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.equicontinuousOn {F : ι → X → α} (H : Equicontinuous F)
(S : Set X) : EquicontinuousOn F S | fun x _ ↦ (H x).equicontinuousWithinAt S | lemma | Equicontinuous.equicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"EquicontinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EquicontinuousOn.mono {F : ι → X → α} {S T : Set X}
(H : EquicontinuousOn F T) (hST : S ⊆ T) : EquicontinuousOn F S | fun x hx ↦ (H x (hST hx)).mono hST | lemma | EquicontinuousOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_univ (F : ι → X → α) :
EquicontinuousOn F univ ↔ Equicontinuous F | by
simp [EquicontinuousOn, Equicontinuous] | lemma | equicontinuousOn_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"EquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_restrict_iff (F : ι → X → α) {S : Set X} :
Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S | by
simp [Equicontinuous, EquicontinuousOn, equicontinuousAt_restrict_iff] | lemma | equicontinuous_restrict_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"EquicontinuousOn",
"Set",
"equicontinuousAt_restrict_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformEquicontinuous.uniformEquicontinuousOn {F : ι → β → α} (H : UniformEquicontinuous F)
(S : Set β) : UniformEquicontinuousOn F S | fun U hU ↦ (H U hU).filter_mono inf_le_left | lemma | UniformEquicontinuous.uniformEquicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformEquicontinuous",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformEquicontinuousOn.mono {F : ι → β → α} {S T : Set β}
(H : UniformEquicontinuousOn F T) (hST : S ⊆ T) : UniformEquicontinuousOn F S | fun U hU ↦ (H U hU).filter_mono <| by gcongr | lemma | UniformEquicontinuousOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_univ (F : ι → β → α) :
UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F | by
simp [UniformEquicontinuousOn, UniformEquicontinuous] | lemma | uniformEquicontinuousOn_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformEquicontinuous",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_restrict_iff (F : ι → β → α) {S : Set β} :
UniformEquicontinuous (S.restrict ∘ F) ↔ UniformEquicontinuousOn F S | by
rw [UniformEquicontinuous, UniformEquicontinuousOn]
conv in _ ⊓ _ => rw [← Subtype.range_val (s := S), ← range_prodMap, ← map_comap]
rfl | lemma | uniformEquicontinuous_restrict_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"Subtype.range_val",
"UniformEquicontinuous",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
EquicontinuousAt F x₀ | fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim) | lemma | equicontinuousAt_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"IsEmpty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_empty [h : IsEmpty ι] (F : ι → X → α) (S : Set X) (x₀ : X) :
EquicontinuousWithinAt F S x₀ | fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim) | lemma | equicontinuousWithinAt_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"IsEmpty",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
Equicontinuous F | equicontinuousAt_empty F | lemma | equicontinuous_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"IsEmpty",
"equicontinuousAt_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_empty [IsEmpty ι] (F : ι → X → α) (S : Set X) :
EquicontinuousOn F S | fun x₀ _ ↦ equicontinuousWithinAt_empty F S x₀ | lemma | equicontinuousOn_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"IsEmpty",
"Set",
"equicontinuousWithinAt_empty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
UniformEquicontinuous F | fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim) | lemma | uniformEquicontinuous_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"IsEmpty",
"UniformEquicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_empty [h : IsEmpty ι] (F : ι → β → α) (S : Set β) :
UniformEquicontinuousOn F S | fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim) | lemma | uniformEquicontinuousOn_empty | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"IsEmpty",
"Set",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ | by
simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff,
UniformSpace.ball, @forall_comm _ ι] | theorem | equicontinuousAt_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Finite",
"UniformSpace.ball",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_finite [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X} :
EquicontinuousWithinAt F S x₀ ↔ ∀ i, ContinuousWithinAt (F i) S x₀ | by
simp [EquicontinuousWithinAt, ContinuousWithinAt,
(nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff, UniformSpace.ball,
@forall_comm _ ι] | theorem | equicontinuousWithinAt_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Finite",
"Set",
"UniformSpace.ball",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_finite [Finite ι] {F : ι → X → α} :
Equicontinuous F ↔ ∀ i, Continuous (F i) | by
simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_comm ι] | theorem | equicontinuous_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Finite",
"continuous_iff_continuousAt",
