Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ D.map (g i ⋯) ≫ 𝟙 (D.obj i) =
    D.map
      (IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ?mY)

pp: case mY
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ i ∈ insert i (Finset.image i' Finset.univ)

pp: case mf
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ ⟨i' j, ⟨i, ⟨⋯, ⟨?mY, f j⟩⟩⟩⟩ ∈ Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ

Tactic:
]

Goal state after:
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ D.map (g i ⋯) ≫ 𝟙 (D.obj i) =
    D.map
      (IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ?mY)

case mY
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ i ∈ insert i (Finset.image i' Finset.univ)

case mf
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofilteredOrEmpty I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
h✝ : Nonempty I
i : I
this✝ : IsCofiltered I
𝒰 : (D.obj i).OpenCover := (D.obj i).affineCover.finiteSubcover
this : ∀ (i' : 𝒰.I₀), IsAffine (𝒰.X i')
i' : 𝒰.I₀ → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  j ∈ insert i (Finset.image i' Finset.univ) →
    (IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ) ⟶
      j) :=
  fun j ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ)
j : 𝒰.I₀
⊢ ⟨i' j, ⟨i, ⟨⋯, ⟨?mY, f j⟩⟩⟩⟩ ∈ Finset.image (fun j ↦ ⟨i' j, ⟨i, ⟨⋯, ⟨⋯, f j⟩⟩⟩⟩) Finset.univ