Theorem: Continuous.uniformContinuous_of_tendsto_cocompact
File: Mathlib/Topology/UniformSpace/Compact.lean

tactic: obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
state_before: α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α

tactic: obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx <| mem_nhds_left _ ht)
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α

tactic: apply
  mem_of_superset
    (symmetrize_mem_uniformity <|
      (hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.continuousAt) <|
        symmetrize_mem_uniformity hr)
state_before: case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
⊢ {x | (f x.fst, f x.snd) ∈ r} ∈ 𝓤 α

tactic: rintro ⟨b₁, b₂⟩ h
state_before: case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
⊢ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r} ⊆ {x | (f x.fst, f x.snd) ∈ r}

tactic: by_cases h₁ : b₁ ∈ s
state_before: case intro.intro.intro.intro.intro.mk
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}

tactic: by_cases h₂ : b₂ ∈ s
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
h₁ : ¬b₁ ∈ s
⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}

tactic: apply htr
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
h₁ : ¬b₁ ∈ s
h₂ : ¬b₂ ∈ s
⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}

tactic: exact ⟨x, htsymm.mk_mem_comm.1 (hst h₁), hst h₂⟩
state_before: case neg.a
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
h₁ : ¬b₁ ∈ s
h₂ : ¬b₂ ∈ s
⊢ (f (b₁, b₂).fst, f (b₁, b₂).snd) ∈ t ○ t

tactic: exact (h.1 h₁).1
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
h₁ : b₁ ∈ s
⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}

tactic: exact (h.2 h₂).2
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type ?u.44329
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
x : β
h_cont : Continuous f
hx : Tendsto f (cocompact α) (𝓝 x)
r : Set (β × β)
hr : r ∈ 𝓤 β
t : Set (β × β)
ht : t ∈ 𝓤 β
htsymm : SymmetricRel t
htr : t ○ t ⊆ r
s : Set α
hs : IsCompact s
hst : sᶜ ⊆ f ⁻¹' {y | (x, y) ∈ t}
b₁ b₂ : α
h : (b₁, b₂) ∈ symmetrizeRel {x | x.fst ∈ s → (f x.fst, f x.snd) ∈ symmetrizeRel r}
h₁ : ¬b₁ ∈ s
h₂ : b₂ ∈ s
⊢ (b₁, b₂) ∈ {x | (f x.fst, f x.snd) ∈ r}