mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
⊢ GaloisConnection (fun 𝓐 ↦ map (UniformFun.phi α β) (𝓐 ×ˢ ⊤)) fun 𝓕 ↦ UniformFun.filter α β 𝓕

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
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 : ι), (F i x, F i y) ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
x : α
⊢ UniformContinuous (Function.eval x ∘ ⇑toFun)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
γ : Type u_3
ι : Type u_4
u : ι → UniformSpace γ
⊢ uniformSpace α γ = ⨅ i, uniformSpace α γ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
x₀ : X
⊢ EquicontinuousAt F x₀ ↔ ∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ x ∈ V, ∀ y ∈ V, ∀ (i : ι), (F i x, F i y) ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
x₀ : X
⊢ EquicontinuousAt F x₀ ↔ EquicontinuousAt Subtype.val x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
γ : Type u_3
u₁ u₂ : UniformSpace γ
⊢ uniformSpace α γ = uniformSpace α γ ⊓ uniformSpace α γ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
⊢ EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt Subtype.val S x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : UniformSpace β
f : γ → β
⊢ uniformSpace α γ = UniformSpace.comap (fun x ↦ f ∘ x) (uniformSpace α β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
h : UniformEquicontinuous F
⊢ UniformEquicontinuous Subtype.val

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : γ → β
hf : UniformContinuous f
⊢ uniformSpace α γ = UniformSpace.comap (fun x ↦ f ∘ x) (uniformSpace α β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
f : γ → β
hf : UniformContinuous f
⊢ UniformContinuous (⇑ofFun ∘ (fun x ↦ f ∘ x) ∘ ⇑toFun)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
S : Set β
h : UniformEquicontinuousOn F S
⊢ UniformEquicontinuousOn Subtype.val S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : UniformSpace β
f : γ → α
⊢ UniformContinuous fun g ↦ ofFun (toFun g ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
⊢ UniformContinuous ⇑toFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
x₀ : X
⊢ EquicontinuousAt F x₀ ↔ ContinuousAt (⇑ofFun ∘ swap F) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
p : Filter ι
inst✝ : UniformSpace β
F : ι → α →ᵤ β
f : α →ᵤ β
⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (⇑toFun ∘ F) (toFun f) p

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
x₀ : X
⊢ EquicontinuousWithinAt F S x₀ ↔ ContinuousWithinAt (⇑ofFun ∘ swap F) S x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
⊢ Equicontinuous F ↔ Continuous[tX, topologicalSpace ι α] (⇑ofFun ∘ swap F)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
S : Set X
⊢ EquicontinuousOn F S ↔ ContinuousOn (⇑ofFun ∘ swap F) S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
⊢ UniformEquicontinuous F ↔ UniformContinuous (⇑ofFun ∘ swap F)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
F : ι → β → α
S : Set β
⊢ UniformEquicontinuousOn F S ↔ UniformContinuousOn (⇑ofFun ∘ swap F) S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
p : Filter ι
inst✝¹ : UniformSpace β
inst✝ : UniformSpace γ
⊢ IsUniformInducing ⇑(Equiv.arrowProdEquivProdArrow α (fun a ↦ β) fun a ↦ γ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
p : Filter ι
inst✝¹ : UniformSpace β
δ : ι → Type u_5
inst✝ : (i : ι) → UniformSpace (δ i)
⊢ IsUniformInducing ⇑(Equiv.piComm fun a ↦ δ)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α' : Type u_7
tX : TopologicalSpace X
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ EquicontinuousWithinAt F S x₀ ↔ ∀ (k : κ), EquicontinuousWithinAt F S x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α' : Type u_7
tX : TopologicalSpace X
u : κ → UniformSpace α'
F : ι → X → α'
x₀ : X
⊢ EquicontinuousAt F x₀ ↔ ∀ (k : κ), EquicontinuousAt F x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace β
inst✝ : TopologicalSpace α
⊢ IsClosed[topologicalSpace α β] {f | Continuous[inst✝, inst✝¹.toTopologicalSpace] (toFun f)}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α' : Type u_7
tX : TopologicalSpace X
u : κ → UniformSpace α'
F : ι → X → α'
⊢ Equicontinuous F ↔ ∀ (k : κ), Equicontinuous F

