟨t, tx, F, hFc, hF⟩
ha... | theorem | continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousWithinAt",
"comp_mem_uniformity_sets",
"comp_symm_of_uniformity",
"refl_mem_uniformity"
] | A function which can be locally uniformly approximated by functions which are continuous
within a set at a point is continuous within this set at this point. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousAt_of_locally_uniform_approx_of_continuousAt
(L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
ContinuousAt f x | by
rw [← continuousWithinAt_univ]
apply continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt (mem_univ _) _
simpa only [exists_prop, nhdsWithin_univ, continuousWithinAt_univ] using L | theorem | continuousAt_of_locally_uniform_approx_of_continuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousAt",
"continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt",
"continuousWithinAt_univ",
"nhdsWithin_univ"
] | A function which can be locally uniformly approximated by functions which are continuous at
a point is continuous at this point. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_of_locally_uniform_approx_of_continuousWithinAt
(L : ∀ x ∈ s, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F,
ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
ContinuousOn f s | fun x hx =>
continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt hx (L x hx) | theorem | continuousOn_of_locally_uniform_approx_of_continuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousOn",
"ContinuousWithinAt",
"continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt"
] | A function which can be locally uniformly approximated by functions which are continuous
on a set is continuous on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuousOn_of_uniform_approx_of_continuousOn
(L : ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) : ContinuousOn f s | continuousOn_of_locally_uniform_approx_of_continuousWithinAt fun _x hx u hu =>
⟨s, self_mem_nhdsWithin, (L u hu).imp fun _F hF => ⟨hF.1.continuousWithinAt hx, hF.2⟩⟩ | theorem | continuousOn_of_uniform_approx_of_continuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousOn",
"continuousOn_of_locally_uniform_approx_of_continuousWithinAt",
"self_mem_nhdsWithin"
] | A function which can be uniformly approximated by functions which are continuous on a set
is continuous on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_of_locally_uniform_approx_of_continuousAt
(L : ∀ x : α, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) :
Continuous f | continuous_iff_continuousAt.2 fun x =>
continuousAt_of_locally_uniform_approx_of_continuousAt (L x) | theorem | continuous_of_locally_uniform_approx_of_continuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"Continuous",
"ContinuousAt",
"continuousAt_of_locally_uniform_approx_of_continuousAt"
] | A function which can be locally uniformly approximated by continuous functions is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
continuous_of_uniform_approx_of_continuous
(L : ∀ u ∈ 𝓤 β, ∃ F, Continuous F ∧ ∀ y, (f y, F y) ∈ u) : Continuous f | continuousOn_univ.mp <|
continuousOn_of_uniform_approx_of_continuousOn <| by
simpa [continuousOn_univ] using L | theorem | continuous_of_uniform_approx_of_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"Continuous",
"continuousOn_of_uniform_approx_of_continuousOn",
"continuousOn_univ"
] | A function which can be uniformly approximated by continuous functions is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformlyOn.continuousOn (h : TendstoLocallyUniformlyOn F f p s)
(hc : ∃ᶠ n in p, ContinuousOn (F n) s) : ContinuousOn f s | by
refine continuousOn_of_locally_uniform_approx_of_continuousWithinAt fun x hx u hu => ?_
rcases h u hu x hx with ⟨t, ht, H⟩
rcases (hc.and_eventually H).exists with ⟨n, hFc, hF⟩
exact ⟨t, ht, ⟨F n, hFc.continuousWithinAt hx, hF⟩⟩ | theorem | TendstoLocallyUniformlyOn.continuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousOn",
"TendstoLocallyUniformlyOn",
"continuousOn_of_locally_uniform_approx_of_continuousWithinAt"
] | A locally uniform limit on a set of functions which are continuous on this set is itself
continuous on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn.continuousOn (h : TendstoUniformlyOn F f p s)
(hc : ∃ᶠ n in p, ContinuousOn (F n) s) : ContinuousOn f s | h.tendstoLocallyUniformlyOn.continuousOn hc | theorem | TendstoUniformlyOn.continuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousOn",
"TendstoUniformlyOn"
] | A uniform limit on a set of functions which are continuous on this set is itself continuous
on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformly.continuous (h : TendstoLocallyUniformly F f p)
(hc : ∃ᶠ n in p, Continuous (F n)) : Continuous f | continuousOn_univ.mp <|
h.tendstoLocallyUniformlyOn.continuousOn <| hc.mono fun _n hn => hn.continuousOn | theorem | TendstoLocallyUniformly.continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"Continuous",