"equicontinuousAt_finite"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_finite [Finite ι] {F : ι → X → α} {S : Set X} :
EquicontinuousOn F S ↔ ∀ i, ContinuousOn (F i) S | by
simp only [EquicontinuousOn, equicontinuousWithinAt_finite, ContinuousOn, @forall_comm ι] | theorem | equicontinuousOn_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"EquicontinuousOn",
"Finite",
"Set",
"equicontinuousWithinAt_finite"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) | by
simp only [UniformEquicontinuous, eventually_all, @forall_comm _ ι]; rfl | theorem | uniformEquicontinuous_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Finite",
"UniformContinuous",
"UniformEquicontinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_finite [Finite ι] {F : ι → β → α} {S : Set β} :
UniformEquicontinuousOn F S ↔ ∀ i, UniformContinuousOn (F i) S | by
simp only [UniformEquicontinuousOn, eventually_all, @forall_comm _ ι]; rfl | theorem | uniformEquicontinuousOn_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Finite",
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} :
EquicontinuousAt F x ↔ ContinuousAt (F default) x | equicontinuousAt_finite.trans Unique.forall_iff | theorem | equicontinuousAt_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_unique [Unique ι] {F : ι → X → α} {S : Set X} {x : X} :
EquicontinuousWithinAt F S x ↔ ContinuousWithinAt (F default) S x | equicontinuousWithinAt_finite.trans Unique.forall_iff | theorem | equicontinuousWithinAt_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_unique [Unique ι] {F : ι → X → α} :
Equicontinuous F ↔ Continuous (F default) | equicontinuous_finite.trans Unique.forall_iff | theorem | equicontinuous_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_unique [Unique ι] {F : ι → X → α} {S : Set X} :
EquicontinuousOn F S ↔ ContinuousOn (F default) S | equicontinuousOn_finite.trans Unique.forall_iff | theorem | equicontinuousOn_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"EquicontinuousOn",
"Set",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (F default) | uniformEquicontinuous_finite.trans Unique.forall_iff | theorem | uniformEquicontinuous_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformContinuous",
"UniformEquicontinuous",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_unique [Unique ι] {F : ι → β → α} {S : Set β} :
UniformEquicontinuousOn F S ↔ UniformContinuousOn (F default) S | uniformEquicontinuousOn_finite.trans Unique.forall_iff | theorem | uniformEquicontinuousOn_unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"Unique",
"Unique.forall_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_iff_pair {F : ι → X → α} {S : Set X} {x₀ : X} (hx₀ : x₀ ∈ S) :
EquicontinuousWithinAt F S x₀ ↔
∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝[S] x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U | by
constructor <;> intro H U hU
· rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩
refine ⟨_, H V hV, fun x hx y hy i => hVU (SetRel.prodMk_mem_comp ?_ (hy i))⟩
exact SetRel.symm V (hx i)
· rcases H U hU with ⟨V, hV, hVU⟩
filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhds... | theorem | equicontinuousWithinAt_iff_pair | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set",
"SetRel.prodMk_mem_comp",
"SetRel.symm",
"comp_symm_mem_uniformity_sets",
"mem_of_mem_nhdsWithin"
] | Reformulation of equicontinuity at `x₀` within a set `S`, comparing two variables near `x₀`
instead of comparing only one with `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousAt_iff_pair {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔
∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ i, (F i x, F i y) ∈ U | by
simp_rw [← equicontinuousWithinAt_univ, equicontinuousWithinAt_iff_pair (mem_univ x₀),
nhdsWithin_univ] | theorem | equicontinuousAt_iff_pair | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"equicontinuousWithinAt_iff_pair",
"equicontinuousWithinAt_univ",
"nhdsWithin_univ"
] | Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
only one with `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuous.equicontinuous {F : ι → β → α} (h : UniformEquicontinuous F) :
Equicontinuous F | fun x₀ U hU ↦
mem_of_superset (ball_mem_nhds x₀ (h U hU)) fun _ hx i ↦ hx i | theorem | UniformEquicontinuous.equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"UniformEquicontinuous"
] | Uniform equicontinuity implies equicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuousOn.equicontinuousOn {F : ι → β → α} {S : Set β}
(h : UniformEquicontinuousOn F S) :
EquicontinuousOn F S | fun _ hx₀ U hU ↦
mem_of_superset (ball_mem_nhdsWithin hx₀ (h U hU)) fun _ hx i ↦ hx i | theorem | UniformEquicontinuousOn.equicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set",
"UniformEquicontinuousOn"
] | Uniform equicontinuity on a subset implies equicontinuity on that subset. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousAt.continuousAt {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (i : ι) :
ContinuousAt (F i) x₀ | (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i | theorem | EquicontinuousAt.continuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"UniformSpace.hasBasis_nhds"
