awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (goal)
Goal:
R M : Type u
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ Eq (α := Quiver.Hom (V := ModuleCat R) (ModuleCat.of R M) (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)))
    (toStalkₗ' R M x ≫ stalkToLocalizationₗ R M x)
    (ModuleCat.ofHom (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M))
have rw : Eq (α := Quiver.Hom (V := ModuleCat R) (ModuleCat.of R M) (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)))
  (CategoryStruct.comp (Y :=
    ModuleCat.of 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 M))
        (op ⊤)))
    (ModuleCat.ofHom (toOpenₗ R M ⊤)) (openToLocalizationₗ R M ⊤ x _root_.trivial))
  (ModuleCat.ofHom (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M))
Tactic: rw [toStalkₗ', Category.assoc, germ_stalkToLocalizationₗ]
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x = 𝟙 (stalk (structurePresheafInModuleCat R M) x)
have ih : ∀ (U : Opens (↑(PrimeSpectrum.Top R) : Type u)) (hxU : x ∈ U),
  germ (structurePresheafInModuleCat R M) U x hxU ≫ stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x =
    germ (structurePresheafInModuleCat R M) U x hxU ≫ 𝟙 (stalk (structurePresheafInModuleCat R M) x)
Tactic: apply stalk_hom_ext
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
⊢ germ (structurePresheafInModuleCat R M) U x hxU ≫ stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x =
    germ (structurePresheafInModuleCat R M) U x hxU ≫ 𝟙 (stalk (structurePresheafInModuleCat R M) x)
have ih : ∀ (s : (↑(Functor.obj (structurePresheafInModuleCat R M) (op U)) : Type u)),
  Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫
              stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫ 𝟙 (stalk (structurePresheafInModuleCat R M) x)) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: ext s
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
U : Opens (↑(PrimeSpectrum.Top R) : Type u)
hxU : x ∈ U
s : (↑(Functor.obj (structurePresheafInModuleCat R M) (op U)) : Type u)
⊢ Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫
              stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫ 𝟙 (stalk (structurePresheafInModuleCat R M) x)) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
have ih : ∀ (g : R) (hxg : Membership.mem (γ := Opens (PrimeSpectrum R)) (basicOpen g) x) (igU : basicOpen g ≤ U) (f : M)
  (hs :
    Eq (α :=
      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 (basicOpen g)))
      (const f g (basicOpen g) sorry)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map
              (ObjectProperty.FullSubcategory.obj (P :=
                CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top R) : Type u)))
                (structureSheafInType R M))
              igU.hom.op) :
          (a : (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op U))) →
            (fun (X : Type u) ↦ X) ((structureSheafInType R M).obj.obj (op (basicOpen g))))
        s)),
  Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫
              stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom
            (germ (structurePresheafInModuleCat R M) U x hxU ≫ 𝟙 (stalk (structurePresheafInModuleCat R M) x)) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op U)) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        s :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: obtain ⟨g, hxg, igU, f, hs⟩ :=
      exists_const _ s x hxU
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ Eq (α :=
    Quiver.Hom (V := ModuleCat R) (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M))
      (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)))
    (localizationtoStalkₗ R M x ≫ stalkToLocalizationₗ R M x)
    (𝟙 (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)))
have hf : Eq (α :=
  (↑(ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)) : Type u) →ₗ[R]
    (↑(ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M)) : Type u))
  (ModuleCat.Hom.hom (localizationtoStalkₗ R M x ≫ stalkToLocalizationₗ R M x))
  (ModuleCat.Hom.hom (𝟙 (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M))))
Tactic: ext1
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ Eq (α := M →ₗ[R] LocalizedModule x.asIdeal.primeCompl M)
    (ModuleCat.Hom.hom (localizationtoStalkₗ R M x ≫ stalkToLocalizationₗ R M x) ∘ₗ
      LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
