fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn.mono | null |
uniformEquicontinuousOn_univ (F : ι → β → α) :
UniformEquicontinuousOn F univ ↔ UniformEquicontinuous F := by
simp [UniformEquicontinuousOn, UniformEquicontinuous] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_univ | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_restrict_iff | null |
@[simp]
equicontinuousAt_empty [h : IsEmpty ι] (F : ι → X → α) (x₀ : X) :
EquicontinuousAt F x₀ :=
fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim)
@[simp] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_empty | null |
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)
@[simp] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_empty | null |
equicontinuous_empty [IsEmpty ι] (F : ι → X → α) :
Equicontinuous F :=
equicontinuousAt_empty F
@[simp] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_empty | null |
equicontinuousOn_empty [IsEmpty ι] (F : ι → X → α) (S : Set X) :
EquicontinuousOn F S :=
fun x₀ _ ↦ equicontinuousWithinAt_empty F S x₀
@[simp] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_empty | null |
uniformEquicontinuous_empty [h : IsEmpty ι] (F : ι → β → α) :
UniformEquicontinuous F :=
fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim)
@[simp] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_empty | null |
uniformEquicontinuousOn_empty [h : IsEmpty ι] (F : ι → β → α) (S : Set β) :
UniformEquicontinuousOn F S :=
fun _ _ ↦ Eventually.of_forall (fun _ ↦ h.elim)
/-! | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_empty | null |
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_swap _ ι] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_finite | null |
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_swap _ ι] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_finite | null |
equicontinuous_finite [Finite ι] {F : ι → X → α} :
Equicontinuous F ↔ ∀ i, Continuous (F i) := by
simp only [Equicontinuous, equicontinuousAt_finite, continuous_iff_continuousAt, @forall_swap ι] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_finite | null |
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_swap ι] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_finite | null |
uniformEquicontinuous_finite [Finite ι] {F : ι → β → α} :
UniformEquicontinuous F ↔ ∀ i, UniformContinuous (F i) := by
simp only [UniformEquicontinuous, eventually_all, @forall_swap _ ι]; rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_finite | null |
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_swap _ ι]; rfl
/-! | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_finite | null |
equicontinuousAt_unique [Unique ι] {F : ι → X → α} {x : X} :
EquicontinuousAt F x ↔ ContinuousAt (F default) x :=
equicontinuousAt_finite.trans Unique.forall_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_unique | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_unique | null |
equicontinuous_unique [Unique ι] {F : ι → X → α} :
Equicontinuous F ↔ Continuous (F default) :=
equicontinuous_finite.trans Unique.forall_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_unique | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_unique | null |
uniformEquicontinuous_unique [Unique ι] {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (F default) :=
uniformEquicontinuous_finite.trans Unique.forall_iff | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_unique | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_unique | null |
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 (prodMk_mem_compRel ?_ (hy i))⟩
exact hVsymm.mk_mem_comm.mp (hx i)
· rcases H U hU with ⟨V, hV, hVU⟩
filter_upwards [hV] using fun x hx i => hVU x₀ (mem_of_mem_nhdsWithin hx₀ hV) x hx i | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_iff_pair | Reformulation of equicontinuity at `x₀` within a set `S`, comparing two variables near `x₀`
instead of comparing only one with `x₀`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_iff_pair | Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
only one with `x₀`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous.equicontinuous | Uniform equicontinuity implies equicontinuity. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn.equicontinuousOn | Uniform equicontinuity on a subset implies equicontinuity on that subset. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt.continuousAt | Each function of a family equicontinuous at `x₀` is continuous at `x₀`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousWithinAt.continuousWithinAt | Each function of a family equicontinuous at `x₀` within `S` is continuous at `x₀` within `S`. |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousAt.continuousAt_of_mem | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousWithinAt.continuousWithinAt_of_mem | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.continuous | Each function of an equicontinuous family is continuous. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousOn.continuousOn | Each function of a family equicontinuous on `S` is continuous on `S`. |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.Equicontinuous.continuous_of_mem | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousOn.continuousOn_of_mem | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous.uniformContinuous | Each function of a uniformly equicontinuous family is uniformly continuous. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn.uniformContinuousOn | Each function of a family uniformly equicontinuous on `S` is uniformly continuous on `S`. |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuous.uniformContinuous_of_mem | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuousOn.uniformContinuousOn_of_mem | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt.comp | Taking sub-families preserves equicontinuity at a point. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousWithinAt.comp | Taking sub-families preserves equicontinuity at a point within a subset. |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousAt.mono | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousWithinAt.mono | null |