] | Each function of a family equicontinuous at `x₀` is continuous at `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousWithinAt.continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X}
(h : EquicontinuousWithinAt F S x₀) (i : ι) :
ContinuousWithinAt (F i) S x₀ | (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i | theorem | EquicontinuousWithinAt.continuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"UniformSpace.hasBasis_nhds"
] | Each function of a family equicontinuous at `x₀` within `S` is continuous at `x₀` within `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousAt.continuousAt_of_mem {H : Set <| X → α} {x₀ : X}
(h : H.EquicontinuousAt x₀) {f : X → α} (hf : f ∈ H) : ContinuousAt f x₀ | h.continuousAt ⟨f, hf⟩ | theorem | Set.EquicontinuousAt.continuousAt_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EquicontinuousWithinAt.continuousWithinAt_of_mem {H : Set <| X → α}
{S : Set X} {x₀ : X} (h : H.EquicontinuousWithinAt S x₀) {f : X → α} (hf : f ∈ H) :
ContinuousWithinAt f S x₀ | h.continuousWithinAt ⟨f, hf⟩ | theorem | Set.EquicontinuousWithinAt.continuousWithinAt_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Equicontinuous.continuous {F : ι → X → α} (h : Equicontinuous F) (i : ι) :
Continuous (F i) | continuous_iff_continuousAt.mpr fun x => (h x).continuousAt i | theorem | Equicontinuous.continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous"
] | Each function of an equicontinuous family is continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.continuousOn {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S)
(i : ι) : ContinuousOn (F i) S | fun x hx ↦ (h x hx).continuousWithinAt i | theorem | EquicontinuousOn.continuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"EquicontinuousOn",
"Set"
] | Each function of a family equicontinuous on `S` is continuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Equicontinuous.continuous_of_mem {H : Set <| X → α} (h : H.Equicontinuous)
{f : X → α} (hf : f ∈ H) : Continuous f | h.continuous ⟨f, hf⟩ | theorem | Set.Equicontinuous.continuous_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EquicontinuousOn.continuousOn_of_mem {H : Set <| X → α} {S : Set X}
(h : H.EquicontinuousOn S) {f : X → α} (hf : f ∈ H) : ContinuousOn f S | h.continuousOn ⟨f, hf⟩ | theorem | Set.EquicontinuousOn.continuousOn_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformEquicontinuous.uniformContinuous {F : ι → β → α} (h : UniformEquicontinuous F)
(i : ι) : UniformContinuous (F i) | fun U hU =>
mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i) | theorem | UniformEquicontinuous.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformContinuous",
"UniformEquicontinuous"
] | Each function of a uniformly equicontinuous family is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuousOn.uniformContinuousOn {F : ι → β → α} {S : Set β}
(h : UniformEquicontinuousOn F S) (i : ι) :
UniformContinuousOn (F i) S | fun U hU =>
mem_map.mpr (mem_of_superset (h U hU) fun _ hxy => hxy i) | theorem | UniformEquicontinuousOn.uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn"
] | Each function of a family uniformly equicontinuous on `S` is uniformly continuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuous.uniformContinuous_of_mem {H : Set <| β → α}
(h : H.UniformEquicontinuous) {f : β → α} (hf : f ∈ H) : UniformContinuous f | h.uniformContinuous ⟨f, hf⟩ | theorem | Set.UniformEquicontinuous.uniformContinuous_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem {H : Set <| β → α}
{S : Set β} (h : H.UniformEquicontinuousOn S) {f : β → α} (hf : f ∈ H) :
UniformContinuousOn f S | h.uniformContinuousOn ⟨f, hf⟩ | theorem | Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EquicontinuousAt.comp {F : ι → X → α} {x₀ : X} (h : EquicontinuousAt F x₀) (u : κ → ι) :
EquicontinuousAt (F ∘ u) x₀ | fun U hU => (h U hU).mono fun _ H k => H (u k) | theorem | EquicontinuousAt.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt"
] | Taking sub-families preserves equicontinuity at a point. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousWithinAt.comp {F : ι → X → α} {S : Set X} {x₀ : X}
(h : EquicontinuousWithinAt F S x₀) (u : κ → ι) :
EquicontinuousWithinAt (F ∘ u) S x₀ | fun U hU ↦ (h U hU).mono fun _ H k => H (u k) | theorem | EquicontinuousWithinAt.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set"
] | Taking sub-families preserves equicontinuity at a point within a subset. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousAt.mono {H H' : Set <| X → α} {x₀ : X}
(h : H.EquicontinuousAt x₀) (hH : H' ⊆ H) : H'.EquicontinuousAt x₀ | h.comp (inclusion hH) | theorem | Set.EquicontinuousAt.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EquicontinuousWithinAt.mono {H H' : Set <| X → α} {S : Set X} {x₀ : X}
(h : H.EquicontinuousWithinAt S x₀) (hH : H' ⊆ H) : H'.EquicontinuousWithinAt S x₀ | h.comp (inclusion hH) | theorem | Set.EquicontinuousWithinAt.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
Equicontinuous (F ∘ u) | fun x => (h x).comp u | theorem | Equicontinuous.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous"
