-- Mathlib/MeasureTheory/Function/UnifTight.lean
theorem ae_tendsto_ae_congr {f f' : ℕ → α → β} {g g' : α → β}
    (hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n) (hgg' : g =ᵐ[μ] g')
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    ∀ᵐ x ∂μ, Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun x_1 => f x_1 x) atTop (𝓝 (g x))

Tactic:
exact hfgx

Result:
no goals

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Tactic:
filter_upwards [hff'', hgg', hfg] with x hff'x hgg'x hfgx

Result:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Tactic:
have hff'' := eventually_countable_forall.mpr hff'

Result:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun x_1 => f x_1 x) atTop (𝓝 (g' x))

Tactic:
rw [← hgg'x]

Result:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun x_1 => f x_1 x) atTop (𝓝 (g x))

Proof state:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n => f' n x) atTop (𝓝 (g' x))

Tactic:
apply Tendsto.congr hff'x

Result:
case h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] f' n
hgg' : g =ᶠ[ae μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
hfgx : Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun x_1 => f x_1 x) atTop (𝓝 (g' x))