    (ModuleCat.Hom.hom (𝟙 (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M))) ∘ₗ
      LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
have hf : ∀ (x_1 : M),
  Eq (α := LocalizedModule x.asIdeal.primeCompl M)
    ((ModuleCat.Hom.hom (localizationtoStalkₗ R M x ≫ stalkToLocalizationₗ R M x) ∘ₗ
            LocalizedModule.mkLinearMap x.asIdeal.primeCompl M :
          (a : M) → LocalizedModule x.asIdeal.primeCompl M)
        x_1 :
      LocalizedModule x.asIdeal.primeCompl M)
    (((ModuleCat.Hom.hom (𝟙 (ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M))) ∘ₗ
              LocalizedModule.mkLinearMap x.asIdeal.primeCompl M :
            M →ₗ[R] LocalizedModule x.asIdeal.primeCompl M) :
          (a : M) → LocalizedModule x.asIdeal.primeCompl M)
        x_1 :
      LocalizedModule x.asIdeal.primeCompl M)
Tactic: ext
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
x_1 : M
⊢ Eq (α := LocalizedModule x.asIdeal.primeCompl M)
    ((ModuleCat.Hom.hom (stalkToLocalizationₗ R M x) :
          (a : (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) → LocalizedModule x.asIdeal.primeCompl M)
        ((IsLocalizedModule.lift x.asIdeal.primeCompl (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
              (ModuleCat.Hom.hom (toStalkₗ' R M x)) ⋯ :
            (a : LocalizedModule x.asIdeal.primeCompl M) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
          (LocalizedModule.mk x_1 1)) :
      LocalizedModule x.asIdeal.primeCompl M)
    (LocalizedModule.mk x_1 1 : LocalizedModule x.asIdeal.primeCompl M)
have hf : Eq (α := LocalizedModule x.asIdeal.primeCompl M)
  ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (X := ModuleCat.of R M)
          (ModuleCat.ofHom (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)) :
        (a : M) → LocalizedModule x.asIdeal.primeCompl M)
      x_1 :
    LocalizedModule x.asIdeal.primeCompl M)
  ((LocalizedModule.mkLinearMap x.asIdeal.primeCompl M : (a : M) → LocalizedModule x.asIdeal.primeCompl M) x_1 :
    LocalizedModule x.asIdeal.primeCompl M)
Tactic: rw [← LocalizedModule.mkLinearMap_apply, IsLocalizedModule.lift_apply,
      elementwise_of% toStalkₗ'_stalkToFiberRingHom (M := M) x]
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ IsLocalizedModule (M := (↑(ModuleCat.of R M) : Type u)) x.asIdeal.primeCompl (ModuleCat.Hom.hom (toStalkₗ' R M x))
have h : Eq (α := (↑(ModuleCat.of R M) : Type u) →ₗ[R] (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
  (ModuleCat.Hom.hom (toStalkₗ' R M x))
  ((↑(LinearEquiv.symm (Iso.toLinearEquiv (stalkIsoₗ R M x))) :
      LocalizedModule x.asIdeal.primeCompl M →ₗ[R] (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) ∘ₗ
    LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
Tactic: convert!
    IsLocalizedModule.of_linearEquiv x.asIdeal.primeCompl
      (LocalizedModule.mkLinearMap x.asIdeal.primeCompl M) (stalkIsoₗ R M x).toLinearEquiv.symm
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ Eq (α := (↑(ModuleCat.of R M) : Type u) →ₗ[R] (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    (ModuleCat.Hom.hom (toStalkₗ' R M x))
    ((↑(LinearEquiv.symm (Iso.toLinearEquiv (stalkIsoₗ R M x))) :
        LocalizedModule x.asIdeal.primeCompl M →ₗ[R] (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) ∘ₗ
      LocalizedModule.mkLinearMap x.asIdeal.primeCompl M)
have h : ∀ (m : (↑(ModuleCat.of R M) : Type u)),
  Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom (toStalkₗ' R M x) :
          (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        m :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    (((↑(LinearEquiv.symm (Iso.toLinearEquiv (stalkIsoₗ R M x))) :
              LocalizedModule x.asIdeal.primeCompl M →ₗ[R] (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) ∘ₗ
            LocalizedModule.mkLinearMap x.asIdeal.primeCompl M :
          (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        m :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: ext m
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
m : (↑(ModuleCat.of R M) : Type u)
⊢ Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom (toStalkₗ' R M x) :
          (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        m :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    (((↑(LinearEquiv.symm (Iso.toLinearEquiv (stalkIsoₗ R M x))) :
              LocalizedModule x.asIdeal.primeCompl M →ₗ[R] (↑((structurePresheafInModuleCat R M).stalk x) : Type u)) ∘ₗ
