-- 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)
hU : IsCompact (↑U : Set (↑(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)
g : (x : ↥U) → R
hxg :
  ∀ (x : ↥U), Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen (g x)) (↑x : (↑(PrimeSpectrum.Top R) : Type u))
igU : ∀ (x : ↥U), basicOpen (g x) ≤ U
f : (x : ↥U) → M
H :
  ∀ (x : ↥U),
    Eq (α :=
      Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
          (structureSheafInType R M))
        (op (basicOpen (g x))))
      (const (f x) (g x) (basicOpen (g x)) ⋯)
      ((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))
              ⋯.hom.op) :
          (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 x)))))
        s)
i j : ↥U
this :
  Eq (α :=
    Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
      (ObjectProperty.FullSubcategory.obj (P :=
        CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
        (structureSheafInType R M))
      (op (basicOpen (g i))))
    (const (f i) (g i) (basicOpen (g i)) ⋯)
    ((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))
            ⋯.hom.op) :
        (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 i)))))
      s)
a : ↥(unop (op (basicOpen (g i * g j))))
⊢ Eq (α := Localizations M (↑a : (↑(PrimeSpectrum.Top R) : Type u)))
    ((↑(const (g i • f j) (g i * g j) (basicOpen (g i * g j)) ⋯) :
          (x : ↥(unop (op (basicOpen (g i * g j))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        a :
      Localizations M (↑a : (↑(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 (homOfLE ⋯))) :
                (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g j))))) →
                  (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen (g i * g j)))))
              (const (f j) (g j) (basicOpen (g j)) ⋯)) :
          (x : ↥(unop (op (basicOpen (g i * g j))))) → Localizations M (↑x : (↑(PrimeSpectrum.Top R) : Type u)))
        a :
      Localizations M (↑a : (↑(PrimeSpectrum.Top R) : Type u)))

Tactic:
exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]; ring_nf⟩

Premises used: Iff.mpr, LocalizedModule.mk_eq, Exists.intro, OfNat.ofNat