Theorem: homology.map_eq_desc'_lift_left
File: Mathlib/CategoryTheory/Abelian/Homology.lean

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
⊢ (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0

tactic: 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
⊢ 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

tactic: simp only [← h, Category.assoc, zero_comp, lift_ι, kernel.lift_ι_assoc]
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
⊢ (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)) ≫
      ι f' g' w' =
    0 ≫ ι f' g' w'

tactic: erw [← reassoc_of% α.w]
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
⊢ f ≫ α.right ≫ cokernel.π f' = 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
⊢ 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
⊢ map w w' α β h =
    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)

tactic: simp only [π'_map, π'_desc']
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 ≫ map w w' α β h =
    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)

tactic: dsimp [π', lift]
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
⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫ π' 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)

tactic: rw [Iso.eq_comp_inv]
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
⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫
      cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv =
    kernel.lift (cokernel.desc 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) ≫
      (homologyIsoKernelDesc f' g' w').inv

tactic: dsimp [homologyIsoKernelDesc]
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
⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫
        cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫
      (homologyIsoKernelDesc f' g' w').hom =
    kernel.lift (cokernel.desc 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)

tactic: apply Limits.equalizer.hom_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
⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫
        cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫
      (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w' =
    kernel.lift (cokernel.desc 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)

tactic: simp [h]
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
⊢ ((kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫
          cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫
        (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w') ≫
      equalizer.ι (cokernel.desc f' g' w') 0 =
    kernel.lift (cokernel.desc 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) ≫
      equalizer.ι (cokernel.desc f' g' w') 0