-- 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))))
simp : (g j * g i) • f j = (g i * g j) • f j
⊢ HSMul.hSMul (α := ↥(↑a : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) 1
      (HSMul.hSMul (α := ↥(↑a : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
        (Subtype.mk (p := fun (x : R) ↦
          Membership.mem (γ := Submonoid R)
            (Ideal.primeCompl
              (PrimeSpectrum.asIdeal
                (↑((Quiver.Hom.unop (Quiver.Hom.op (homOfLE ⋯)) :
                        (a : ↥(unop (op (basicOpen (g i * g j))))) → ↥(unop (op (basicOpen (g j)))))
                      a) :
                  (↑(PrimeSpectrum.Top R) : Type u))))
            x)
          (g j) ⋯)
        (g i • f j)) =
    HSMul.hSMul (α := ↥(↑a : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) 1
      (HSMul.hSMul (α := ↥(↑a : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl)
        (Subtype.mk (p := fun (x : R) ↦ x ∈ (↑a : (↑(PrimeSpectrum.Top R) : Type u)).asIdeal.primeCompl) (g i * g j) ⋯)
        (f j))

Tactic:
simp [Submonoid.smul_def, ← smul_assoc]

Premises used: smul_assoc, Submonoid.smul_def