Equicontinuous.comp {F : ι → X → α} (h : Equicontinuous F) (u : κ → ι) :
Equicontinuous (F ∘ u) := fun x => (h x).comp u | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.comp | Taking sub-families preserves equicontinuity. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousOn.comp | Taking sub-families preserves equicontinuity on a subset. |
protected Set.Equicontinuous.mono {H H' : Set <| X → α} (h : H.Equicontinuous)
(hH : H' ⊆ H) : H'.Equicontinuous :=
h.comp (inclusion hH) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.Equicontinuous.mono | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousOn.mono | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous.comp | Taking sub-families preserves uniform equicontinuity. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn.comp | Taking sub-families preserves uniform equicontinuity on a subset. |
protected Set.UniformEquicontinuous.mono {H H' : Set <| β → α} (h : H.UniformEquicontinuous)
(hH : H' ⊆ H) : H'.UniformEquicontinuous :=
h.comp (inclusion hH) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuous.mono | null |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuousOn.mono | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_iff_range | 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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_iff_range | 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`. |
equicontinuous_iff_range {F : ι → X → α} :
Equicontinuous F ↔ Equicontinuous ((↑) : range F → X → α) :=
forall_congr' fun _ => equicontinuousAt_iff_range | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_iff_range | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_iff_range | 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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_iff_continuousAt | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_iff_continuousWithinAt | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_iff_continuous | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_iff_continuousOn | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_iff_uniformContinuous | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_iff_uniformContinuousOn | 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 purposes. |
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 only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace]
unfold ContinuousWithinAt
rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_iInf_rng | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_iInf_rng | null |
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 [equicontinuous_iff_continuous (uα := _), UniformFun.topologicalSpace]
rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, continuous_iInf_rng] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_iInf_rng | null |
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_swap _ κ] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_iInf_rng | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_iInf_rng | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_iInf_rng | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousWithinAt_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuous_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | uniformEquicontinuousOn_iInf_dom | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousAt_iff_left | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousWithinAt_iff_left | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousAt_iff_right | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousWithinAt_iff_right | null |
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) ∈ s₂ k₂ := by
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousAt_iff | null |
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 ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂ k₂ := by
rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.equicontinuousWithinAt_iff | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuous_iff_left | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuousOn_iff_left | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuous_iff_right | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuousOn_iff_right | null |
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 i x, F i y) ∈ s₂ k₂ := by
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
simp only [Prod.forall]
rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuous_iff | null |
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 ∧ ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x, F i y) ∈ s₂ k₂ := by
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity_of_basis ι α hα)]
simp only [Prod.forall]
rfl | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Filter.HasBasis.uniformEquicontinuousOn_iff | null |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.equicontinuousAt_iff | 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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.equicontinuousWithinAt_iff | 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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.equicontinuous_iff | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.equicontinuousOn_iff | 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`. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.uniformEquicontinuous_iff | 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. |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | IsUniformInducing.uniformEquicontinuousOn_iff | 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`. |
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)
exact (continuous_apply ⟨x, hxS⟩).comp hu₁ | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousWithinAt.closure' | 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 conditions. See also `Set.EquicontinuousWithinAt.closure`
for a more familiar (but weaker) statement.