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α' : Type u_7
tX : TopologicalSpace X
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
⊢ EquicontinuousOn F S ↔ ∀ (k : κ), EquicontinuousOn F S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Group.Defs

Goal state before:
pp: G✝ : Type u_1
α : Type u_2
X : Type u_3
inst✝³ : TopologicalSpace X
G : Type u
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
⊢ Continuous[instTopologicalSpaceProd, inst✝²] fun p ↦ p.1 / p.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α' : Type u_7
β : Type u_8
uβ : UniformSpace β
u : κ → UniformSpace α'
F : ι → β → α'
⊢ UniformEquicontinuous F ↔ ∀ (k : κ), UniformEquicontinuous F

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α' : Type u_7
β : Type u_8
uβ : UniformSpace β
u : κ → UniformSpace α'
F : ι → β → α'
S : Set β
⊢ UniformEquicontinuousOn F S ↔ ∀ (k : κ), UniformEquicontinuousOn F S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
δ₁ : Type u_6
δ₂ : Type u_7
φ₁ : δ₁ → α
φ₂ : δ₂ → α
h_cover : range φ₁ ∪ range φ₂ = univ
⊢ uniformSpace α β =
    UniformSpace.comap (⇑ofFun ∘ (fun x ↦ x ∘ φ₁) ∘ ⇑toFun) (uniformSpace δ₁ β) ⊓
      UniformSpace.comap (⇑ofFun ∘ (fun x ↦ x ∘ φ₂) ∘ ⇑toFun) (uniformSpace δ₂ β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
S : Set X'
x₀ : X'
k : κ
hk : EquicontinuousWithinAt F S x₀
⊢ EquicontinuousWithinAt F S x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
x₀ : X'
k : κ
hk : EquicontinuousAt F x₀
⊢ EquicontinuousAt F x₀

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_6
β' : Type u_9
uα : UniformSpace α
u : κ → UniformSpace β'
F : ι → β' → α
k : κ
hk : UniformEquicontinuous F
⊢ UniformEquicontinuous F

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Map

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
⊢ comap (⇑e) atTop = atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Map

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
⊢ map (⇑e) atTop = atTop

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Map

Goal state before:
pp: α : Type u_3
β : Type u_4
γ : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
l : Filter γ
f : γ → α
e : α ≃o β
⊢ Tendsto (fun x ↦ e (f x)) l atTop ↔ Tendsto f l atTop

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_6
β' : Type u_9
uα : UniformSpace α
u : κ → UniformSpace β'
F : ι → β' → α
S : Set β'
k : κ
hk : UniformEquicontinuousOn F S
⊢ UniformEquicontinuousOn F S