"TendstoLocallyUniformly"
] | A locally uniform limit of continuous functions is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly.continuous (h : TendstoUniformly F f p)
(hc : ∃ᶠ n in p, Continuous (F n)) : Continuous f | h.tendstoLocallyUniformly.continuous hc | theorem | TendstoUniformly.continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"Continuous",
"TendstoUniformly"
] | A uniform limit of continuous functions is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_comp_of_locally_uniform_limit_within (h : ContinuousWithinAt f s x)
(hg : Tendsto g p (𝓝[s] x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | by
refine Uniform.tendsto_nhds_right.2 fun u₀ hu₀ => ?_
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀
rcases hunif u₁ h₁ with ⟨s, sx, hs⟩
have A : ∀ᶠ n in p, g n ∈ s := hg sx
have B : ∀ᶠ n in p, (f x, f (g n)) ∈ u₁ := hg (Uniform.continuousWithinAt_iff'_right.1 h h₁)
exact B.... | theorem | tendsto_comp_of_locally_uniform_limit_within | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousWithinAt",
"comp_mem_uniformity_sets"
] | If `Fₙ` converges locally uniformly on a neighborhood of `x` within a set `s` to a function `f`
which is continuous at `x` within `s`, and `gₙ` tends to `x` within `s`, then `Fₙ (gₙ)` tends
to `f x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_comp_of_locally_uniform_limit (h : ContinuousAt f x) (hg : Tendsto g p (𝓝 x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | by
rw [← continuousWithinAt_univ] at h
rw [← nhdsWithin_univ] at hunif hg
exact tendsto_comp_of_locally_uniform_limit_within h hg hunif | theorem | tendsto_comp_of_locally_uniform_limit | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousAt",
"continuousWithinAt_univ",
"nhdsWithin_univ",
"tendsto_comp_of_locally_uniform_limit_within"
] | If `Fₙ` converges locally uniformly on a neighborhood of `x` to a function `f` which is
continuous at `x`, and `gₙ` tends to `x`, then `Fₙ (gₙ)` tends to `f x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformlyOn.tendsto_comp (h : TendstoLocallyUniformlyOn F f p s)
(hf : ContinuousWithinAt f s x) (hx : x ∈ s) (hg : Tendsto g p (𝓝[s] x)) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit_within hf hg fun u hu => h u hu x hx | theorem | TendstoLocallyUniformlyOn.tendsto_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousWithinAt",
"TendstoLocallyUniformlyOn",
"tendsto_comp_of_locally_uniform_limit_within"
] | If `Fₙ` tends locally uniformly to `f` on a set `s`, and `gₙ` tends to `x` within `s`, then
`Fₙ gₙ` tends to `f x` if `f` is continuous at `x` within `s` and `x ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn.tendsto_comp (h : TendstoUniformlyOn F f p s)
(hf : ContinuousWithinAt f s x) (hg : Tendsto g p (𝓝[s] x)) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit_within hf hg fun u hu => ⟨s, self_mem_nhdsWithin, h u hu⟩ | theorem | TendstoUniformlyOn.tendsto_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousWithinAt",
"TendstoUniformlyOn",
"self_mem_nhdsWithin",
"tendsto_comp_of_locally_uniform_limit_within"
] | If `Fₙ` tends uniformly to `f` on a set `s`, and `gₙ` tends to `x` within `s`, then `Fₙ gₙ`
tends to `f x` if `f` is continuous at `x` within `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformly.tendsto_comp (h : TendstoLocallyUniformly F f p)
(hf : ContinuousAt f x) (hg : Tendsto g p (𝓝 x)) : Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | tendsto_comp_of_locally_uniform_limit hf hg fun u hu => h u hu x | theorem | TendstoLocallyUniformly.tendsto_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousAt",
"TendstoLocallyUniformly",
"tendsto_comp_of_locally_uniform_limit"
] | If `Fₙ` tends locally uniformly to `f`, and `gₙ` tends to `x`, then `Fₙ gₙ` tends to `f x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly.tendsto_comp (h : TendstoUniformly F f p) (hf : ContinuousAt f x)
(hg : Tendsto g p (𝓝 x)) : Tendsto (fun n => F n (g n)) p (𝓝 (f x)) | h.tendstoLocallyUniformly.tendsto_comp hf hg | theorem | TendstoUniformly.tendsto_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"ContinuousAt",
"TendstoUniformly"
] | If `Fₙ` tends uniformly to `f`, and `gₙ` tends to `x`, then `Fₙ gₙ` tends to `f x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn
(h : ∀ u ∈ 𝓤 β, ∃ F : α → β, UniformContinuousOn F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) :
UniformContinuousOn f s | by
simp_rw [uniformContinuousOn_iff_restrict, uniformContinuous_def] at h ⊢
intro u hu
obtain ⟨v, hv, hvsymm, hvu⟩ := comp_comp_symm_mem_uniformity_sets hu
obtain ⟨F, hF, hFv⟩ := h v hv
filter_upwards [hF v hv] with x hx
exact hvu <| prodMk_mem_comp (prodMk_mem_comp (hFv _ x.1.prop) hx)
<| hvsymm.symm... | theorem | uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"UniformContinuousOn",
"comp_comp_symm_mem_uniformity_sets",
"uniformContinuousOn_iff_restrict",
"uniformContinuous_def"
] | A function which can be uniformly approximated by functions which are uniformly continuous on a
