-- 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
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)
rw :
  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))
⊢ 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) (toStalk R x) :
            (a : (↑(CommRingCat.of R) : Type u)) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
          ((RingHom.id R : (a : R) → R) r : R) :
        (↑(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))

Tactic:
rw [← algebraMap_germ_apply U x hxU]

Premises used: AlgebraicGeometry.StructureSheaf.algebraMap_germ_apply