Theorem: CategoryTheory.ShortComplex.Exact.epi_f
File: Mathlib/Algebra/Homology/ShortComplex/Exact.lean

tactic: have := hS.hasHomology
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
⊢ Epi S.f
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this : S.HasHomology
⊢ Epi S.f

tactic: have := hS.mono_fromOpcycles
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this : S.HasHomology
⊢ Epi S.f
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝ : S.HasHomology
this : Mono S.fromOpcycles
⊢ Epi S.f

tactic: have : S.pOpcycles = 0 := by rw [← cancel_mono S.fromOpcycles, zero_comp, p_fromOpcycles, hg]
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝ : S.HasHomology
this : Mono S.fromOpcycles
⊢ Epi S.f
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
⊢ Epi S.f

tactic: apply Preadditive.epi_of_cancel_zero
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
⊢ Epi S.f
state_after: case h
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
⊢ ∀ {R : C} (g : S.X₂ ⟶ R), S.f ≫ g = 0 → g = 0

tactic: intro A x₂ hx₂
state_before: case h
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
⊢ ∀ {R : C} (g : S.X₂ ⟶ R), S.f ≫ g = 0 → g = 0
state_after: case h
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
A : C
x₂ : S.X₂ ⟶ A
hx₂ : S.f ≫ x₂ = 0
⊢ x₂ = 0

tactic: rw [← S.p_descOpcycles x₂ hx₂, this, zero_comp]
state_before: case h
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝¹ : S.HasHomology
this✝ : Mono S.fromOpcycles
this : S.pOpcycles = 0
A : C
x₂ : S.X₂ ⟶ A
hx₂ : S.f ≫ x₂ = 0
⊢ x₂ = 0
state_after: no goals

tactic: rw [← cancel_mono S.fromOpcycles, zero_comp, p_fromOpcycles, hg]
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{?u.69371, u_2} D
inst✝¹ : Preadditive C
inst✝ : Preadditive D
S : ShortComplex C
hS : S.Exact
hg : S.g = 0
this✝ : S.HasHomology
this : Mono S.fromOpcycles
⊢ S.pOpcycles = 0
state_after: no goals