set is uniformly continuous on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_of_uniform_approx_of_uniformContinuous
(h : ∀ u ∈ 𝓤 β, ∃ F : α → β, UniformContinuous F ∧ ∀ y, (f y, F y) ∈ u) :
UniformContinuous f | uniformContinuousOn_univ.mp <| uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn
<| by simpa [uniformContinuousOn_univ] using h | theorem | uniformContinuous_of_uniform_approx_of_uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"UniformContinuous",
"uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn",
"uniformContinuousOn_univ"
] | A function which can be uniformly approximated by uniformly continuous functions is uniformly
continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn.uniformContinuousOn (h : TendstoUniformlyOn F f p s)
(hc : ∃ᶠ n in p, UniformContinuousOn (F n) s) : UniformContinuousOn f s | uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn fun u hu ↦
let ⟨i, hF⟩ := (hc.and_eventually (h u hu)).exists
⟨F i, hF⟩ | theorem | TendstoUniformlyOn.uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"TendstoUniformlyOn",
"UniformContinuousOn",
"uniformContinuousOn_of_uniform_approx_of_uniformContinuousOn"
] | A uniform limit on a set of functions which are uniformly continuous on this set is itself
uniformly continuous on this set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly.uniformContinuous (h : TendstoUniformly F f p)
(hc : ∃ᶠ n in p, UniformContinuous (F n)) : UniformContinuous f | uniformContinuous_of_uniform_approx_of_uniformContinuous fun u hu ↦
let ⟨i, hF⟩ := (hc.and_eventually (h u hu)).exists
⟨F i, hF⟩ | theorem | TendstoUniformly.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformApproximation.lean | [] | [
"TendstoUniformly",
"UniformContinuous",
"uniformContinuous_of_uniform_approx_of_uniformContinuous"
] | A uniform limit of uniformly continuous functions is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOnFilter (F : ι → α → β) (f : α → β) (p : Filter ι) (p' : Filter α) | ∀ u ∈ 𝓤 β, ∀ᶠ n : ι × α in p ×ˢ p', (f n.snd, F n.fst n.snd) ∈ u | def | TendstoUniformlyOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter"
] | A sequence of functions `Fₙ` converges uniformly on a filter `p'` to a limiting function `f`
with respect to the filter `p` if, for any entourage of the diagonal `u`, one has
`p ×ˢ p'`-eventually `(f x, Fₙ x) ∈ u`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOnFilter_iff_tendsto :
TendstoUniformlyOnFilter F f p p' ↔
Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ p') (𝓤 β) | Iff.rfl | theorem | tendstoUniformlyOnFilter_iff_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter"
] | A sequence of functions `Fₙ` converges uniformly on a filter `p'` to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ˢ p'` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit besides it being in `p'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn (F : ι → α → β) (f : α → β) (p : Filter ι) (s : Set α) | ∀ u ∈ 𝓤 β, ∀ᶠ n in p, ∀ x : α, x ∈ s → (f x, F n x) ∈ u | def | TendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set"
] | A sequence of functions `Fₙ` converges uniformly on a set `s` to a limiting function `f` with
respect to the filter `p` if, for any entourage of the diagonal `u`, one has `p`-eventually
`(f x, Fₙ x) ∈ u` for all `x ∈ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_iff_tendstoUniformlyOnFilter :
TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s) | by
simp only [TendstoUniformlyOn, TendstoUniformlyOnFilter]
apply forall₂_congr
simp_rw [eventually_prod_principal_iff]
simp
alias ⟨TendstoUniformlyOn.tendstoUniformlyOnFilter, TendstoUniformlyOnFilter.tendstoUniformlyOn⟩ :=
tendstoUniformlyOn_iff_tendstoUniformlyOnFilter | theorem | tendstoUniformlyOn_iff_tendstoUniformlyOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"TendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformlyOn_iff_tendsto :
TendstoUniformlyOn F f p s ↔
Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β) | by
simp [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, tendstoUniformlyOnFilter_iff_tendsto] | theorem | tendstoUniformlyOn_iff_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"tendstoUniformlyOnFilter_iff_tendsto",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter"
] | A sequence of functions `Fₙ` converges uniformly on a set `s` to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ˢ 𝓟 s` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit besides it being in `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly (F : ι → α → β) (f : α → β) (p : Filter ι) | ∀ u ∈ 𝓤 β, ∀ᶠ n in p, ∀ x : α, (f x, F n x) ∈ u | def | TendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter"
] | A sequence of functions `Fₙ` converges uniformly to a limiting function `f` with respect to a
filter `p` if, for any entourage of the diagonal `u`, one has `p`-eventually
`(f x, Fₙ x) ∈ u` for all `x`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_univ : TendstoUniformlyOn F f p univ ↔ TendstoUniformly F f p | by
