-- 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))
obtain :
  ∀ (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 :
      ∀ (x : ↥V),
        Eq (α := at (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
          ((fun (x : ↥V) ↦
              (↑s :
                  (x : ↥(unop (op U))) →
                    (fun (x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)) ↦ at x)
                      (↑x : (↑(ProjectiveSpectrum.top 𝒜) : Type u_1)))
                ((i : (a : ↥V) → ↥(unop (op U))) x))
            x)
          (HomogeneousLocalization.mk { deg := n, num := a, den := b, den_mem := ⋯ })),
    Eq (α := (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
      (homogeneousLocalizationToStalk 𝒜 x
        ((CommRingCat.Hom.hom (stalkToFiberRingHom 𝒜 x) :
            (a : (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1)) → (↑(CommRingCat.of (at x)) : Type u_1))
          (((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 U))
                  (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
                  (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                    Functor.obj
                        (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                        (op U) ⟶
                      Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
                (↑(Functor.obj
                        (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                        (op U)) :
                    Type u_1) →+*
                  (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
              (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
                (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
            s)))
      (((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 U))
              (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
              (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                    (op U) ⟶
                  Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
            (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                    (op U)) :
                Type u_1) →+*
              (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
            (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
        s)
⊢ Eq (α := (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1))
    (homogeneousLocalizationToStalk 𝒜 x
      ((CommRingCat.Hom.hom (stalkToFiberRingHom 𝒜 x) :
          (a : (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1)) → (↑(CommRingCat.of (at x)) : Type u_1))
        (((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 U))
                (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
                (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
                  Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U) ⟶
                    Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
              (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                      (op U)) :
                  Type u_1) →+*
                (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
            (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
              (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
          s)))
    (((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 U))
            (Y := Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x)
            (Presheaf.germ (Sheaf.presheaf (Proj.structureSheaf 𝒜)) U x hxU :
              Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U) ⟶
                Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) :
          (↑(Functor.obj (Sheaf.presheaf (Proj.structureSheaf 𝒜) : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜))
                  (op U)) :
              Type u_1) →+*
            (↑(Presheaf.stalk (Sheaf.presheaf (Proj.structureSheaf 𝒜)) x) : Type u_1)) :
        (a : (↑((Proj.structureSheaf 𝒜).presheaf.obj (op U)) : Type u_1)) →
          (↑((Proj.structureSheaf 𝒜).presheaf.stalk x) : Type u_1))
      s)

Tactic:
obtain ⟨V, hxV, i, n, a, b, h, e⟩ := s.2 ⟨x, hxU⟩

Premises used: Subtype.mk, Subtype.property