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 : IsCofiltered I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : ↑Γ(c.pt, ⊤)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : CompactSpace ↥c.pt
i : ↥c.pt → I
U : (x : ↥c.pt) → (D.obj (i x)).Opens
hU : ∀ (x : ↥c.pt), IsAffineOpen (U x)
hxU : ∀ (x : ↥c.pt), (c.π.app (i x)) x ∈ U x
t : (x : ↥c.pt) → ↑Γ(D.obj (i x), U x)
ht : ∀ (x : ↥c.pt), (ConcreteCategory.hom (Scheme.Hom.app (c.π.app (i x)) (U x))) (t x) = (s |_ c.π.app (i x) ⁻¹ᵁ U x) ⋯
x y : ↥c.pt
j : I
fjx : j ⟶ i x
fjy : j ⟶ i y
k : I
fkj : k ⟶ j
H : D.map (fkj ≫ fjx) ⁻¹ᵁ U x ⊓ D.map (fkj ≫ fjy) ⁻¹ᵁ U y ≤ D.map fkj ⁻¹ᵁ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)
hk :
  (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE H).op))
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map fkj) (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)))
        (((ConcreteCategory.hom (Scheme.Hom.app (D.map fjx) (U x))) (t x) |_ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y))
            ⋯ -
          ((ConcreteCategory.hom (Scheme.Hom.app (D.map fjy) (U y))) (t y) |_ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y))
            ⋯)) =
    (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE H).op)) 0
⊢ ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fkj ≫ fjx)) (U x))) (t x) |_
        (D.map (fkj ≫ fjx) ⁻¹ᵁ U x ⊓ D.map (fkj ≫ fjy) ⁻¹ᵁ U y))
      ⋯ =
    ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fkj ≫ fjy)) (U y))) (t y) |_
        (D.map (fkj ≫ fjx) ⁻¹ᵁ U x ⊓ D.map (fkj ≫ fjy) ⁻¹ᵁ U y))
      ⋯

Tactic:
dsimp [TopCat.Presheaf.restrictOpen, TopCat.Presheaf.restrict] at hk ⊢

Goal state after:
I : Type u
inst✝⁴ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝³ : IsCofiltered I
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
s : ↑Γ(c.pt, ⊤)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
this : CompactSpace ↥c.pt
i : ↥c.pt → I
U : (x : ↥c.pt) → (D.obj (i x)).Opens
hU : ∀ (x : ↥c.pt), IsAffineOpen (U x)
hxU : ∀ (x : ↥c.pt), (c.π.app (i x)) x ∈ U x
t : (x : ↥c.pt) → ↑Γ(D.obj (i x), U x)
ht : ∀ (x : ↥c.pt), (ConcreteCategory.hom (Scheme.Hom.app (c.π.app (i x)) (U x))) (t x) = (s |_ c.π.app (i x) ⁻¹ᵁ U x) ⋯
x y : ↥c.pt
j : I
fjx : j ⟶ i x
fjy : j ⟶ i y
k : I
fkj : k ⟶ j
H : D.map (fkj ≫ fjx) ⁻¹ᵁ U x ⊓ D.map (fkj ≫ fjy) ⁻¹ᵁ U y ≤ D.map fkj ⁻¹ᵁ (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)
hk :
  (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE H).op))
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map fkj) (D.map fjx ⁻¹ᵁ U x ⊓ D.map fjy ⁻¹ᵁ U y)))
        ((ConcreteCategory.hom ((D.obj j).presheaf.map (homOfLE ⋯).op))
            ((ConcreteCategory.hom (Scheme.Hom.app (D.map fjx) (U x))) (t x)) -
          (ConcreteCategory.hom ((D.obj j).presheaf.map (homOfLE ⋯).op))
            ((ConcreteCategory.hom (Scheme.Hom.app (D.map fjy) (U y))) (t y)))) =
    (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE H).op)) 0
⊢ (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE ⋯).op))
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fkj ≫ fjx)) (U x))) (t x)) =
    (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE ⋯).op))
      ((ConcreteCategory.hom (Scheme.Hom.app (D.map (fkj ≫ fjy)) (U y))) (t y))