-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
rw :
  Eq (α := (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X := Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤))
            (Scheme.Hom.appTop ((opensCone D c i U).π.app (Over.mk (𝟙 i)))) :
          (a : (↑Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤) : Type u)) →
            (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    ((CommRingCat.Hom.hom (c.pt.presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(c.pt.presheaf.obj (Opposite.op (c.π.app i ⁻¹ᵁ U))) : Type u)) →
            (↑(c.pt.presheaf.obj (Opposite.op ((c.π.app i ⁻¹ᵁ U).ι ''ᵁ ⊤))) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
            s :
          (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u)) :
      (↑(c.pt.presheaf.obj (Opposite.op ((c.π.app i ⁻¹ᵁ U).ι ''ᵁ ⊤))) : Type u))
⊢ Eq (α := (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X := Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤))
            (Scheme.Hom.appTop ((opensCone D c i U).π.app (Over.mk (𝟙 i)))) :
          (a : (↑Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤) : Type u)) →
            (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    (0 : (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))

Tactic:
rw [← map_zero (c.pt.presheaf.map (homOfLE
        (show (c.π.app i ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ c.π.app i ⁻¹ᵁ U by simp)).op).hom, ← hs]

Premises used: CommRingCat.Hom.hom, Quiver.Hom.op, map_zero, CategoryTheory.Limits.Cone.pt, CategoryTheory.Functor.map, AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace, CategoryTheory.homOfLE, AlgebraicGeometry.PresheafedSpace.presheaf, AlgebraicGeometry.SheafedSpace.toPresheafedSpace, AlgebraicGeometry.Scheme.toLocallyRingedSpace