] | Taking sub-families preserves equicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.comp {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (u : κ → ι) :
EquicontinuousOn (F ∘ u) S | fun x hx ↦ (h x hx).comp u | theorem | EquicontinuousOn.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set"
] | Taking sub-families preserves equicontinuity on a subset. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
(hH : H' ⊆ H) : H'.Equicontinuous | h.comp (inclusion hH) | theorem | Set.Equicontinuous.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.EquicontinuousOn.mono {H H' : Set <| X → α} {S : Set X}
(h : H.EquicontinuousOn S) (hH : H' ⊆ H) : H'.EquicontinuousOn S | h.comp (inclusion hH) | theorem | Set.EquicontinuousOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformEquicontinuous.comp {F : ι → β → α} (h : UniformEquicontinuous F) (u : κ → ι) :
UniformEquicontinuous (F ∘ u) | fun U hU => (h U hU).mono fun _ H k => H (u k) | theorem | UniformEquicontinuous.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformEquicontinuous"
] | Taking sub-families preserves uniform equicontinuity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformEquicontinuousOn.comp {F : ι → β → α} {S : Set β} (h : UniformEquicontinuousOn F S)
(u : κ → ι) : UniformEquicontinuousOn (F ∘ u) S | fun U hU ↦ (h U hU).mono fun _ H k => H (u k) | theorem | UniformEquicontinuousOn.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformEquicontinuousOn"
] | Taking sub-families preserves uniform equicontinuity on a subset. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
(hH : H' ⊆ H) : H'.UniformEquicontinuous | h.comp (inclusion hH) | theorem | Set.UniformEquicontinuous.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Set.UniformEquicontinuousOn.mono {H H' : Set <| β → α} {S : Set β}
(h : H.UniformEquicontinuousOn S) (hH : H' ⊆ H) : H'.UniformEquicontinuousOn S | h.comp (inclusion hH) | theorem | Set.UniformEquicontinuousOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ | by
simp only [EquicontinuousAt, forall_subtype_range_iff] | theorem | equicontinuousAt_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousWithinAt_iff_range {F : ι → X → α} {S : Set X} {x₀ : X} :
EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((↑) : range F → X → α) S x₀ | by
simp only [EquicontinuousWithinAt, forall_subtype_range_iff] | theorem | equicontinuousWithinAt_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff `range 𝓕` is equicontinuous
at `x₀` within `S`, i.e the family `(↑) : range F → X → α` is equicontinuous at `x₀` within `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuous_iff_range {F : ι → X → α} :
Equicontinuous F ↔ Equicontinuous ((↑) : range F → X → α) | forall_congr' fun _ => equicontinuousAt_iff_range | theorem | equicontinuous_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"equicontinuousAt_iff_range"
] | A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
i.e the family `(↑) : range F → X → α` is equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousOn_iff_range {F : ι → X → α} {S : Set X} :
EquicontinuousOn F S ↔ EquicontinuousOn ((↑) : range F → X → α) S | forall_congr' fun _ ↦ forall_congr' fun _ ↦ equicontinuousWithinAt_iff_range | theorem | equicontinuousOn_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set",
"equicontinuousWithinAt_iff_range"
] | A family `𝓕 : ι → X → α` is equicontinuous on `S` iff `range 𝓕` is equicontinuous on `S`,
i.e the family `(↑) : range F → X → α` is equicontinuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformEquicontinuous_iff_range {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformEquicontinuous ((↑) : range F → β → α) | ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h =>
h.comp (rangeFactorization F)⟩ | theorem | uniformEquicontinuous_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformEquicontinuous"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
i.e the family `(↑) : range F → β → α` is uniformly equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformEquicontinuousOn_iff_range {F : ι → β → α} {S : Set β} :
UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((↑) : range F → β → α) S | ⟨fun h => by rw [← comp_rangeSplitting F]; exact h.comp _, fun h =>
h.comp (rangeFactorization F)⟩ | theorem | uniformEquicontinuousOn_iff_range | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformEquicontinuousOn"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff `range 𝓕` is uniformly
equicontinuous on `S`, i.e the family `(↑) : range F → β → α` is uniformly equicontinuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ ContinuousAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) x₀ | by
rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff]
rfl | theorem | equicontinuousAt_iff_continuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Function.swap",
"UniformFun.hasBasis_nhds"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developing the equicontinuity API, but it should not be used directly for other
purposes. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousWithinAt_iff_continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} :