simp [TendstoUniformlyOn, TendstoUniformly] | theorem | tendstoUniformlyOn_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"TendstoUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformly_iff_tendstoUniformlyOnFilter :
TendstoUniformly F f p ↔ TendstoUniformlyOnFilter F f p ⊤ | by
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, principal_univ] | theorem | tendstoUniformly_iff_tendstoUniformlyOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"TendstoUniformlyOnFilter",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter",
"tendstoUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformly.tendstoUniformlyOnFilter (h : TendstoUniformly F f p) :
TendstoUniformlyOnFilter F f p ⊤ | by rwa [← tendstoUniformly_iff_tendstoUniformlyOnFilter] | theorem | TendstoUniformly.tendstoUniformlyOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"TendstoUniformlyOnFilter",
"tendstoUniformly_iff_tendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformlyOn_iff_tendstoUniformly_comp_coe :
TendstoUniformlyOn F f p s ↔ TendstoUniformly (fun i (x : s) => F i x) (f ∘ (↑)) p | forall₂_congr fun u _ => by simp | theorem | tendstoUniformlyOn_iff_tendstoUniformly_comp_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"TendstoUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformlyOn_iff_restrict {K : Set α} : TendstoUniformlyOn F f p K ↔
TendstoUniformly (fun n : ι => K.restrict (F n)) (K.restrict f) p | tendstoUniformlyOn_iff_tendstoUniformly_comp_coe | lemma | tendstoUniformlyOn_iff_restrict | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Set",
"TendstoUniformly",
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendstoUniformly_comp_coe"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformly_iff_tendsto :
TendstoUniformly F f p ↔ Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ ⊤) (𝓤 β) | by
simp [tendstoUniformly_iff_tendstoUniformlyOnFilter, tendstoUniformlyOnFilter_iff_tendsto] | theorem | tendstoUniformly_iff_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"tendstoUniformlyOnFilter_iff_tendsto",
"tendstoUniformly_iff_tendstoUniformlyOnFilter"
] | A sequence of functions `Fₙ` converges uniformly to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ˢ ⊤` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOnFilter.tendsto_at (h : TendstoUniformlyOnFilter F f p p')
(hx : 𝓟 {x} ≤ p') : Tendsto (fun n => F n x) p <| 𝓝 (f x) | by
refine Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr ?_
filter_upwards [(h u hu).curry]
intro i h
simpa using h.filter_mono hx | theorem | TendstoUniformlyOnFilter.tendsto_at | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter"
] | Uniform convergence implies pointwise convergence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn.tendsto_at (h : TendstoUniformlyOn F f p s) (hx : x ∈ s) :
Tendsto (fun n => F n x) p <| 𝓝 (f x) | h.tendstoUniformlyOnFilter.tendsto_at
(le_principal_iff.mpr <| mem_principal.mpr <| singleton_subset_iff.mpr <| hx) | theorem | TendstoUniformlyOn.tendsto_at | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn"
] | Uniform convergence implies pointwise convergence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly.tendsto_at (h : TendstoUniformly F f p) (x : α) :
Tendsto (fun n => F n x) p <| 𝓝 (f x) | h.tendstoUniformlyOnFilter.tendsto_at le_top | theorem | TendstoUniformly.tendsto_at | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"le_top"
] | Uniform convergence implies pointwise convergence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOnFilter.mono_left {p'' : Filter ι} (h : TendstoUniformlyOnFilter F f p p')
(hp : p'' ≤ p) : TendstoUniformlyOnFilter F f p'' p' | fun u hu =>
(h u hu).filter_mono (p'.prod_mono_left hp) | theorem | TendstoUniformlyOnFilter.mono_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOnFilter.mono_right {p'' : Filter α} (h : TendstoUniformlyOnFilter F f p p')
(hp : p'' ≤ p') : TendstoUniformlyOnFilter F f p p'' | fun u hu =>
(h u hu).filter_mono (p.prod_mono_right hp) | theorem | TendstoUniformlyOnFilter.mono_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.mono (h : TendstoUniformlyOn F f p s) (h' : s' ⊆ s) :
TendstoUniformlyOn F f p s' | tendstoUniformlyOn_iff_tendstoUniformlyOnFilter.mpr
(h.tendstoUniformlyOnFilter.mono_right (le_principal_iff.mpr <| mem_principal.mpr h')) | theorem | TendstoUniformlyOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOnFilter.congr_inseparable {F' : ι → α → β}
(hf : TendstoUniformlyOnFilter F f p p')
(hff' : ∀ᶠ n : ι × α in p ×ˢ p', Inseparable (F n.fst n.snd) (F' n.fst n.snd)) :
TendstoUniformlyOnFilter F' f p p' | by
rw [tendstoUniformlyOnFilter_iff_tendsto, uniformity_hasBasis_open.tendsto_right_iff] at hf ⊢
exact fun i hi => (hf i hi).congr (hff'.mono fun x hx =>
(Inseparable.rfl.prod hx).mem_open_iff hi.2) | theorem | TendstoUniformlyOnFilter.congr_inseparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Inseparable",
"TendstoUniformlyOnFilter",
