-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
s :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
x : (↑(PrimeSpectrum.Top R) : Type u)
hx : x ∈ U
V : Opens (↑(PrimeSpectrum.Top R) : Type u)
hxV :
  (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦ x ∈ unop (op U)) x hx) :
      (↑(PrimeSpectrum.Top R) : Type u)) ∈
    V
iVU : V ⟶ unop (op U)
f : M
g : R
hfg :
  ∀ (x : ↥V),
    ∃ (hs : g ∉ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal),
      Eq (α := Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        ((fun (x : ↥V) ↦
            (↑s : (x : ↥(unop (op U))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
              ((iVU : (a : ↥V) → ↥(unop (op U))) x))
          x)
        (LocalizedModule.mk f
          (Subtype.mk (p := fun (x_2 : R) ↦ x_2 ∈ (↑x : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) g hs))
g' : R
hxg' :
  (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦ x ∈ unop (op U)) x hx) :
      (↑(PrimeSpectrum.Top R) : Type u)) ∈
    (↑(basicOpen g') : Set (PrimeSpectrum R))
hg'U : (↑(basicOpen g') : Set (PrimeSpectrum R)) ⊆ (↑V : Set (↑(PrimeSpectrum.Top R) : Type u))
x_1 : ↥(unop (op (basicOpen (g' * g))))
y : (↑(PrimeSpectrum.Top R) : Type u)
hy : Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) y
refine_3 :
  Eq (α := Localizations M y)
    ((↑(const (g' • f) (g' * g) (basicOpen (g' * g)) ⋯) :
          (x : ↥(basicOpen (g' * g))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
          Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (basicOpen (g' * g)) x) y hy) :
      Localizations M
        (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (basicOpen (g' * g)) x) y hy) :
          (↑(PrimeSpectrum.Top R) : Type u)))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (CategoryTheory.Functor.map
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                      (structureSheafInType R M))
                    (Quiver.Hom.op (LE.le.hom ⋯))) :
                (a : (structureSheafInType R M).obj.obj (op U)) →
                  (structureSheafInType R M).obj.obj (op (basicOpen (g' * g))))
              s) :
          (x : ↥(basicOpen (g' * g))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
          Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (basicOpen (g' * g)) x) y hy) :
      Localizations M
        (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (basicOpen (g' * g)) x) y hy) :
          (↑(PrimeSpectrum.Top R) : Type u)))
⊢ Eq (α :=
    Localizations M
      (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
            Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
        (↑(PrimeSpectrum.Top R) : Type u)))
    ((↑(const (g' • f) (g' * g) (basicOpen (g' * g)) ⋯) :
          (x : ↥(unop (op (basicOpen (g' * g))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
          Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
      Localizations M
        (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
          (↑(PrimeSpectrum.Top R) : Type u)))
    ((↑((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (CategoryTheory.Functor.map
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                      (structureSheafInType R M))
                    (Quiver.Hom.op (LE.le.hom ⋯))) :
                (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
                  (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g' * g)))))
              s) :
          (x : ↥(unop (op (basicOpen (g' * g))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        (Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
          Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
      Localizations M
        (↑(Subtype.mk (p := fun (x : (↑(PrimeSpectrum.Top R) : Type u)) ↦
              Membership.mem (γ := Opens (↑(PrimeSpectrum.Top R) : Type u)) (unop (op (basicOpen (g' * g)))) x) y hy) :
          (↑(PrimeSpectrum.Top R) : Type u)))

Tactic:
simp only [PrimeSpectrum.basicOpen_mul]

Premises used: PrimeSpectrum.basicOpen_mul