-- 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)
r : R
m : M
refine :
  Eq (α := (↑(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))
        (HSMul.hSMul (β :=
          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 ⊤))
          r
          ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) m :
            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 _root_.trivial) :
          (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op ⊤)) : Type u)) →
            (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
        (HSMul.hSMul (α :=
          (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top R) : Type u))ᵒᵖ) (structurePresheafInCommRingCat R) (op ⊤)) :
            Type u))
          (β := (↑(PresheafOfModules.obj (moduleStructurePresheaf R M) (op ⊤)) : Type u))
          ((CommRingCat.Hom.hom'
                  (CommRingCat.ofHom
                    (algebraMap R
                      (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 R))
                        (op ⊤)))) :
                (a : (↑(CommRingCat.of R) : Type u)) →
                  (↑(CommRingCat.of ((structureSheafInType R R).obj.obj (op ⊤))) : Type u))
              r :
            (↑(CommRingCat.of
                  (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 R))
                    (op ⊤))) :
              Type u))
          ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) m :
            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))
      (HSMul.hSMul (β :=
        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 ⊤))
        r
        ((toOpenₗ R M ⊤ : (a : M) → (structureSheafInType R M).obj.obj (op ⊤)) m :
          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 ⊤))) =
    HSMul.hSMul (α := (↑(stalk (structurePresheafInCommRingCat R) x) : Type u)) (β :=
      (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (toStalk R x) :
            (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
          r :
        (↑(stalk (structurePresheafInCommRingCat R) 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 ⊤)) m :
            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:
refine .trans ?_ ((moduleStructurePresheaf R M).germ_smul ..)