Theorem: homology.map_eq_lift_desc'_left
File: Mathlib/CategoryTheory/Abelian/Homology.lean

tactic: simp only [kernel.lift_ι_assoc, ← h]
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0

tactic: erw [← reassoc_of% α.w]
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ f ≫ α.right ≫ cokernel.π f' = 0

tactic: simp
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0

tactic: apply homology.hom_from_ext
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f') (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
      cokernel.desc f' g' w' =
    0

tactic: simp
state_before: case h
A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ π' f g w ≫
      desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
          (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
        cokernel.desc f' g' w' =
    π' f g w ≫ 0

tactic: rw [map_eq_desc'_lift_left]
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ map w w' α β h =
    lift f' g' w'
      (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
        (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))
      (_ :
        desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
              (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
            cokernel.desc f' g' w' =
          0)

tactic: apply homology.hom_to_ext
state_before: A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ desc' f g w
      (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
        (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))
      (_ :
        kernel.lift g f w ≫
            lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
              (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =
          0) =
    lift f' g' w'
      (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
        (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))
      (_ :
        desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
              (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
            cokernel.desc f' g' w' =
          0)

tactic: apply homology.hom_from_ext
state_before: case h
A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ desc' f g w
        (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
          (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))
        (_ :
          kernel.lift g f w ≫
              lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
                (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =
            0) ≫
      ι f' g' w' =
    lift f' g' w'
        (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
          (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))
        (_ :
          desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
                (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
              cokernel.desc f' g' w' =
            0) ≫
      ι f' g' w'

tactic: simp
state_before: case h.h
A : Type u
inst✝¹ : Category A
inst✝ : Abelian A
X Y Z : A
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
X' Y' Z' : A
f' : X' ⟶ Y'
g' : Y' ⟶ Z'
w' : f' ≫ g' = 0
α : Arrow.mk f ⟶ Arrow.mk f'
β : Arrow.mk g ⟶ Arrow.mk g'
h : α.right = β.left
⊢ π' f g w ≫
      desc' f g w
          (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
            (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))
          (_ :
            kernel.lift g f w ≫
                lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')
                  (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =
              0) ≫
        ι f' g' w' =
    π' f g w ≫
      lift f' g' w'
          (desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
            (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0))
          (_ :
            desc' f g w (kernel.ι g ≫ β.left ≫ cokernel.π f')
                  (_ : kernel.lift g f w ≫ kernel.ι g ≫ β.left ≫ cokernel.π f' = 0) ≫
                cokernel.desc f' g' w' =
              0) ≫
        ι f' g' w'