"tendstoUniformlyOnFilter_iff_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOnFilter.congr {F' : ι → α → β} (hf : TendstoUniformlyOnFilter F f p p')
(hff' : ∀ᶠ n : ι × α in p ×ˢ p', F n.fst n.snd = F' n.fst n.snd) :
TendstoUniformlyOnFilter F' f p p' | hf.congr_inseparable (hff'.mono fun _ h => .of_eq h) | theorem | TendstoUniformlyOnFilter.congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.congr_inseparable {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s)
(hff' : ∀ᶠ n in p, ∀ x ∈ s, Inseparable (F n x) (F' n x)) : TendstoUniformlyOn F' f p s | by
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at hf ⊢
refine hf.congr_inseparable ?_
rwa [eventually_prod_principal_iff] | theorem | TendstoUniformlyOn.congr_inseparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Inseparable",
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.congr {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s)
(hff' : ∀ᶠ n in p, Set.EqOn (F n) (F' n) s) : TendstoUniformlyOn F' f p s | hf.congr_inseparable (hff'.mono fun _ h _ hx => .of_eq (h hx)) | theorem | TendstoUniformlyOn.congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Set.EqOn",
"TendstoUniformlyOn",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformly_congr_inseparable {F' : ι → α → β}
(hF : ∀ᶠ x in p, ∀ y, Inseparable (F x y) (F' x y)) :
TendstoUniformly F f p ↔ TendstoUniformly F' f p | by
rw [← tendstoUniformlyOn_univ, ← tendstoUniformlyOn_univ]
exact ⟨fun h => h.congr_inseparable (hF.mono fun _ hx y _ => hx y),
fun h => h.congr_inseparable (hF.mono fun _ hx y _ => (hx y).symm)⟩ | lemma | tendstoUniformly_congr_inseparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Inseparable",
"TendstoUniformly",
"symm",
"tendstoUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformly_congr {F' : ι → α → β} (hF : F =ᶠ[p] F') :
TendstoUniformly F f p ↔ TendstoUniformly F' f p | tendstoUniformly_congr_inseparable (hF.mono fun _ hx y => .of_eq (congrFun hx y)) | lemma | tendstoUniformly_congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"of_eq",
"tendstoUniformly_congr_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.congr_inseparable_right {g : α → β} (hf : TendstoUniformlyOn F f p s)
(hfg : ∀ x ∈ s, Inseparable (f x) (g x)) : TendstoUniformlyOn F g p s | by
rw [tendstoUniformlyOn_iff_tendsto, uniformity_hasBasis_open.tendsto_right_iff] at hf ⊢
refine forall₂_imp (fun i hi hf => ?_) hf
rw [eventually_prod_principal_iff] at hf ⊢
exact hf.mono fun x hx y hy => (((hfg y hy).prod .rfl).mem_open_iff hi.2).mp (hx y hy) | theorem | TendstoUniformlyOn.congr_inseparable_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Inseparable",
"TendstoUniformlyOn",
"forall₂_imp",
"tendstoUniformlyOn_iff_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.congr_right {g : α → β} (hf : TendstoUniformlyOn F f p s)
(hfg : EqOn f g s) : TendstoUniformlyOn F g p s | hf.congr_inseparable_right fun _ hx => .of_eq (hfg hx) | theorem | TendstoUniformlyOn.congr_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformly.tendstoUniformlyOn (h : TendstoUniformly F f p) :
TendstoUniformlyOn F f p s | (tendstoUniformlyOn_univ.2 h).mono (subset_univ s) | theorem | TendstoUniformly.tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"TendstoUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOnFilter.comp (h : TendstoUniformlyOnFilter F f p p') (g : γ → α) :
TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p (p'.comap g) | by
rw [tendstoUniformlyOnFilter_iff_tendsto] at h ⊢
exact h.comp (tendsto_id.prodMap tendsto_comap) | theorem | TendstoUniformlyOnFilter.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter",
"tendstoUniformlyOnFilter_iff_tendsto"
] | Composing on the right by a function preserves uniform convergence on a filter | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOn.comp (h : TendstoUniformlyOn F f p s) (g : γ → α) :
TendstoUniformlyOn (fun n => F n ∘ g) (f ∘ g) p (g ⁻¹' s) | by
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at h ⊢
simpa [TendstoUniformlyOn, comap_principal] using TendstoUniformlyOnFilter.comp h g | theorem | TendstoUniformlyOn.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"TendstoUniformlyOnFilter.comp",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter"
] | Composing on the right by a function preserves uniform convergence on a set | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformly.comp (h : TendstoUniformly F f p) (g : γ → α) :
TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p | by
rw [tendstoUniformly_iff_tendstoUniformlyOnFilter] at h ⊢
simpa [principal_univ, comap_principal] using h.comp g | theorem | TendstoUniformly.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"tendstoUniformly_iff_tendstoUniformlyOnFilter"
] | Composing on the right by a function preserves uniform convergence | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuous.comp_tendstoUniformlyOnFilter [UniformSpace γ] {g : β → γ}
(hg : UniformContinuous g) (h : TendstoUniformlyOnFilter F f p p') :
