-- Module: Mathlib.AlgebraicGeometry.StructureSheaf
Goal:
R M A : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : CommRing A
inst_4 : Algebra R A
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ ∀ (x_1 y : M),
    (ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier) (X :=
            Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
              (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op ⊤))
            (Y := stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
            (germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) ⊤ x ⋯) :
          (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op ⊤)) : Type u)) →
            (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
        ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) (x_1 + y) :
          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 ⊤)) =
      HAdd.hAdd (α := (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u)) (β :=
        (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                AddCommGrpCat.carrier) (X :=
                Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                  (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op ⊤))
                (Y := stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
                (germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) ⊤ x ⋯) :
              (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op ⊤)) : Type u)) →
                (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
            ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) x_1 :
              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 ⊤)) :
          (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
                AddCommGrpCat.carrier) (X :=
                Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ)
                  (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op ⊤))
                (Y := stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
                (germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) ⊤ x ⋯) :
              (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op ⊤)) : Type u)) →
                (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
            ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) y :
              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 ⊤)) :
          (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))

Tactic:
simp