EquicontinuousWithinAt F S x₀ ↔
ContinuousWithinAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) S x₀ | by
rw [ContinuousWithinAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff]
rfl | theorem | equicontinuousWithinAt_iff_continuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Function.swap",
"Set",
"UniformFun.hasBasis_nhds"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff the function
`swap 𝓕 : X → ι → α` is continuous at `x₀` within `S`
*when `ι → α` is equipped with the topology of uniform convergence*. This is very useful for
developing the equicontinuity API, but it should not be used directly for other purposes. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuous_iff_continuous {F : ι → X → α} :
Equicontinuous F ↔ Continuous (ofFun ∘ Function.swap F : X → ι →ᵤ α) | by
simp_rw [Equicontinuous, continuous_iff_continuousAt, equicontinuousAt_iff_continuousAt] | theorem | equicontinuous_iff_continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"Equicontinuous",
"Function.swap",
"continuous_iff_continuousAt",
"equicontinuousAt_iff_continuousAt"
] | A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developing the equicontinuity API, but it should not be used directly for other
purposes. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousOn_iff_continuousOn {F : ι → X → α} {S : Set X} :
EquicontinuousOn F S ↔ ContinuousOn (ofFun ∘ Function.swap F : X → ι →ᵤ α) S | by
simp_rw [EquicontinuousOn, ContinuousOn, equicontinuousWithinAt_iff_continuousWithinAt] | theorem | equicontinuousOn_iff_continuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousOn",
"EquicontinuousOn",
"Function.swap",
"Set",
"equicontinuousWithinAt_iff_continuousWithinAt"
] | A family `𝓕 : ι → X → α` is equicontinuous on `S` iff the function `swap 𝓕 : X → ι → α` is
continuous on `S` *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developing the equicontinuity API, but it should not be used directly for other
purposes. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) | by
rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl | theorem | uniformEquicontinuous_iff_uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Function.swap",
"UniformContinuous",
"UniformEquicontinuous",
"UniformFun.hasBasis_uniformity"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
This is very useful for developing the equicontinuity API, but it should not be used directly
for other purposes. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformEquicontinuousOn_iff_uniformContinuousOn {F : ι → β → α} {S : Set β} :
UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S | by
rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl | theorem | uniformEquicontinuousOn_iff_uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Function.swap",
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformFun.hasBasis_uniformity"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous on `S` iff the function
`swap 𝓕 : β → ι → α` is uniformly continuous on `S`
*when `ι → α` is equipped with the uniform structure of uniform convergence*. This is very useful
for developing the equicontinuity API, but it should not be used directly for other purpose... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equicontinuousWithinAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
{S : Set X} {x₀ : X} : EquicontinuousWithinAt (uα := ⨅ k, u k) F S x₀ ↔
∀ k, EquicontinuousWithinAt (uα := u k) F S x₀ | by
simp +instances only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace]
unfold ContinuousWithinAt
rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf] | theorem | equicontinuousWithinAt_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"UniformFun.iInf_eq",
"UniformSpace",
"equicontinuousWithinAt_iff_continuousWithinAt",
"nhds_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
{x₀ : X} :
EquicontinuousAt (uα := ⨅ k, u k) F x₀ ↔ ∀ k, EquicontinuousAt (uα := u k) F x₀ | by
simp only [← equicontinuousWithinAt_univ (uα := _), equicontinuousWithinAt_iInf_rng] | theorem | equicontinuousAt_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"UniformSpace",
"equicontinuousWithinAt_iInf_rng",
"equicontinuousWithinAt_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'} :
Equicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, Equicontinuous (uα := u k) F | by
simp_rw +instances [equicontinuous_iff_continuous (uα := _), UniformFun.topologicalSpace]
rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng] | theorem | equicontinuous_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"UniformFun.iInf_eq",
"UniformFun.topologicalSpace",
"UniformSpace",
"continuous_iInf_rng",
"equicontinuous_iff_continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → X → α'}
{S : Set X} :
EquicontinuousOn (uα := ⨅ k, u k) F S ↔ ∀ k, EquicontinuousOn (uα := u k) F S | by
simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_comm _ κ] | theorem | equicontinuousOn_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set",
"UniformSpace",
"equicontinuousWithinAt_iInf_rng"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'} :
UniformEquicontinuous (uα := ⨅ k, u k) F ↔ ∀ k, UniformEquicontinuous (uα := u k) F | by