            LocalizedModule.mkLinearMap x.asIdeal.primeCompl M :
          (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        m :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
have h : Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
  ((ModuleCat.Hom.hom (toStalkₗ' R M x) :
        (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      m :
    (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
  ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen (↑1 : R)))) (Y :=
          stalk (structurePresheafInModuleCat R M) x)
          (germ (structurePresheafInModuleCat R M) (basicOpen (↑1 : R)) x sorry) :
        (a : (↑((structurePresheafInModuleCat R M).obj (op (basicOpen (↑1 : R)))) : Type u)) →
          (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      (const m (↑1 : R) (basicOpen (↑1 : R)) sorry) :
    (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: refine .trans ?_ (localizationtoStalkₗ_mk ..).symm
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
m : (↑(ModuleCat.of R M) : Type u)
⊢ Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ModuleCat.Hom.hom (toStalkₗ' R M x) :
          (a : (↑(ModuleCat.of R M) : Type u)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        m :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen (↑1 : R)))) (Y :=
            stalk (structurePresheafInModuleCat R M) x)
            (germ (structurePresheafInModuleCat R M) (basicOpen (↑1 : R)) x ⋯) :
          (a : (↑((structurePresheafInModuleCat R M).obj (op (basicOpen (↑1 : R)))) : Type u)) →
            (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
        (const m (↑1 : R) (basicOpen (↑1 : R)) ⋯) :
      (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
have h : Eq (α := (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
  (((ModuleCat.Hom.hom (germ (structurePresheafInModuleCat R M) ⊤ x _root_.trivial) :
          (↑(ModuleCat.of 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 M))
                    (op ⊤))) :
              Type u) →ₗ[R]
            (↑(stalk (structurePresheafInModuleCat R M) x) : Type u)) :
        (a : (structureSheafInType R M).obj.obj (op ⊤)) → (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      (const m 1 ⊤ sorry) :
    (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
  (((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (X := Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen 1))) (Y :=
            stalk (structurePresheafInModuleCat R M) x) (germ (structurePresheafInModuleCat R M) (basicOpen 1) x sorry) :
          (↑(Functor.obj (structurePresheafInModuleCat R M) (op (basicOpen 1))) : Type u) →ₗ[R]
            (↑(stalk (structurePresheafInModuleCat R M) x) : Type u)) :
        (a : (structureSheafInType R M).obj.obj (op (basicOpen 1))) →
          (↑((structurePresheafInModuleCat R M).stalk x) : Type u))
      (const m 1 (basicOpen 1) sorry) :
    (↑(stalk (structurePresheafInModuleCat R M) x) : Type u))
Tactic: dsimp +instances [toStalkₗ', toOpenₗ]
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ IsLocalizedModule (M' := (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
    x.asIdeal.primeCompl (toStalkₗ R M x)
have h : Eq (α := M →ₗ[R] (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u)) (toStalkₗ R M x)
  ((↑(LinearEquiv.symm (modulePresheafStalkIso R M x)) :
      (↑((structurePresheafInModuleCat R M).stalk x) : Type u) →ₗ[R]
        (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u)) ∘ₗ
    ModuleCat.Hom.hom (toStalkₗ' R M x))
Tactic: convert!
    IsLocalizedModule.of_linearEquiv x.asIdeal.primeCompl (toStalkₗ' R M x).hom
      (modulePresheafStalkIso R M x).symm
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ Eq (α := M →ₗ[R] (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u)) (toStalkₗ R M x)
    ((↑(LinearEquiv.symm (modulePresheafStalkIso R M x)) :
        (↑((structurePresheafInModuleCat R M).stalk x) : Type u) →ₗ[R]
          (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u)) ∘ₗ
      ModuleCat.Hom.hom (toStalkₗ' R M x))
have h : ∀ (m : M),
  Eq (α := (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
    ((toStalkₗ R M x : (a : M) → (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u)) m :
      (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
    (((↑(LinearEquiv.symm (modulePresheafStalkIso R M x)) :
              (↑((structurePresheafInModuleCat R M).stalk x) : Type u) →ₗ[R]