Note: This could *technically* be called `EquicontinuousWithinAt.closure` without name clashes
with `Set.EquicontinuousWithinAt.closure`, but we don't do it because, even with a `protected`
marker, it would introduce ambiguities while working in namespace `Set` (e.g, in the proof of
any theorem called `Set.something`). |
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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt.closure' | 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.EquicontinuousAt.closure` for a more familiar statement. |
protected Set.EquicontinuousAt.closure {A : Set (X → α)} {x₀ : X}
(hA : A.EquicontinuousAt x₀) : (closure A).EquicontinuousAt x₀ :=
hA.closure' (u := id) continuous_id | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousAt.closure | If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
also equicontinuous at `x₀`. |
protected 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 | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousWithinAt.closure | 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'`. |
Equicontinuous.closure' {A : Set Y} {u : Y → X → α}
(hA : Equicontinuous (u ∘ (↑) : A → X → α)) (hu : Continuous u) :
Equicontinuous (u ∘ (↑) : closure A → X → α) := fun x ↦ (hA x).closure' hu | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.closure' | If a set of functions is equicontinuous, the same is true for its closure in *any*
topology for which evaluation at any point is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `X → α` satisfying the right
continuity conditions. See also `Set.Equicontinuous.closure` for a more familiar statement. |
EquicontinuousOn.closure' {A : Set Y} {u : Y → X → α} {S : Set X}
(hA : EquicontinuousOn (u ∘ (↑) : A → X → α) S) (hu : Continuous (S.restrict ∘ u)) :
EquicontinuousOn (u ∘ (↑) : closure A → X → α) S :=
fun x hx ↦ (hA x hx).closure' hu <| by exact continuous_apply ⟨x, hx⟩ |>.comp hu | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousOn.closure' | If a set of functions is equicontinuous on a set `S`, the same is true for its closure in *any*
topology for which evaluation at any `x ∈ S` is continuous. Since this will be applied to
`DFunLike` types, we state it for any topological space with a map to `X → α` satisfying the right
continuity conditions. See also `Set.EquicontinuousOn.closure` for a more familiar
(but weaker) statement. |
protected Set.Equicontinuous.closure {A : Set <| X → α} (hA : A.Equicontinuous) :
(closure A).Equicontinuous := fun x ↦ Set.EquicontinuousAt.closure (hA x) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.Equicontinuous.closure | If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. |
protected Set.EquicontinuousOn.closure {A : Set <| X → α} {S : Set X}
(hA : A.EquicontinuousOn S) : (closure A).EquicontinuousOn S :=
fun x hx ↦ Set.EquicontinuousWithinAt.closure (hA x hx) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousOn.closure | If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. This would also be true for the coarser topology of pointwise convergence on `S`,
see `EquicontinuousOn.closure'`. |
UniformEquicontinuousOn.closure' {A : Set Y} {u : Y → β → α} {S : Set β}
(hA : UniformEquicontinuousOn (u ∘ (↑) : A → β → α) S) (hu : Continuous (S.restrict ∘ u)) :
UniformEquicontinuousOn (u ∘ (↑) : closure A → β → α) S := by
intro U hU
rcases mem_uniformity_isClosed hU with ⟨V, hV, hVclosed, hVU⟩
filter_upwards [hA V hV, mem_inf_of_right (mem_principal_self _)]
rintro ⟨x, y⟩ hxy ⟨hxS, hyS⟩
rw [SetCoe.forall] at *
change A ⊆ (fun f => (u f x, u f y)) ⁻¹' V at hxy
refine (closure_minimal hxy <| hVclosed.preimage <| .prodMk ?_ ?_).trans (preimage_mono hVU)
· exact (continuous_apply ⟨x, hxS⟩).comp hu
· exact (continuous_apply ⟨y, hyS⟩).comp hu | theorem | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn.closure' | If a set of functions is uniformly equicontinuous on a set `S`, the same is true for its
closure in *any* topology for which evaluation at any `x ∈ S` i continuous. Since this will be
applied to `DFunLike` types, we state it for any topological space with a map to `β → α` satisfying
the right continuity conditions. See also `Set.UniformEquicontinuousOn.closure` for a more familiar
(but weaker) statement. |
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