TendstoUniformlyOnFilter (fun i => g ∘ F i) (g ∘ f) p p' | fun _u hu => h _ (hg hu) | theorem | UniformContinuous.comp_tendstoUniformlyOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter",
"UniformContinuous",
"UniformSpace"
] | Composing on the left by a uniformly continuous function preserves
uniform convergence on a filter | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuous.comp_tendstoUniformlyOn [UniformSpace γ] {g : β → γ}
(hg : UniformContinuous g) (h : TendstoUniformlyOn F f p s) :
TendstoUniformlyOn (fun i => g ∘ F i) (g ∘ f) p s | fun _u hu => h _ (hg hu) | theorem | UniformContinuous.comp_tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"UniformContinuous",
"UniformSpace"
] | Composing on the left by a uniformly continuous function preserves
uniform convergence on a set | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuous.comp_tendstoUniformly [UniformSpace γ] {g : β → γ}
(hg : UniformContinuous g) (h : TendstoUniformly F f p) :
TendstoUniformly (fun i => g ∘ F i) (g ∘ f) p | fun _u hu => h _ (hg hu) | theorem | UniformContinuous.comp_tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"UniformContinuous",
"UniformSpace"
] | Composing on the left by a uniformly continuous function preserves uniform convergence | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoUniformlyOnFilter.prodMap {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
{f' : α' → β'} {q : Filter ι'} {q' : Filter α'} (h : TendstoUniformlyOnFilter F f p p')
(h' : TendstoUniformlyOnFilter F' f' q q') :
TendstoUniformlyOnFilter (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f'... | by
rw [tendstoUniformlyOnFilter_iff_tendsto] at h h' ⊢
rw [uniformity_prod_eq_comap_prod, tendsto_comap_iff, ← map_swap4_prod, tendsto_map'_iff]
simpa using! h.prodMap h' | theorem | TendstoUniformlyOnFilter.prodMap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter",
"UniformSpace",
"tendstoUniformlyOnFilter_iff_tendsto",
"uniformity_prod_eq_comap_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.prodMap {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
{f' : α' → β'} {p' : Filter ι'} {s' : Set α'} (h : TendstoUniformlyOn F f p s)
(h' : TendstoUniformlyOn F' f' p' s') :
TendstoUniformlyOn (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p')
(s ×ˢ s'... | by
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at h h' ⊢
simpa only [prod_principal_principal] using h.prodMap h' | theorem | TendstoUniformlyOn.prodMap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOn",
"UniformSpace",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformly.prodMap {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
{f' : α' → β'} {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
TendstoUniformly (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p') | by
rw [← tendstoUniformlyOn_univ, ← univ_prod_univ] at *
exact h.prodMap h' | theorem | TendstoUniformly.prodMap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformly",
"UniformSpace",
"tendstoUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOnFilter.prodMk {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'}
{f' : α → β'} {q : Filter ι'} (h : TendstoUniformlyOnFilter F f p p')
(h' : TendstoUniformlyOnFilter F' f' q p') :
TendstoUniformlyOnFilter (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a))
(p ×ˢ q) p... | fun u hu => ((h.prodMap h') u hu).diag_of_prod_right | theorem | TendstoUniformlyOnFilter.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformlyOn.prodMk {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'}
{f' : α → β'} {p' : Filter ι'} (h : TendstoUniformlyOn F f p s)
(h' : TendstoUniformlyOn F' f' p' s) :
TendstoUniformlyOn (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p ×ˢ p')
s | (congr_arg _ s.inter_self).mp ((h.prodMap h').comp fun a => (a, a)) | theorem | TendstoUniformlyOn.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOn",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoUniformly.prodMk {ι' β' : Type*} [UniformSpace β'] {F' : ι' → α → β'} {f' : α → β'}
{p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
TendstoUniformly (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p ×ˢ p') | (h.prodMap h').comp fun a => (a, a) | theorem | TendstoUniformly.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformly",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_prod_filter_iff {c : β} :
Tendsto ↿F (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun _ => c) p p' | by
simp_rw [nhds_eq_comap_uniformity, tendsto_comap_iff]
rfl | theorem | tendsto_prod_filter_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOnFilter",
"nhds_eq_comap_uniformity"
] | Uniform convergence on a filter `p'` to a constant function is equivalent to convergence in
`p ×ˢ p'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_prod_principal_iff {c : β} :
Tendsto ↿F (p ×ˢ 𝓟 s) (𝓝 c) ↔ TendstoUniformlyOn F (fun _ => c) p s | by
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter]
exact tendsto_prod_filter_iff | theorem | tendsto_prod_principal_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendstoUniformlyOnFilter",
"tendsto_prod_filter_iff"
] | Uniform convergence on a set `s` to a constant function is equivalent to convergence in
`p ×ˢ 𝓟 s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_prod_top_iff {c : β} :
Tendsto ↿F (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun _ => c) p | by
rw [tendstoUniformly_iff_tendstoUniformlyOnFilter]
exact tendsto_prod_filter_iff | theorem | tendsto_prod_top_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"tendstoUniformly_iff_tendstoUniformlyOnFilter",
"tendsto_prod_filter_iff"
] | Uniform convergence to a constant function is equivalent to convergence in `p ×ˢ ⊤`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_empty : TendstoUniformlyOn F f p ∅ | fun u _ => by simp | theorem | tendstoUniformlyOn_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn"
] | Uniform convergence on the empty set is vacuously true | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_singleton_iff_tendsto :
TendstoUniformlyOn F f p {x} ↔ Tendsto (fun n : ι => F n x) p (𝓝 (f x)) | by
simp_rw [tendstoUniformlyOn_iff_tendsto, Uniform.tendsto_nhds_right, tendsto_def]
exact forall₂_congr fun u _ => by simp [preimage] | theorem | tendstoUniformlyOn_singleton_iff_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformlyOn",
"Uniform.tendsto_nhds_right",
"tendstoUniformlyOn_iff_tendsto"
] | Uniform convergence on a singleton is equivalent to regular convergence | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.tendstoUniformlyOnFilter_const {g : ι → β} {b : β} (hg : Tendsto g p (𝓝 b))
(p' : Filter α) :
TendstoUniformlyOnFilter (fun n : ι => fun _ : α => g n) (fun _ : α => b) p p' | by
simpa only [nhds_eq_comap_uniformity, tendsto_comap_iff] using! hg.comp (tendsto_fst (g := p')) | theorem | Filter.Tendsto.tendstoUniformlyOnFilter_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"TendstoUniformlyOnFilter",
"nhds_eq_comap_uniformity"
] | If a sequence `g` converges to some `b`, then the sequence of constant functions
`fun n ↦ fun a ↦ g n` converges to the constant function `fun a ↦ b` on any set `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.tendstoUniformlyOn_const {g : ι → β} {b : β} (hg : Tendsto g p (𝓝 b))
(s : Set α) : TendstoUniformlyOn (fun n : ι => fun _ : α => g n) (fun _ : α => b) p s | tendstoUniformlyOn_iff_tendstoUniformlyOnFilter.mpr (hg.tendstoUniformlyOnFilter_const (𝓟 s)) | theorem | Filter.Tendsto.tendstoUniformlyOn_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Set",
"TendstoUniformlyOn"
] | If a sequence `g` converges to some `b`, then the sequence of constant functions
`fun n ↦ fun a ↦ g n` converges to the constant function `fun a ↦ b` on any set `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.Tendsto.tendstoUniformly_const {g : ι → β} {b : β} (hg : Tendsto g p (𝓝 b)) :
TendstoUniformly (fun n : ι => fun _ : α => g n) (fun _ : α => b) p | tendstoUniformly_iff_tendstoUniformlyOnFilter.mpr (hg.tendstoUniformlyOnFilter_const _) | theorem | Filter.Tendsto.tendstoUniformly_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly"
] | If a sequence `g` converges to some `b`, then the sequence of constant functions
`fun n ↦ fun a ↦ g n` converges to the constant function `fun a ↦ b`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuousOn.tendstoUniformlyOn [UniformSpace α] [UniformSpace γ] {U : Set α}
{V : Set β} {F : α → β → γ} (hF : UniformContinuousOn ↿F (U ×ˢ V)) (hU : x ∈ U) :
TendstoUniformlyOn F (F x) (𝓝[U] x) V | by
set φ := fun q : α × β => ((x, q.2), q)
rw [tendstoUniformlyOn_iff_tendsto]
change Tendsto (Prod.map ↿F ↿F ∘ φ) (𝓝[U] x ×ˢ 𝓟 V) (𝓤 γ)
simp only [nhdsWithin, Filter.prod_eq_inf, comap_inf, inf_assoc, comap_principal, inf_principal]
refine Tendsto.comp hF
(Tendsto.inf ?_ <| tendsto_principal_principal... | theorem | UniformContinuousOn.tendstoUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter.prod_eq_inf",
"Set",
"TendstoUniformlyOn",
"UniformContinuousOn",
"UniformSpace",
"nhdsWithin",
"nhds_eq_comap_uniformity",
"tendstoUniformlyOn_iff_tendsto",
"tendsto_diag_uniformity",
"uniformity_prod_eq_comap_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.tendstoUniformly [UniformSpace α] [UniformSpace γ] {U : Set α}
(hU : U ∈ 𝓝 x) {F : α → β → γ} (hF : UniformContinuousOn ↿F (U ×ˢ (univ : Set β))) :
TendstoUniformly F (F x) (𝓝 x) | by
simpa only [tendstoUniformlyOn_univ, nhdsWithin_eq_nhds.2 hU]
using hF.tendstoUniformlyOn (mem_of_mem_nhds hU) | theorem | UniformContinuousOn.tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Set",
"TendstoUniformly",
"UniformContinuousOn",
"UniformSpace",
"mem_of_mem_nhds",
"tendstoUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous₂.tendstoUniformly [UniformSpace α] [UniformSpace γ] {f : α → β → γ}
(h : UniformContinuous₂ f) : TendstoUniformly f (f x) (𝓝 x) | UniformContinuousOn.tendstoUniformly univ_mem <| by rwa [univ_prod_univ, uniformContinuousOn_univ] | theorem | UniformContinuous₂.tendstoUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"TendstoUniformly",