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
δ : ι → Type u_6
φ : (i : ι) → δ i → α
h_cover : ∃ I, I.Finite ∧ ⋃ i ∈ I, range (φ i) = univ
⊢ uniformSpace α β = ⨅ i, UniformSpace.comap (⇑ofFun ∘ (fun x ↦ x ∘ φ i) ∘ ⇑toFun) (uniformSpace (δ i) β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
𝔖 : Set (Set α)
S : Set α
V : Set (β × β)
⊢ UniformOnFun.gen 𝔖 S V =
    Prod.map (S.domRestrict ∘ ⇑UniformFun.toFun) (S.domRestrict ∘ ⇑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
⊢ topologicalSpace α β 𝔖 =
    ⨅ s ∈ 𝔖,
      TopologicalSpace.induced (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(toFun 𝔖)) (UniformFun.topologicalSpace (↑s) β)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
𝔖 : Set (Set α)
p : ι → Prop
s : ι → Set (β × β)
hb : (𝓤 β).HasBasis p s
S : Set α
⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis p fun i ↦ UniformOnFun.gen 𝔖 S (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
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 : ι), (F i x₀, F i x) ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_4
inst✝ : UniformSpace β
𝔖 : Set (Set α)
h : 𝔖.Nonempty
h' : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) 𝔖
p : ι → Prop
s : ι → Set (β × β)
hb : (𝓤 β).HasBasis p s
⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun Si ↦ Si.1 ∈ 𝔖 ∧ p Si.2) fun Si ↦ UniformOnFun.gen 𝔖 Si.1 (s Si.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
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 : ι), (F i x₀, F i x) ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
ι' : Type u_6
inst✝ : Nonempty ι
t : ι → Set α
p : ι' → Prop
V : ι' → Set (β × β)
ht : ∀ (i : ι), t i ∈ 𝔖
hdir : Directed (fun x1 x2 ↦ x1 ⊆ x2) t
hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i
hb : (𝓤 β).HasBasis p V
hne : 𝔖.Nonempty
s₁ : Set α
hs₁ : s₁ ∈ 𝔖
s₂ : Set α
hs₂ : s₂ ∈ 𝔖
⊢ ∃ z ∈ 𝔖, (fun x1 x2 ↦ x1 ⊆ x2) s₁ z ∧ (fun x1 x2 ↦ x1 ⊆ x2) s₂ z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ EquicontinuousAt F x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
ι' : Type u_6
inst✝ : Nonempty ι
t : ι → Set α
p : ι' → Prop
V : ι' → Set (β × β)
ht : ∀ (i : ι), t i ∈ 𝔖
hdir : Directed (fun x1 x2 ↦ x1 ⊆ x2) t
hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i
hb : (𝓤 β).HasBasis p V
⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i ↦ p i.2) fun i ↦ UniformOnFun.gen 𝔖 (t i.1) (V i.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
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 : κ), p k → ∀ᶠ (x : X) in 𝓝[S] x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
m✝ : α → β → γ
f✝ f₁ f₂ : Filter α
g✝ g₁ g₂ : Filter β
h : Filter γ
s : Set α
t : Set β
u : Set γ
a : α
b : β
m : α → β → γ
f : Filter α
g : Filter β
x✝ : Set γ
⊢ x✝ ∈ {s | ∃ u ∈ f, ∃ v ∈ g, image2 m u v ⊆ s} ↔ x✝ ∈ map (uncurry m) (f ×ˢ g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : IsDirectedOrder ι
t : ι → Set α
V : ι → Set (β × β)
ht : ∀ (n : ι), t n ∈ 𝔖
hmono : Monotone t
hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n
hb : (𝓤 β).HasAntitoneBasis V
⊢ (𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n ↦ UniformOnFun.gen 𝔖 (t n) (V n)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ map (fun p ↦ m p.1 p.2) (f ×ˢ g) = map₂ m f g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
f : Filter α
g : Filter β
⊢ map₂ Prod.mk f g = f ×ˢ g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
h : s ∈ 𝔖
⊢ UniformContinuous (⇑UniformFun.ofFun ∘ s.domRestrict ∘ ⇑(toFun 𝔖))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
f : Filter α
g : Filter β
ι : Type u_11
ι' : Type u_12
p : ι → Prop
q : ι' → Prop
s : ι → Set α
t : ι' → Set β
m : α → β → γ
hf : f.HasBasis p s
hg : g.HasBasis q t
⊢ (map₂ m f g).HasBasis (fun i ↦ p i.1 ∧ q i.2) fun i ↦ image2 m (s i.1) (t i.2)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
κ₁ : Type u_11
κ₂ : Type u_12
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 : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ (i : ι), (F i x₀, F i x) ∈ s₂ k₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ map₂ m f g = ⊥ ↔ f = ⊥ ∨ g = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ (map₂ m f g).NeBot ↔ f.NeBot ∧ g.NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
⊢ IsUniformEmbedding ⇑(toFun {s | s.Finite})

