awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasWeakSheafify J AddCommGrpCat
F : SheafOfModules.{v, v', u', u} R
simp :
  IsIso (X := (SheafOfModules.toSheaf R).obj ((sheafification α).obj (F.val.restrictScalarsObj α)))
    (NatTrans.app (Adjunction.counit (CategoryTheory.sheafificationAdjunction J AddCommGrpCat))
      ((SheafOfModules.toSheaf R).obj F))
⊢ IsIso (X := (SheafOfModules.toSheaf R).obj (((SheafOfModules.forget R ⋙ restrictScalars α) ⋙ sheafification α).obj F))
    ((SheafOfModules.toSheaf R).map (NatTrans.app (Adjunction.counit (sheafificationAdjunction α)) F))

Tactic:
simp only [Functor.comp_obj, SheafOfModules.forget_obj, restrictScalars_obj, Functor.id_obj,
    toSheaf_map_sheafificationAdjunction_counit_app]

Premises used: CategoryTheory.Functor.comp_obj, PresheafOfModules.toSheaf_map_sheafificationAdjunction_counit_app, PresheafOfModules.restrictScalars_obj, CategoryTheory.Functor.id_obj, SheafOfModules.forget_obj
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasWeakSheafify J AddCommGrpCat
F : SheafOfModules.{v, v', u', u} R
⊢ IsIso (X := (SheafOfModules.toSheaf R).obj ((sheafification α).obj (F.val.restrictScalarsObj α)))
    (NatTrans.app (Adjunction.counit (CategoryTheory.sheafificationAdjunction J AddCommGrpCat))
      ((SheafOfModules.toSheaf R).obj F))

Tactic:
infer_instance
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasWeakSheafify J AddCommGrpCat
⊢ (SheafOfModules.forget R ⋙ restrictScalars α).Full

Tactic:
(sheafificationAdjunction.{v} α).fullyFaithfulROfIsIsoCounit.full

Premises used: CategoryTheory.Adjunction.fullyFaithfulROfIsIsoCounit, CategoryTheory.Functor.FullyFaithful.full, PresheafOfModules.sheafificationAdjunction
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasWeakSheafify J AddCommGrpCat
⊢ (SheafOfModules.forget R ⋙ restrictScalars α).Faithful

Tactic:
(sheafificationAdjunction.{v} α).fullyFaithfulROfIsIsoCounit.faithful

Premises used: CategoryTheory.Adjunction.fullyFaithfulROfIsIsoCounit, CategoryTheory.Functor.FullyFaithful.faithful, PresheafOfModules.sheafificationAdjunction
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasSheafify J AddCommGrpCat
⊢ PreservesFiniteLimits (sheafification α ⋙ SheafOfModules.toSheaf R)

Tactic:
comp_preservesFiniteLimits (toPresheaf.{v} R₀) (presheafToSheaf J AddCommGrpCat)

Premises used: PresheafOfModules.sheafification, CategoryTheory.Functor.comp, CategoryTheory.presheafToSheaf, AddCommGrpCat, AddMonoidHom.mk', PresheafOfModules.toPresheaf, PresheafOfModules.sheafify, CategoryTheory.Limits.comp_preservesFiniteLimits, PresheafOfModules.presheaf
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasSheafify J AddCommGrpCat
⊢ (SheafOfModules.toSheaf R).ReflectsIsomorphisms

Tactic:
reflectsIsomorphisms_of_comp (SheafOfModules.toSheaf.{v} R) (sheafToPresheaf J _)

Premises used: SheafOfModules.toSheaf, CategoryTheory.sheafToPresheaf, CategoryTheory.reflectsIsomorphisms_of_comp
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafification
Goal:
C : Type u'
inst : Category.{v', u'} C
J : GrothendieckTopology.{v', u'} C
R₀ : CategoryTheory.Functor.{v', u, u', u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
inst_3 :
  J.WEqualsLocallyBijective AddCommGrpCat (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v))
    (CA := AddCommGrpCat.carrier)
inst_4 : HasSheafify J AddCommGrpCat
⊢ PreservesFiniteLimits (sheafification α)

Tactic:
preservesFiniteLimits_of_reflects_of_preserves
    (sheafification.{v} α) (SheafOfModules.toSheaf.{v} R)

