-- Mathlib/Topology/DenseEmbedding.lean
theorem continuousAt_extend [T3Space γ] {b : β} {f : α → γ} (di : IsDenseInducing i)
    (hf : ∀ᶠ x in 𝓝 b, ∃ c, Tendsto f (comap i <| 𝓝 x) (𝓝 c)) : ContinuousAt (di.extend f) b

Proof state:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
V₂ : Set β
V₂_in : V₂ ∈ 𝓝 b
V₂_op : IsOpen V₂
hV₂ : ∀ x ∈ i ⁻¹' V₂, f x ∈ V'
⊢ φ ⁻¹' V' ∈ 𝓝 b

Tactic:
suffices ∀ x ∈ V₁ ∩ V₂, φ x ∈ V' by filter_upwards [inter_mem V₁_in V₂_in] using this

Result:
case intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
V₂ : Set β
V₂_in : V₂ ∈ 𝓝 b
V₂_op : IsOpen V₂
hV₂ : ∀ x ∈ i ⁻¹' V₂, f x ∈ V'
⊢ ∀ x ∈ V₁ ∩ V₂, φ x ∈ V'

Proof state:
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
x : β
c : γ
hc : Tendsto f (comap i (𝓝 x)) (𝓝 c)
⊢ x ∈ V₁

Tactic:
rwa [← di.extend_eq_of_tendsto hc] at hc

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this✝ : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
V₂ : Set β
V₂_in : V₂ ∈ 𝓝 b
V₂_op : IsOpen V₂
hV₂ : ∀ x ∈ i ⁻¹' V₂, f x ∈ V'
this : ∀ x ∈ V₁ ∩ V₂, φ x ∈ V'
⊢ φ ⁻¹' V' ∈ 𝓝 b

Tactic:
filter_upwards [inter_mem V₁_in V₂_in] using this

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
⊢ ContinuousAt φ b

Tactic:
suffices ∀ V' ∈ 𝓝 (φ b), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 b by
  simpa [ContinuousAt, (closed_nhds_basis (φ b)).tendsto_right_iff]

Result:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
⊢ ∀ V' ∈ 𝓝 (φ b), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 b

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
⊢ ∀ (a : β), (∃ c, Tendsto f (comap i (𝓝 a)) (𝓝 c)) → a ∈ V₁

Tactic:
rintro x ⟨c, hc⟩

Result:
case h.intro
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
x : β
c : γ
hc : Tendsto f (comap i (𝓝 x)) (𝓝 c)
⊢ x ∈ V₁

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
⊢ φ ⁻¹' V' ∈ 𝓝 b

Tactic:
set V₁ := { x | Tendsto f (comap i <| 𝓝 x) (𝓝 <| φ x) }

Result:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
⊢ φ ⁻¹' V' ∈ 𝓝 b

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
⊢ ∃ V₂ ∈ 𝓝 b, IsOpen V₂ ∧ ∀ x ∈ i ⁻¹' V₂, f x ∈ V'

Tactic:
simpa [and_assoc] using
  ((nhds_basis_opens' b).comap i).tendsto_left_iff.mp (mem_of_mem_nhds V₁_in : b ∈ V₁) V' V'_in

Result:
no goals

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
V₂ : Set β
V₂_in : V₂ ∈ 𝓝 b
V₂_op : IsOpen V₂
hV₂ : ∀ x ∈ i ⁻¹' V₂, f x ∈ V'
x : β
x_in₁ : x ∈ V₁
x_in₂ : x ∈ V₂
hV₂x : V₂ ∈ 𝓝 x
⊢ V₂ ∈ 𝓝 x ∧ i ⁻¹' V₂ ⊆ {x | (fun x => f x ∈ V') x}

Tactic:
tauto

Result:
no goals