"UniformContinuousOn.tendstoUniformly",
"UniformContinuous₂",
"UniformSpace",
"uniformContinuousOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoUniformlyOnFilter_iff_of_uniformity {F : X → α → β} {f : α → β}
{l : Filter X} {l' : Filter α} {pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)}
(hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUniformlyOnFilter F f l l' ↔
(∀ i, pβ i → ∀ᶠ n in l ×ˢ l', (f n.2, F n.1 n.2) ∈ sβ i) | by
rw [tendstoUniformlyOnFilter_iff_tendsto, hβ.tendsto_right_iff] | lemma | Filter.HasBasis.tendstoUniformlyOnFilter_iff_of_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOnFilter",
"tendstoUniformlyOnFilter_iff_tendsto",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_right_iff` for `TendstoUniformlyOnFilter`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOnFilter_iff {F : X → α → β} {f : α → β}
{l : Filter X} {l' : Filter α} {pX : ιX → Prop} {sX : ιX → Set X}
{pα : ια → Prop} {sα : ια → Set α} {pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)}
(hl : l.HasBasis pX sX) (hl' : l'.HasBasis pα sα)
(hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUnifo... | by
simp [hβ.tendstoUniformlyOnFilter_iff_of_uniformity, (hl.prod hl').eventually_iff] | lemma | Filter.HasBasis.tendstoUniformlyOnFilter_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOnFilter",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_iff` for `TendstoUniformlyOnFilter`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_iff_of_uniformity {F : X → α → β} {f : α → β}
{l : Filter X} {s : Set α} {pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)}
(hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUniformlyOn F f l s ↔
(∀ i, pβ i → ∀ᶠ n in l, ∀ x ∈ s, (f x, F n x) ∈ sβ i) | by
simp_rw [tendstoUniformlyOn_iff_tendsto, hβ.tendsto_right_iff, eventually_prod_principal_iff] | lemma | Filter.HasBasis.tendstoUniformlyOn_iff_of_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendsto",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_right_iff` for `TendstoUniformlyOn`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformlyOn_iff {F : X → α → β} {f : α → β}
{l : Filter X} {s : Set α} {pX : ιX → Prop} {sX : ιX → Set X} {pβ : ιβ → Prop}
{sβ : ιβ → Set (β × β)} (hl : l.HasBasis pX sX) (hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUniformlyOn F f l s ↔
(∀ i, pβ i → ∃ j, pX j ∧ ∀ ⦃x⦄, x ∈ sX j → ∀ a ∈ s, (f a... | by
simp [hβ.tendstoUniformlyOn_iff_of_uniformity, hl.eventually_iff] | lemma | Filter.HasBasis.tendstoUniformlyOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformlyOn",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_iff` for `TendstoUniformlyOn`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformly_iff_of_uniformity {F : X → α → β} {f : α → β}
{l : Filter X} {pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)}
(hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUniformly F f l ↔
(∀ i, pβ i → ∀ᶠ n in l, ∀ x, (f x, F n x) ∈ sβ i) | by
simp_rw [← tendstoUniformlyOn_univ, hβ.tendstoUniformlyOn_iff_of_uniformity, mem_univ,
true_imp_iff] | lemma | Filter.HasBasis.tendstoUniformly_iff_of_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformly",
"tendstoUniformlyOn_univ",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_right_iff` for `TendstoUniformly`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoUniformly_iff {F : X → α → β} {f : α → β}
{l : Filter X} {pX : ιX → Prop} {sX : ιX → Set X} (hl : l.HasBasis pX sX)
{pβ : ιβ → Prop} {sβ : ιβ → Set (β × β)} (hβ : (uniformity β).HasBasis pβ sβ) :
TendstoUniformly F f l ↔
(∀ i, pβ i → ∃ j, pX j ∧ ∀ ⦃x⦄, x ∈ sX j → ∀ a, (f a, F x a) ∈ sβ i) | by
simp only [hβ.tendstoUniformly_iff_of_uniformity, hl.eventually_iff] | lemma | Filter.HasBasis.tendstoUniformly_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set",
"TendstoUniformly",
"uniformity"
] | An analogue of `Filter.HasBasis.tendsto_iff` for `TendstoUniformly`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformCauchySeqOnFilter (F : ι → α → β) (p : Filter ι) (p' : Filter α) : Prop | ∀ u ∈ 𝓤 β, ∀ᶠ m : (ι × ι) × α in (p ×ˢ p) ×ˢ p', (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u | def | UniformCauchySeqOnFilter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter"
] | A sequence is uniformly Cauchy if eventually all of its pairwise differences are
uniformly bounded | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformCauchySeqOn (F : ι → α → β) (p : Filter ι) (s : Set α) : Prop | ∀ u ∈ 𝓤 β, ∀ᶠ m : ι × ι in p ×ˢ p, ∀ x : α, x ∈ s → (F m.fst x, F m.snd x) ∈ u | def | UniformCauchySeqOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/UniformConvergence.lean | [] | [
"Filter",
"Set"
] | A sequence is uniformly Cauchy if eventually all of its pairwise differences are
uniformly bounded | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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