-- Mathlib/CategoryTheory/Limits/FilteredColimitCommutesProduct.lean
theorem Types.isIso_colimitPointwiseProductToProductColimit (F : ∀ i, I i ⥤ Type u) :
    IsIso (colimitPointwiseProductToProductColimit F)

Proof state:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
k' : (s : α) → I s
f : (s : α) → k s ⟶ k' s
hk' :
  ∀ (s : α),
    (F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk) = (F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk')
⊢ colimit.ι (pointwiseProduct F) k yk = colimit.ι (pointwiseProduct F) k yk'

Tactic:
apply Types.colimit_sound' f f

Result:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : ∏ᶜ fun s => (F s).obj (k s) := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = colimit.ι (pointwiseProduct F) k yk'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk')
k' : (s : α) → I s
f : (s : α) → k s ⟶ k' s
hk' :
  ∀ (s : α),
    (F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk) = (F s).map (f s) (Pi.π (fun s => (F s).obj (k s)) s yk')
⊢ (pointwiseProduct F).map f yk = (pointwiseProduct F).map f yk'

Proof state:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = y
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = y'
⊢ y = y'

Tactic:
let k := IsFiltered.max ky ky'

Result:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = y
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = y'
k : (i : α) → I i := IsFiltered.max ky ky'
⊢ y = y'

Proof state:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = y
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = y'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
⊢ y = y'

Tactic:
obtain rfl : y = colimit.ι (pointwiseProduct F) k yk := by
  simp only [k, yk, Types.Colimit.w_apply', hyk₀]

Result:
case refine_1.intro.intro.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y' : colimit (pointwiseProduct F)
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
hyk₀' : colimit.ι (pointwiseProduct F) ky' yk₀' = y'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.leftToMax ky ky') yk₀
yk' : (pointwiseProduct F).obj k := (pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀'
hy :
  colimitPointwiseProductToProductColimit F (colimit.ι (pointwiseProduct F) k yk) =
    colimitPointwiseProductToProductColimit F y'
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = colimit.ι (pointwiseProduct F) k yk
⊢ colimit.ι (pointwiseProduct F) k yk = y'

Proof state:
case refine_1
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
⊢ y = y'

Tactic:
obtain ⟨ky, yk₀, hyk₀⟩ := Types.jointly_surjective' y

Result:
case refine_1.intro.intro
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
y y' : colimit (pointwiseProduct F)
hy : colimitPointwiseProductToProductColimit F y = colimitPointwiseProductToProductColimit F y'
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
hyk₀ : colimit.ι (pointwiseProduct F) ky yk₀ = y
⊢ y = y'