Premises used: PresheafOfModules.sheafification, CategoryTheory.Limits.preservesFiniteLimits_of_reflects_of_preserves, SheafOfModules.toSheaf
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ R : Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
α : R₀ ⟶ R
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Functor.{v₁, v, u₁, v + 1} Cᵒᵖ AddCommGrpCat
φ : M₀.presheaf ⟶ A
inst_2 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
hA :
  Presheaf.IsSeparated J A (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier)
Y : C
r₀ r₀' : (↑(R₀.obj (Opposite.op Y)) : Type u)
m₀ m₀' : (↑(M₀.obj (Opposite.op Y)) : Type v)
hr₀ :
  Eq (α := (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀ :
      (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀' :
      (↑(R.obj (Opposite.op Y)) : Type u))
hm₀ :
  Eq (α := (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀ :
      (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀' :
      (↑(A.obj (Opposite.op Y)) : Type v))
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop (Opposite.op Y))))
    ((J : (a : C) → Set (Sieve a)) (Opposite.unop (Opposite.op Y)))
    (Presheaf.equalizerSieve (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
      r₀ r₀')

Tactic:
Presheaf.equalizerSieve_mem J α _ _ hr₀

Premises used: CategoryTheory.Presheaf.equalizerSieve_mem
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ R : Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
α : R₀ ⟶ R
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Functor.{v₁, v, u₁, v + 1} Cᵒᵖ AddCommGrpCat
φ : M₀.presheaf ⟶ A
inst_2 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
hA :
  Presheaf.IsSeparated J A (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier)
Y : C
r₀ r₀' : (↑(R₀.obj (Opposite.op Y)) : Type u)
m₀ m₀' : (↑(M₀.obj (Opposite.op Y)) : Type v)
hr₀ :
  Eq (α := (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀ :
      (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀' :
      (↑(R.obj (Opposite.op Y)) : Type u))
hm₀ :
  Eq (α := (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀ :
      (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀' :
      (↑(A.obj (Opposite.op Y)) : Type v))
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop (Opposite.op Y))))
    ((J : (a : C) → Set (Sieve a)) (Opposite.unop (Opposite.op Y)))
    (Presheaf.equalizerSieve (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
      r₀ r₀')

Tactic:
exact Presheaf.equalizerSieve_mem J α _ _ hr₀

Premises used: CategoryTheory.Presheaf.equalizerSieve_mem
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ R : Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
α : R₀ ⟶ R
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Functor.{v₁, v, u₁, v + 1} Cᵒᵖ AddCommGrpCat
φ : M₀.presheaf ⟶ A
inst_2 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
hA :
  Presheaf.IsSeparated J A (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier)
Y : C
r₀ r₀' : (↑(R₀.obj (Opposite.op Y)) : Type u)
m₀ m₀' : (↑(M₀.obj (Opposite.op Y)) : Type v)
hr₀ :
  Eq (α := (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀ :
      (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀' :
      (↑(R.obj (Opposite.op Y)) : Type u))
hm₀ :
  Eq (α := (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀ :
      (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀' :
      (↑(A.obj (Opposite.op Y)) : Type v))
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop (Opposite.op Y))))
    ((J : (a : C) → Set (Sieve a)) (Opposite.unop (Opposite.op Y)))
    (Presheaf.equalizerSieve (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
      AddCommGrpCat.carrier) (F := M₀.presheaf) m₀ m₀')

