-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
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
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
x : (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
r : R
m :
  Functor.obj
    (ObjectProperty.FullSubcategory.obj (P :=
      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
      (structureSheafInType R M))
    (op U)
⊢ ZeroHom.toFun
      (↑(AddCommGrpCat.Hom.hom (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU)) :
        ZeroHom (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)
          (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
      (r • m) =
    HSMul.hSMul (α := (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)) (β :=
      (↑(Presheaf.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) (X :=
                CommRingCat.of
                  (Functor.obj
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                      (structureSheafInType R R))
                    (op U)))
                (Y := Presheaf.stalk (structurePresheafInCommRingCat R) x)
                (Presheaf.germ (structurePresheafInCommRingCat R) U x hxU) :
              (↑(CommRingCat.of
                      (Functor.obj
                        (ObjectProperty.FullSubcategory.obj (P :=
                          CategoryTheory.Presheaf.IsSheaf
                            (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                          (structureSheafInType R R))
                        (op U))) :
                  Type u) →+*
                (↑(Presheaf.stalk (structurePresheafInCommRingCat R) x) : Type u)) :
            (a : (structureSheafInType R R).obj.obj (op U)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
          ((algebraMap R
                  (Functor.obj
                    (ObjectProperty.FullSubcategory.obj (P :=
                      CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                      (structureSheafInType R R))
                    (op U)) :
                (a : R) → (structureSheafInType R R).obj.obj (op U))
              ((RingHom.id R : (a : R) → R) r) :
            Functor.obj
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R R))
              (op U)) :
        (↑(Presheaf.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 (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op U)) (Y :=
              Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
              (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU) :
            (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)) →
              (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
          m :
        (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
have refine : Eq (α := (↑(Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
  (ZeroHom.toFun
    (↑(AddCommGrpCat.Hom.hom (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU)) :
      ZeroHom (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)
        (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
    (r • m))
  ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
        AddCommGrpCat.carrier) (X := Functor.obj (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) (op U))
        (Y := Presheaf.stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x)
        (Presheaf.germ (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) U x hxU) :
      (a : (↑((moduleStructurePresheaf R M).presheaf.obj (op U)) : Type u)) →
        (↑(Presheaf.stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
    (HSMul.hSMul (α := (↑(Functor.obj (structurePresheafInCommRingCat R) (op U)) : Type u))
      ((algebraMap R
              (Functor.obj
                (ObjectProperty.FullSubcategory.obj (P :=
                  CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                  (structureSheafInType R R))
                (op U)) :
            (a : R) → (structureSheafInType R R).obj.obj (op U))
          ((RingHom.id R : (a : R) → R) r) :
        Functor.obj
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
            (structureSheafInType R R))
          (op U))
      m))
Tactic: refine .trans ?_ (PresheafOfModules.germ_smul ..)