simp_rw [uniformEquicontinuous_iff_uniformContinuous (uα := _)]
rw [UniformFun.iInf_eq, uniformContinuous_iInf_rng] | theorem | uniformEquicontinuous_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformEquicontinuous",
"UniformFun.iInf_eq",
"UniformSpace",
"uniformContinuous_iInf_rng",
"uniformEquicontinuous_iff_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_iInf_rng {u : κ → UniformSpace α'} {F : ι → β → α'}
{S : Set β} : UniformEquicontinuousOn (uα := ⨅ k, u k) F S ↔
∀ k, UniformEquicontinuousOn (uα := u k) F S | by
simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uα := _)]
unfold UniformContinuousOn
rw [UniformFun.iInf_eq, iInf_uniformity, tendsto_iInf] | theorem | uniformEquicontinuousOn_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformFun.iInf_eq",
"UniformSpace",
"iInf_uniformity",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
{S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) :
EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ | by
simp only [equicontinuousWithinAt_iff_continuousWithinAt (tX := _)] at hk ⊢
unfold ContinuousWithinAt nhdsWithin at hk ⊢
rw [nhds_iInf]
exact hk.mono_left <| inf_le_inf_right _ <| iInf_le _ k | theorem | equicontinuousWithinAt_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"TopologicalSpace",
"equicontinuousWithinAt_iff_continuousWithinAt",
"nhdsWithin",
"nhds_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
{x₀ : X'} {k : κ} (hk : EquicontinuousAt (tX := t k) F x₀) :
EquicontinuousAt (tX := ⨅ k, t k) F x₀ | by
rw [← equicontinuousWithinAt_univ (tX := _)] at hk ⊢
exact equicontinuousWithinAt_iInf_dom hk | theorem | equicontinuousAt_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"TopologicalSpace",
"equicontinuousWithinAt_iInf_dom",
"equicontinuousWithinAt_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuous_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
{k : κ} (hk : Equicontinuous (tX := t k) F) :
Equicontinuous (tX := ⨅ k, t k) F | fun x ↦ equicontinuousAt_iInf_dom (hk x) | theorem | equicontinuous_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"TopologicalSpace",
"equicontinuousAt_iInf_dom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousOn_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
{S : Set X'} {k : κ} (hk : EquicontinuousOn (tX := t k) F S) :
EquicontinuousOn (tX := ⨅ k, t k) F S | fun x hx ↦ equicontinuousWithinAt_iInf_dom (hk x hx) | theorem | equicontinuousOn_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set",
"TopologicalSpace",
"equicontinuousWithinAt_iInf_dom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuous_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
{k : κ} (hk : UniformEquicontinuous (uβ := u k) F) :
UniformEquicontinuous (uβ := ⨅ k, u k) F | by
simp_rw [uniformEquicontinuous_iff_uniformContinuous (uβ := _)] at hk ⊢
exact uniformContinuous_iInf_dom hk | theorem | uniformEquicontinuous_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"UniformEquicontinuous",
"UniformSpace",
"uniformContinuous_iInf_dom",
"uniformEquicontinuous_iff_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformEquicontinuousOn_iInf_dom {u : κ → UniformSpace β'} {F : ι → β' → α}
{S : Set β'} {k : κ} (hk : UniformEquicontinuousOn (uβ := u k) F S) :
UniformEquicontinuousOn (uβ := ⨅ k, u k) F S | by
simp_rw [uniformEquicontinuousOn_iff_uniformContinuousOn (uβ := _)] at hk ⊢
unfold UniformContinuousOn
rw [iInf_uniformity]
exact hk.mono_left <| inf_le_inf_right _ <| iInf_le _ k | theorem | uniformEquicontinuousOn_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformSpace",
"iInf_uniformity",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousAt_iff_left {p : κ → Prop} {s : κ → Set X}
{F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U | by
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
rfl | theorem | Filter.HasBasis.equicontinuousAt_iff_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Set",
"UniformFun.hasBasis_nhds",
"equicontinuousAt_iff_continuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousWithinAt_iff_left {p : κ → Prop} {s : κ → Set X}
{F : ι → X → α} {S : Set X} {x₀ : X} (hX : (𝓝[S] x₀).HasBasis p s) :
EquicontinuousWithinAt F S x₀ ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U | by
rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds ι α _)]
rfl | theorem | Filter.HasBasis.equicontinuousWithinAt_iff_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"UniformFun.hasBasis_nhds",
"equicontinuousWithinAt_iff_continuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k | by
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
rfl | theorem | Filter.HasBasis.equicontinuousAt_iff_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Set",
"UniformFun.hasBasis_nhds_of_basis",
"equicontinuousAt_iff_continuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k | by
rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
rfl | theorem | Filter.HasBasis.equicontinuousWithinAt_iff_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"UniformFun.hasBasis_nhds_of_basis",
"equicontinuousWithinAt_iff_continuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