Tactic:
Presheaf.equalizerSieve_mem J φ _ _ hm₀

Premises used: CategoryTheory.Presheaf.equalizerSieve_mem
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ R : Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
α : R₀ ⟶ R
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Functor.{v₁, v, u₁, v + 1} Cᵒᵖ AddCommGrpCat
φ : M₀.presheaf ⟶ A
inst_2 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
hA :
  Presheaf.IsSeparated J A (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier)
Y : C
r₀ r₀' : (↑(R₀.obj (Opposite.op Y)) : Type u)
m₀ m₀' : (↑(M₀.obj (Opposite.op Y)) : Type v)
hr₀ :
  Eq (α := (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀ :
      (↑(R.obj (Opposite.op Y)) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := RingCat.carrier)
            (X := R₀.obj (Opposite.op Y)) (α.app (Opposite.op Y)) :
          (a : (↑(R₀.obj (Opposite.op Y)) : Type u)) → (↑(R.obj (Opposite.op Y)) : Type u))
        r₀' :
      (↑(R.obj (Opposite.op Y)) : Type u))
hm₀ :
  Eq (α := (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀ :
      (↑(A.obj (Opposite.op Y)) : Type v))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CC :=
            AddCommGrpCat.carrier) (X := M₀.presheaf.obj (Opposite.op Y)) (φ.app (Opposite.op Y)) :
          (a : (↑(M₀.presheaf.obj (Opposite.op Y)) : Type v)) → (↑(A.obj (Opposite.op Y)) : Type v))
        m₀' :
      (↑(A.obj (Opposite.op Y)) : Type v))
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop (Opposite.op Y))))
    ((J : (a : C) → Set (Sieve a)) (Opposite.unop (Opposite.op Y)))
    (Presheaf.equalizerSieve (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
      AddCommGrpCat.carrier) (F := M₀.presheaf) m₀ m₀')

Tactic:
exact Presheaf.equalizerSieve_mem J φ _ _ hm₀

Premises used: CategoryTheory.Presheaf.equalizerSieve_mem
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj (P := Presheaf.IsSheaf J)
inst_3 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
inst_4 :
  Presheaf.IsLocallySurjective J (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier) φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : (↑(R.obj (P := Presheaf.IsSheaf J).obj X) : Type u)
m m' : (↑(A.obj (P := Presheaf.IsSheaf J).obj X) : Type v)
S : Sieve.{v₁, u₁} (Opposite.unop X) :=
  Presheaf.imageSieve (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA := RingCat.carrier) α r ⊓
    Presheaf.imageSieve (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
      AddCommGrpCat.carrier) φ m
rj :
  Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop X))) ((J : (a : C) → Set (Sieve a)) (Opposite.unop X))
    (Presheaf.imageSieve (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA := RingCat.carrier) α r)
sj :
  Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop X))) ((J : (a : C) → Set (Sieve a)) (Opposite.unop X))
    (Presheaf.imageSieve (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
      AddCommGrpCat.carrier) φ m)
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop X))) ((J : (a : C) → Set (Sieve a)) (Opposite.unop X)) S

Tactic:
apply J.intersection_covering

Premises used: CategoryTheory.GrothendieckTopology.intersection_covering
-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
R₀ : CategoryTheory.Functor.{v₁, u, u₁, u + 1} Cᵒᵖ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj (P := Presheaf.IsSheaf J)
inst_1 :
  Presheaf.IsLocallyInjective (FD := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CD := RingCat.carrier)
    J α
inst_2 :
  Presheaf.IsLocallySurjective J (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA :=
    RingCat.carrier) α
M₀ : PresheafOfModules.{v, v₁, u₁, u} R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj (P := Presheaf.IsSheaf J)
inst_3 :
  Presheaf.IsLocallyInjective (FD := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CD :=
    AddCommGrpCat.carrier) J φ
inst_4 :
  Presheaf.IsLocallySurjective J (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
    AddCommGrpCat.carrier) φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : (↑(R.obj (P := Presheaf.IsSheaf J).obj X) : Type u)
m m' : (↑(A.obj (P := Presheaf.IsSheaf J).obj X) : Type v)
S : Sieve.{v₁, u₁} (Opposite.unop X) :=
  Presheaf.imageSieve (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA := RingCat.carrier) α r ⊓
    Presheaf.imageSieve (FA := fun (x1 x2 : AddCommGrpCat.{v}) ↦ (↑x1 : Type v) →+ (↑x2 : Type v)) (CA :=
      AddCommGrpCat.carrier) φ m
⊢ Membership.mem (γ := Set (Sieve.{v₁, u₁} (Opposite.unop X))) ((J : (a : C) → Set (Sieve a)) (Opposite.unop X))
    (Presheaf.imageSieve (FA := fun (R S : RingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CA := RingCat.carrier) α r)

Tactic:
apply Presheaf.imageSieve_mem

Premises used: CategoryTheory.Presheaf.imageSieve_mem