-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
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
x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
U : Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
hxU : x ∈ U
s :
  ToType (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC := CommRingCat.carrier)
    (Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) (op U))
V : Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
hxV :
  (↑(Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op U)) x hxU) :
      (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ∈
    V
i : V ⟶ unop (op U)
n : ℕ
a b : ↥(𝒜 n)
h : ∀ (x : ↥V), (↑b : A) ∉ (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)).asHomogeneousIdeal
e :
  ∀ (a_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) (b_1 : a_1 ∈ V),
    (↑s : (x : ↥U) → at (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ U) a_1 ⋯) =
      HomogeneousLocalization.mk { deg := n, num := a, den := b, den_mem := ⋯ }
t : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)
ht : t ∈ unop (op V)
a₁ :
  HomogeneousLocalization.val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(sectionInBasicOpen 𝒜 x { deg := n, num := a, den := b, den_mem := ⋯ }) :
          (x_1 :
              ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑{ deg := n, num := a, den := b, den_mem := ⋯ }.den : A))))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
              (↑x_1 : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        ((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
            (a_1 : ↥(unop (op V))) →
              ↥(unop (op (ProjectiveSpectrum.basicOpen 𝒜 (↑{ deg := n, num := a, den := b, den_mem := ⋯ }.den : A)))))
          (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op V)) t ht))) =
    HomogeneousLocalization.val
      ((↑s :
          (x : ↥(unop (op (unop (op U))))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x) (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        ((i.op.unop : (a : ↥(unop (op V))) → ↥(unop (op (unop (op U)))))
          (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op V)) t ht)))
⊢ Eq (α :=
    Localization (M := A)
      (↑(Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op V)) t ht) :
              (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl)
    (HomogeneousLocalization.val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                    CommRingCat.carrier) (X :=
                    Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op
                          (ProjectiveSpectrum.basicOpen 𝒜
                            (↑(HomogeneousLocalization.NumDenSameDeg.den
                                  { deg := n, num := a, den := b, den_mem := ⋯ }) :
                              A)) :
                        (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ))
                    (Y :=
                    Functor.obj
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) (op V))
                    (CategoryTheory.Functor.map (Sheaf.presheaf (Proj.structureSheaf 𝒜)) (Quiver.Hom.op (homOfLE ⋯)) :
                      Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜
                                (↑(HomogeneousLocalization.NumDenSameDeg.den
                                      { deg := n, num := a, den := b, den_mem := ⋯ :
                                        HomogeneousLocalization.NumDenSameDeg 𝒜
                                          x.asHomogeneousIdeal.toIdeal.primeCompl }) :
                                  A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ) ⟶
                        Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op V)) :
                  (↑(Functor.obj (C := (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op
                              (ProjectiveSpectrum.basicOpen 𝒜
                                (↑(HomogeneousLocalization.NumDenSameDeg.den
                                      { deg := n, num := a, den := b, den_mem := ⋯ }) :
                                  A)) :
                            (Opens (↑(ProjectiveSpectrum.top 𝒜) : Type u_1))ᵒᵖ)) :
                      Type u_1) →+*
                    (↑(Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op V)) :
                      Type u_1)) :
                (a :
                    (↑((Proj.structureSheaf 𝒜).presheaf.obj
                          (op
                            (ProjectiveSpectrum.basicOpen 𝒜
                              (↑{ deg := n, num := a, den := b, den_mem := ⋯ }.den : A)))) :
                      Type u_1)) →
                  (↑((Proj.structureSheaf 𝒜).presheaf.obj (op V)) : Type u_1))
              (sectionInBasicOpen 𝒜 x { deg := n, num := a, den := b, den_mem := ⋯ })) :
          (x : ↥(unop (op V))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x) (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op V)) t ht)))
    (HomogeneousLocalization.val.{0, u_1, u_2} (ι := ℕ) (σ := σ) (𝒜 := 𝒜)
      ((↑(((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u_1}) ↦ (↑R : Type u_1) →+* (↑S : Type u_1)) (CC :=
                    CommRingCat.carrier) (X :=
                    Functor.obj
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op (unop (op U))))
                    (Y :=
                    Functor.obj
                      (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜)) (op V))
                    (CategoryTheory.Functor.map (Sheaf.presheaf (Proj.structureSheaf 𝒜)) i.op :
                      Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (unop (op U))) ⟶
                        Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op V)) :
                  (↑(Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op (unop (op U)))) :
                      Type u_1) →+*
                    (↑(Functor.obj
                          (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                          (op V)) :
                      Type u_1)) :
                (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op (unop (op U)))) : Type u_1)) →
                  (↑((Proj.structureSheaf 𝒜).presheaf.obj (op V)) : Type u_1))
              s) :
          (x : ↥(unop (op V))) →
            (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x) (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
        (Subtype.mk (p := fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ x ∈ unop (op V)) t ht)))

Tactic:
rw [Proj.res_apply, Proj.res_apply]

Premises used: AlgebraicGeometry.Proj.res_apply