{p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
(hα : (𝓤 α).HasBasis p₂ s₂) :
EquicontinuousAt F x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈... | by
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl | theorem | Filter.HasBasis.equicontinuousAt_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousAt",
"EquicontinuousAt",
"Set",
"UniformFun.hasBasis_nhds_of_basis",
"equicontinuousAt_iff_continuousAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.equicontinuousWithinAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
{s₁ : κ₁ → Set X} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X}
(hX : (𝓝[S] x₀).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
EquicontinuousWithinAt F S x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ ... | by
rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl | theorem | Filter.HasBasis.equicontinuousWithinAt_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"ContinuousWithinAt",
"EquicontinuousWithinAt",
"Set",
"UniformFun.hasBasis_nhds_of_basis",
"equicontinuousWithinAt_iff_continuousWithinAt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuous_iff_left {p : κ → Prop}
{s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
UniformEquicontinuous F ↔
∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U | by
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl | theorem | Filter.HasBasis.uniformEquicontinuous_iff_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuous",
"UniformEquicontinuous",
"UniformFun.hasBasis_uniformity",
"uniformEquicontinuous_iff_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuousOn_iff_left {p : κ → Prop}
{s : κ → Set (β × β)} {F : ι → β → α} {S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p s) :
UniformEquicontinuousOn F S ↔
∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U | by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl | theorem | Filter.HasBasis.uniformEquicontinuousOn_iff_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformFun.hasBasis_uniformity",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuous_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → β → α} (hα : (𝓤 α).HasBasis p s) :
UniformEquicontinuous F ↔ ∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k | by
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl | theorem | Filter.HasBasis.uniformEquicontinuous_iff_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuous",
"UniformEquicontinuous",
"UniformFun.hasBasis_uniformity_of_basis",
"uniformEquicontinuous_iff_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (𝓤 α).HasBasis p s) :
UniformEquicontinuousOn F S ↔
∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ s k | by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl | theorem | Filter.HasBasis.uniformEquicontinuousOn_iff_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformFun.hasBasis_uniformity_of_basis",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
{s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
(hβ : (𝓤 β).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
UniformEquicontinuous F ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F ... | by
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
simp only [Prod.forall]
rfl | theorem | Filter.HasBasis.uniformEquicontinuous_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuous",
"UniformEquicontinuous",
"UniformFun.hasBasis_uniformity_of_basis",
"uniformEquicontinuous_iff_uniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformEquicontinuousOn_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
{s₁ : κ₁ → Set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → β → α}
{S : Set β} (hβ : (𝓤 β ⊓ 𝓟 (S ×ˢ S)).HasBasis p₁ s₁) (hα : (𝓤 α).HasBasis p₂ s₂) :
UniformEquicontinuousOn F S ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ... | by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
simp only [Prod.forall]
rfl | theorem | Filter.HasBasis.uniformEquicontinuousOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set",
"UniformContinuousOn",
"UniformEquicontinuousOn",
"UniformFun.hasBasis_uniformity_of_basis",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
(hu : IsUniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ | by
have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).isInducing
rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
rfl | theorem | IsUniformInducing.equicontinuousAt_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"IsUniformInducing",
"UniformFun.postcomp_isUniformInducing",
"equicontinuousAt_iff_continuousAt"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
`x₀ : X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
(hu : IsUniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ | by
have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).isInducing
simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff]
rfl | lemma | IsUniformInducing.equicontinuousWithinAt_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"IsUniformInducing",
"Set",
"UniformFun.postcomp_isUniformInducing",
"equicontinuousWithinAt_iff_continuousWithinAt"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
