-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).toPresheafedSpace :
        TopCat) :
    Type u_1)
simp :
  Eq (α :=
    CommRingCat.of (A⁰_ f) ⟶
      CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : ProjectiveSpectrum 𝒜)) : Type u_1))
    (awayToSection 𝒜 f ≫
        CategoryStruct.comp (Y :=
          LocallyRingedSpace.Γ.obj
            (op
              ({ carrier := TopCat.of (ProjectiveSpectrum 𝒜),
                    presheaf :=
                      (ObjectProperty.FullSubcategory.obj (P :=
                          (CategoryTheory.Presheaf.IsSheaf
                              (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                            (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                          (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat),
                    IsSheaf := ⋯, isLocalRing := ⋯ }.restrict
                (U :=
                Functor.obj (C := Opens (ProjectiveSpectrum 𝒜)) (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜)))
                  (pbo f))
                (f :=
                (Opens.inclusion' (pbo f) :
                  Functor.obj (C := Opens (↑(TopCat.of (ProjectiveSpectrum 𝒜)) : Type u_1))
                      (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))) (pbo f) ⟶
                    TopCat.of (ProjectiveSpectrum 𝒜)))
                ⋯)))
          (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)))
              (Quiver.Hom.op (homOfLE ⋯)) :
            Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                (op (pbo f) : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
              Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                (Functor.obj (C :=
                    (Opens
                        (↑(Functor.obj (C := Opens (ProjectiveSpectrum 𝒜))
                              (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))) (pbo f)) :
                          Type u_1))ᵒᵖ)
                    (Functor.op (Topology.IsOpenEmbedding.functor ⋯)) (op ⊤) :
                  (Opens (ProjectiveSpectrum 𝒜))ᵒᵖ))
          (Presheaf.germ
              (ObjectProperty.FullSubcategory.obj (P :=
                (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)))
              (Functor.obj (C :=
                Opens
                  (↑(Functor.obj (C := Opens (ProjectiveSpectrum 𝒜)) (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜)))
                        (pbo f)) :
                    Type u_1))
                (Topology.IsOpenEmbedding.functor ⋯) ⊤)
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (X :=
                    Functor.obj (C := Opens (ProjectiveSpectrum 𝒜)) (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜)))
                      (pbo f))
                    (Opens.inclusion' (pbo f)) :
                  (a : (↑((Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))).obj (pbo f)) : Type u_1)) →
                    (↑{ carrier := TopCat.of (ProjectiveSpectrum 𝒜), presheaf := (structureSheaf 𝒜).obj, IsSheaf := ⋯,
                            isLocalRing := ⋯ }.toTopCat :
                      Type u_1))
                x)
              ⋯ :
            Functor.obj (C := (Opens (↑(TopCat.of (ProjectiveSpectrum 𝒜)) : Type u_1))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                (op
                    (Functor.obj (C :=
                      Opens
                        (↑(Functor.obj (C := Opens (ProjectiveSpectrum 𝒜))
                              (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))) (pbo f)) :
                          Type u_1))
                      (Topology.IsOpenEmbedding.functor ⋯) ⊤) :
                  (Opens (↑(TopCat.of (ProjectiveSpectrum 𝒜)) : Type u_1))ᵒᵖ) ⟶
              Presheaf.stalk (X := TopCat.of (ProjectiveSpectrum 𝒜))
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (X :=
                        Functor.obj (C := Opens (ProjectiveSpectrum 𝒜))
                          (Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))) (pbo f))
                        (Opens.inclusion' (pbo f)) :
                      (a : (↑((Opens.toTopCat (TopCat.of (ProjectiveSpectrum 𝒜))).obj (pbo f)) : Type u_1)) →
                        (↑{ carrier := TopCat.of (ProjectiveSpectrum 𝒜), presheaf := (structureSheaf 𝒜).obj,
                                IsSheaf := ⋯, isLocalRing := ⋯ }.toTopCat :
                          Type u_1))
                    x :
                  (↑{ carrier := TopCat.of (ProjectiveSpectrum 𝒜),
                          presheaf :=
                            (ObjectProperty.FullSubcategory.obj (P :=
                                (CategoryTheory.Presheaf.IsSheaf
                                    (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                              (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat),
                          IsSheaf := ⋯, isLocalRing := ⋯ }.toTopCat :
                    Type u_1))) :
      CommRingCat.of (A⁰_ f) ⟶
        CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : ProjectiveSpectrum 𝒜)) : Type u_1))
    (CategoryStruct.comp (Y :=
        CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal.primeCompl))
        (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯))
        (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : ProjectiveSpectrum 𝒜)))) :
      CommRingCat.of (HomogeneousLocalization 𝒜 (Submonoid.powers f)) ⟶
        CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : ProjectiveSpectrum 𝒜)) : Type u_1))
⊢ Eq (α :=
    CommRingCat.of (A⁰_ f) ⟶
      CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1))
    (awayToSection 𝒜 f ≫
        CategoryStruct.comp (Y :=
          LocallyRingedSpace.Γ.obj
            (op (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯)))
          (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                  (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)))
              (Quiver.Hom.op (homOfLE ⋯)) :
            Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                (op (pbo f) : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
              Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                    (CategoryTheory.Presheaf.IsSheaf
                        (Opens.grothendieckTopology (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) :
                      (P : (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat) → Prop))
                    (structureSheaf 𝒜 : TopCat.Sheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) :
                  (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ ⥤ CommRingCat)
                (Functor.obj (C :=
                    (Opens
                        (↑(Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) :
                          Type u_1))ᵒᵖ)
                    (Functor.op (Topology.IsOpenEmbedding.functor ⋯)) (op ⊤) :
                  (Opens (↑Proj.T : Type u_1))ᵒᵖ))
          (Proj.presheaf.germ
              (Functor.obj (C :=
                Opens (↑(Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) : Type u_1))
                (Topology.IsOpenEmbedding.functor ⋯) ⊤)
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (X := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    (Opens.inclusion' (pbo f)) :
                  (a : (↑((Opens.toTopCat Proj.T).obj (pbo f)) : Type u_1)) → (↑Proj.toTopCat : Type u_1))
                x)
              ⋯ :
            Proj.presheaf.obj
                (op
                  (Functor.obj (C :=
                    Opens (↑(Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) : Type u_1))
                    (Topology.IsOpenEmbedding.functor ⋯) ⊤)) ⟶
              Proj.presheaf.stalk
                ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                      carrier) (X := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      (Opens.inclusion' (pbo f)) :
                    (a : (↑((Opens.toTopCat Proj.T).obj (pbo f)) : Type u_1)) → (↑Proj.toTopCat : Type u_1))
                  x)) :
      CommRingCat.of (A⁰_ f) ⟶
        CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1))
    (CategoryStruct.comp (Y :=
        CommRingCat.of (HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl))
        (CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 ⋯))
        (Iso.inv (RingEquiv.toCommRingCatIso («Proj».stalkIso' 𝒜 (↑x : (↑Proj.T : Type u_1))))) :
      CommRingCat.of (HomogeneousLocalization 𝒜 (Submonoid.powers f)) ⟶
        CommRingCat.of (↑(Presheaf.stalk (Sheaf.presheaf (structureSheaf 𝒜)) (↑x : (↑Proj.T : Type u_1))) : Type u_1))

Tactic:
simp only [Proj.toLocallyRingedSpace, Proj.toSheafedSpace]

Premises used: AlgebraicGeometry.Proj.toSheafedSpace, AlgebraicGeometry.Proj.toLocallyRingedSpace