Theorem: CategoryTheory.Sheaf.isLocallySurjective_iff_epi'
File: Mathlib/CategoryTheory/Sites/EpiMono.lean

tactic: constructor
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
⊢ IsLocallySurjective φ ↔ Epi φ
state_after: case mp
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
⊢ IsLocallySurjective φ → Epi φ

case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
⊢ Epi φ → IsLocallySurjective φ

tactic: intro
state_before: case mp
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
⊢ IsLocallySurjective φ → Epi φ
state_after: case mp
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : IsLocallySurjective φ
⊢ Epi φ

tactic: infer_instance
state_before: case mp
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : IsLocallySurjective φ
⊢ Epi φ
state_after: no goals

tactic: intro
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
⊢ Epi φ → IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
⊢ IsLocallySurjective φ

tactic: let data := (locallySurjective J A).factorizationData (locallyInjective J A) φ
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
⊢ IsLocallySurjective φ

tactic: have : IsLocallySurjective data.i := data.hi
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this : IsLocallySurjective data.i
⊢ IsLocallySurjective φ

tactic: have : IsLocallyInjective data.p := data.hp
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this : IsLocallySurjective data.i
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝ : IsLocallySurjective data.i
this : IsLocallyInjective data.p
⊢ IsLocallySurjective φ

tactic: have : Epi data.p := epi_of_epi_fac data.fac
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝ : IsLocallySurjective data.i
this : IsLocallyInjective data.p
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝¹ : IsLocallySurjective data.i
this✝ : IsLocallyInjective data.p
this : Epi data.p
⊢ IsLocallySurjective φ

tactic: have := mono_of_isLocallyInjective data.p
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝¹ : IsLocallySurjective data.i
this✝ : IsLocallyInjective data.p
this : Epi data.p
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝² : IsLocallySurjective data.i
this✝¹ : IsLocallyInjective data.p
this✝ : Epi data.p
this : Mono data.p
⊢ IsLocallySurjective φ

tactic: have := isIso_of_mono_of_epi data.p
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝² : IsLocallySurjective data.i
this✝¹ : IsLocallyInjective data.p
this✝ : Epi data.p
this : Mono data.p
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝³ : IsLocallySurjective data.i
this✝² : IsLocallyInjective data.p
this✝¹ : Epi data.p
this✝ : Mono data.p
this : IsIso data.p
⊢ IsLocallySurjective φ

tactic: rw [← data.fac]
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝³ : IsLocallySurjective data.i
this✝² : IsLocallyInjective data.p
this✝¹ : Epi data.p
this✝ : Mono data.p
this : IsIso data.p
⊢ IsLocallySurjective φ
state_after: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝³ : IsLocallySurjective data.i
this✝² : IsLocallyInjective data.p
this✝¹ : Epi data.p
this✝ : Mono data.p
this : IsIso data.p
⊢ IsLocallySurjective (data.i ≫ data.p)

tactic: infer_instance
state_before: case mpr
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : ConcreteCategory A
inst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A
inst✝³ : J.WEqualsLocallyBijective A
inst✝² : HasSheafify J A
inst✝¹ : J.HasSheafCompose (forget A)
inst✝ : Balanced (Sheaf J A)
F G : Sheaf J A
φ : F ⟶ G
a✝ : Epi φ
data : (locallySurjective J A).MapFactorizationData (locallyInjective J A) φ :=
  (locallySurjective J A).factorizationData (locallyInjective J A) φ
this✝³ : IsLocallySurjective data.i
this✝² : IsLocallyInjective data.p
this✝¹ : Epi data.p
this✝ : Mono data.p
this : IsIso data.p
⊢ IsLocallySurjective (data.i ≫ data.p)
state_after: no goals