                (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u)) ∘ₗ
            ModuleCat.Hom.hom (toStalkₗ' R M x) :
          (a : M) → (↑(stalk (moduleStructurePresheaf R M).presheaf x) : Type u))
        m :
      (↑(stalk (PresheafOfModules.presheaf (moduleStructurePresheaf R M)) x) : Type u))
Tactic: ext m
-- 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
x : (↑(PrimeSpectrum.Top R) : Type u)
⊢ IsLocalization.AtPrime (↑(stalk (structurePresheafInCommRingCat R) x) : Type u) x.asIdeal
have refine : IsLocalizedModule (M' := (↑(stalk (structurePresheafInCommRingCat R) x) : Type u)) x.asIdeal.primeCompl
  (Algebra.linearMap R (↑(stalk (structurePresheafInCommRingCat R) x) : Type u))
Tactic: refine (isLocalizedModule_iff_isLocalization' _ _).mp ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.StructureSheaf (hypothesis)
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
S : Type u
inst_5 : CommRing S
N : Type u
inst_6 : AddCommGroup N
inst_7 : Module S N
σ : R →+* S
f : M →ₛₗ[σ] N
y : (↑(PrimeSpectrum.Top S) : Type u)
this_1 : Module R N := Module.compHom N σ
this_2 : Algebra R S := σ.toAlgebra
this : IsScalarTower R S N
f' : M →ₗ[R] N :=
  let __spread.0 := f;
  { toAddHom := __spread.0.toAddHom, map_smul' := ⋯ }
x : ↥(PrimeSpectrum.comap σ y).asIdeal.primeCompl
⊢ IsUnit (M := Module.End R (LocalizedModule y.asIdeal.primeCompl N))
    ((algebraMap R (Module.End R (LocalizedModule y.asIdeal.primeCompl N)) :
          (a : R) → Module.End R (LocalizedModule y.asIdeal.primeCompl N))
        (↑x : R) :
      Module.End R (LocalizedModule y.asIdeal.primeCompl N))
have this₁ : IsUnit (M := Module.End S (LocalizedModule y.asIdeal.primeCompl N))
  ((algebraMap S (Module.End S (LocalizedModule y.asIdeal.primeCompl N)) :
        (a : S) → Module.End S (LocalizedModule y.asIdeal.primeCompl N))
      (↑(Subtype.mk (p := fun (x : S) ↦ x ∈ y.asIdeal.primeCompl) ((σ : (a : R) → S) (↑x : R)) sorry) : S) :
    Module.End S (LocalizedModule y.asIdeal.primeCompl N))
Tactic: have := IsLocalizedModule.map_units (S := y.asIdeal.primeCompl)
        (LocalizedModule.mkLinearMap y.asIdeal.primeCompl N) ⟨σ x, x.2⟩
-- 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
S : Type u
inst_5 : CommRing S
N : Type u
inst_6 : AddCommGroup N
inst_7 : Module S N
σ : R →+* S
f : M →ₛₗ[σ] N
y : (↑(PrimeSpectrum.Top S) : Type u)
this_1 : Module R N := Module.compHom N σ
this_2 : Algebra R S := σ.toAlgebra
this_3 : IsScalarTower R S N
f' : M →ₗ[R] N :=
  let __spread.0 := f;
  { toAddHom := __spread.0.toAddHom, map_smul' := ⋯ }
x : ↥(PrimeSpectrum.comap σ y).asIdeal.primeCompl
this :
  IsUnit (M := Module.End S (LocalizedModule y.asIdeal.primeCompl N))
    ((algebraMap S (Module.End S (LocalizedModule y.asIdeal.primeCompl N)) :
          (a : S) → Module.End S (LocalizedModule y.asIdeal.primeCompl N))
        (↑(Subtype.mk (p := fun (x : S) ↦ x ∈ y.asIdeal.primeCompl) ((σ : (a : R) → S) (↑x : R)) ⋯) : S) :
      Module.End S (LocalizedModule y.asIdeal.primeCompl N))
⊢ IsUnit (M := Module.End R (LocalizedModule y.asIdeal.primeCompl N))
    ((algebraMap R (Module.End R (LocalizedModule y.asIdeal.primeCompl N)) :
          (a : R) → Module.End R (LocalizedModule y.asIdeal.primeCompl N))
        (↑x : R) :
      Module.End R (LocalizedModule y.asIdeal.primeCompl N))
have rw : ∀
  (this :
    Function.Bijective (α := LocalizedModule y.asIdeal.primeCompl N) (β := LocalizedModule y.asIdeal.primeCompl N)
      (⇑((algebraMap S (Module.End S (LocalizedModule y.asIdeal.primeCompl N)) :
                (a : S) → Module.End S (LocalizedModule y.asIdeal.primeCompl N))
              (↑(Subtype.mk (p := fun (x : S) ↦ x ∈ y.asIdeal.primeCompl) ((σ : (a : R) → S) (↑x : R)) sorry) : S) :
            Module.End S (LocalizedModule y.asIdeal.primeCompl N)) :
        (a : LocalizedModule y.asIdeal.primeCompl N) → LocalizedModule y.asIdeal.primeCompl N)),
  Function.Bijective (α := LocalizedModule y.asIdeal.primeCompl N) (β := LocalizedModule y.asIdeal.primeCompl N)
    (⇑((algebraMap R (Module.End R (LocalizedModule y.asIdeal.primeCompl N)) :
              (a : R) → Module.End R (LocalizedModule y.asIdeal.primeCompl N))
            (↑x : R) :
          Module.End R (LocalizedModule y.asIdeal.primeCompl N)) :
      (a : LocalizedModule y.asIdeal.primeCompl N) → LocalizedModule y.asIdeal.primeCompl N)
Tactic: rw [Module.End.isUnit_iff] at this ⊢
Cleared hypotheses: this