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f₁ f₂ : Filter α
g : Filter β
⊢ map₂ m (f₁ ⊔ f₂) g = map₂ m f₁ g ⊔ map₂ m f₂ g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g₁ g₂ : Filter β
⊢ map₂ m f (g₁ ⊔ g₂) = map₂ m f g₁ ⊔ map₂ m f g₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
κ₁ : Type u_11
κ₂ : Type u_12
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 : κ₂), p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ (i : ι), (F i x₀, F i x) ∈ s₂ k₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
g : Filter β
a : α
⊢ map₂ m (pure a) g = map (m a) g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
b : β
⊢ map₂ m f (pure b) = map (fun x ↦ m x b) f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
a : α
b : β
⊢ map₂ m (pure a) (pure b) = pure (m a b)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
m : α → β → γ
f : Filter α
g : Filter β
⊢ map₂ m f g = map₂ (fun a b ↦ m b a) g f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
f : Filter α
g : Filter β
inst✝ : g.NeBot
⊢ map₂ (fun x x_1 ↦ x) f g = f

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
f : Filter α
g : Filter β
inst✝ : f.NeBot
⊢ map₂ (fun x y ↦ y) f g = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.Equicontinuity

Goal state before:
pp: ι : Type u_1
κ : Type u_2
α : Type u_6
β : Type u_8
uα : UniformSpace α
uβ : UniformSpace β
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 : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
ι : Sort u_5
p : ι → Prop
V : ι → Set (β × β)
h : (𝓤 β).HasBasis p V
⊢ ⨅ i ∈ 𝔖,
      ⨅ i_1,
        ⨅ (_ : p i_1),
          𝓟
            {a |
              ∀ (a_1 : α) (b : a_1 ∈ i),
                (UniformFun.toFun
                      (Prod.map (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖))
                          (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖)) a).1
                      ⟨a_1, b⟩,
                    UniformFun.toFun
                      (Prod.map (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖))
                          (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖)) a).2
                      ⟨a_1, b⟩) ∈
                  V i_1} =
    ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 {uv | ∀ x ∈ s, ((toFun 𝔖) uv.1 x, (toFun 𝔖) uv.2 x) ∈ V i}

Tactic:
Prod.map_fst,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
ι : Sort u_5
p : ι → Prop
V : ι → Set (β × β)
h : (𝓤 β).HasBasis p V
⊢ ⨅ i ∈ 𝔖,
      ⨅ i_1,
        ⨅ (_ : p i_1),
          𝓟
            {a |
              ∀ (a_1 : α) (b : a_1 ∈ i),
                (UniformFun.toFun ((⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖)) a.1) ⟨a_1, b⟩,
                    UniformFun.toFun
                      (Prod.map (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖))
                          (⇑UniformFun.ofFun ∘ i.domRestrict ∘ ⇑(toFun 𝔖)) a).2
                      ⟨a_1, b⟩) ∈
                  V i_1} =
    ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 {uv | ∀ x ∈ s, ((toFun 𝔖) uv.1 x, (toFun 𝔖) uv.2 x) ∈ V i}
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
f : Filter α
g : Filter β
m : α → β → γ
n : γ → δ
⊢ map n (map₂ m f g) = map₂ (fun a b ↦ n (m a b)) f g

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
f : Filter α
g : Filter β
m : γ → β → δ
n : α → γ
⊢ map₂ m (map n f) g = map₂ (fun a b ↦ m (n a) b) f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
ι : Sort u_5
p : ι → Prop
V : ι → Set (β × β)
h : (𝓤 β).HasBasis p V
⊢ 𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 (UniformOnFun.gen 𝔖 s (V i))

Tactic:
by

Goal state after:
Module: Mathlib.Order.Filter.NAry

Goal state before:
pp: α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
f : Filter α
g : Filter β
m : α → γ → δ
n : β → γ
⊢ map₂ m f (map n g) = map₂ (fun a b ↦ m a (n b)) f g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
s : Set α
inst✝ : UniformSpace β
𝔖 : Set (Set α)
hs : s ∈ 𝔖
V : Set (β × β)
hV : V ∈ 𝓤 β
⊢ UniformOnFun.gen 𝔖 s V ∈ 𝓤 (α →ᵤ[𝔖] β)

Tactic:
by

Goal state after: