-- 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)
e_5 : Eq.{u + 1} (α := AddCommMonoid R) NonUnitalNonAssocSemiring.toAddCommMonoid AddCommGroup.toAddCommMonoid
e_7 : HEq.{u + 1} (α := Module R R) Semiring.toModule (β := Module R R) Semiring.toModule
α : stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) x ≅
  Functor.obj
    (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
        CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
        RingCat.carrier) ⋙
      forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
        AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
        AddCommGrpCat.carrier))
    (stalk (structurePresheafInCommRingCat R) x) :=
  Limits.colimit.isoColimitCocone
    {
      cocone :=
        Functor.mapCocone
          (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
              RingCat.carrier) ⋙
            forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
              AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
              AddCommGrpCat.carrier))
          (Limits.colimit.cocone ((OpenNhds.inclusion x).op ⋙ structurePresheafInCommRingCat R)),
      isColimit :=
        Limits.isColimitOfPreserves
          (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
              RingCat.carrier) ⋙
            forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
              AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
              AddCommGrpCat.carrier))
          (Limits.colimit.isColimit ((OpenNhds.inclusion x).op ⋙ structurePresheafInCommRingCat R)) }
this :
  germ (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) ⊤ x ⋯ ≫ α.hom =
    CategoryTheory.Functor.map
      (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) RingCat (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD :=
          RingCat.carrier) ⋙
        forget₂ RingCat (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
          AddCommGrpCat (FD := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CD :=
          AddCommGrpCat.carrier))
      (germ (structurePresheafInCommRingCat R) ⊤ x ⋯)
h :
  Eq (α := (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
    (1 : (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
    ((commRingCatStalkEquivModuleStalk R x :
          (a : (↑(stalk (moduleStructurePresheaf R R).presheaf x) : Type u)) →
            (↑((structurePresheafInCommRingCat R).stalk 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 R)) (op ⊤))
                (Y := stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) x)
                (germ (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) ⊤ x ⋯) :
              (a : (↑((moduleStructurePresheaf R R).obj (op ⊤)) : Type u)) →
                (↑(stalk (moduleStructurePresheaf R R).presheaf x) : Type u))
            ((toOpenₗ R R ⊤ : (a : R) → (structureSheafInType R R).obj.obj (op ⊤)) 1) :
          (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) x) : Type u)) :
      (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
⊢ Eq (α := (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
    ((algebraMap R (↑(stalk (structurePresheafInCommRingCat R) x) : Type u) :
          (a : R) → (↑((structurePresheafInCommRingCat R).stalk x) : Type u))
        1 :
      (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
    ((commRingCatStalkEquivModuleStalk R x :
          (a : (↑(stalk (moduleStructurePresheaf R R).presheaf x) : Type u)) →
            (↑((structurePresheafInCommRingCat R).stalk 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 R)) (op ⊤))
                (Y := stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) x)
                (germ (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) ⊤ x ⋯) :
              (a : (↑((moduleStructurePresheaf R R).obj (op ⊤)) : Type u)) →
                (↑(stalk (moduleStructurePresheaf R R).presheaf x) : Type u))
            ((toOpenₗ R R ⊤ : (a : R) → (structureSheafInType R R).obj.obj (op ⊤)) 1) :
          (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R R)) x) : Type u)) :
      (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))

Tactic:
simp only [map_one]

Premises used: map_one