`x₀ : X` within a subset `S : Set X` iff the family `𝓕'`, obtained by composing each function
of `𝓕` by `u`, is equicontinuous at `x₀` within `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : IsUniformInducing u) :
Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) | by
congrm ∀ x, ?_
rw [hu.equicontinuousAt_iff] | lemma | IsUniformInducing.equicontinuous_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"IsUniformInducing"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β}
(hu : IsUniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S | by
congrm ∀ x ∈ S, ?_
rw [hu.equicontinuousWithinAt_iff] | theorem | IsUniformInducing.equicontinuousOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"IsUniformInducing",
"Set"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous on a
subset `S : Set X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
equicontinuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
(hu : IsUniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) | by
have := UniformFun.postcomp_isUniformInducing (α := ι) hu
simp only [uniformEquicontinuous_iff_uniformContinuous, this.uniformContinuous_iff]
rfl | theorem | IsUniformInducing.uniformEquicontinuous_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"IsUniformInducing",
"UniformEquicontinuous",
"UniformFun.postcomp_isUniformInducing",
"uniformEquicontinuous_iff_uniformContinuous"
] | Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is uniformly
equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsUniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
(hu : IsUniformInducing u) :
UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S | by
have := UniformFun.postcomp_isUniformInducing (α := ι) hu
simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
rfl | theorem | IsUniformInducing.uniformEquicontinuousOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"IsUniformInducing",
"Set",
"UniformEquicontinuousOn",
"UniformFun.postcomp_isUniformInducing",
"uniformEquicontinuousOn_iff_uniformContinuousOn"
] | Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
on a subset `S : Set β` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`,
is uniformly equicontinuous on `S`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousWithinAt.closure' {A : Set Y} {u : Y → X → α} {S : Set X} {x₀ : X}
(hA : EquicontinuousWithinAt (u ∘ (↑) : A → X → α) S x₀) (hu₁ : Continuous (S.restrict ∘ u))
(hu₂ : Continuous (eval x₀ ∘ u)) :
EquicontinuousWithinAt (u ∘ (↑) : closure A → X → α) S x₀ | by
intro U hU
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [hA V hV, eventually_mem_nhdsWithin] with x hx hxS
rw [SetCoe.forall] at *
change A ⊆ (fun f => (u f x₀, u f x)) ⁻¹' V at hx
refine (closure_minimal hx <| hVclosed.preimage <| hu₂.prodMk ?_).trans (preimage_mono hVU)
... | theorem | EquicontinuousWithinAt.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"EquicontinuousWithinAt",
"Set",
"SetCoe.forall",
"closure",
"closure_minimal",
"continuous_apply",
"eventually_mem_nhdsWithin",
"mem_uniformity_isClosed",
"trans"
] | If a set of functions is equicontinuous at some `x₀` within a set `S`, the same is true for its
closure in *any* topology for which evaluation at any `x ∈ S ∪ {x₀}` 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 ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousAt.closure' {A : Set Y} {u : Y → X → α} {x₀ : X}
(hA : EquicontinuousAt (u ∘ (↑) : A → X → α) x₀) (hu : Continuous u) :
EquicontinuousAt (u ∘ (↑) : closure A → X → α) x₀ | by
rw [← equicontinuousWithinAt_univ] at hA ⊢
exact hA.closure' (Pi.continuous_restrict _ |>.comp hu) (continuous_apply x₀ |>.comp hu) | theorem | EquicontinuousAt.closure' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Continuous",
"EquicontinuousAt",
"Pi.continuous_restrict",
"Set",
"closure",
"continuous_apply",
"equicontinuousWithinAt_univ"
] | If a set of functions is equicontinuous at some `x₀`, 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.... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X}
(hA : A.EquicontinuousAt x₀) : (closure A).EquicontinuousAt x₀ | hA.closure' (u := id) continuous_id | theorem | Set.EquicontinuousAt.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"Set",
"closure",
"continuous_id"
] | If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
also equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousWithinAt.closure {A : Set (X → α)} {S : Set X} {x₀ : X}
(hA : A.EquicontinuousWithinAt S x₀) :
(closure A).EquicontinuousWithinAt S x₀ | hA.closure' (u := id) (Pi.continuous_restrict _) (continuous_apply _) | theorem | Set.EquicontinuousWithinAt.closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Pi.continuous_restrict",
"Set",
"closure",
"continuous_apply"
] | If a set of functions is equicontinuous at some `x₀` within a set `S`, its closure for the
product topology is also equicontinuous at `x₀` within `S`. This would also be true for the coarser
topology of pointwise convergence on `S ∪ {x₀}`, see `Set.EquicontinuousWithinAt.closure'`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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