Theorem: CategoryTheory.Limits.pullbackAssoc_hom_snd_fst
File: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean

tactic: trans l₂ ≫ pullback.fst
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ pullback.snd
state_after: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd ≫ pullback.fst =
    pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst

C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst = pullback.fst ≫ pullback.snd

tactic: rw [← Category.assoc]
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd ≫ pullback.fst =
    pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst
state_after: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ ((pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd) ≫ pullback.fst =
    pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst

tactic: congr 1
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ ((pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd) ≫ pullback.fst =
    pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst
state_after: case e_a
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd = pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯

tactic: exact IsLimit.conePointUniqueUpToIso_hom_comp _ _ WalkingCospan.right
state_before: case e_a
C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).hom ≫ pullback.snd = pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯
state_after: no goals

tactic: exact pullback.lift_fst _ _ _
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (pullback.snd ≫ sorryAx (sorryAx C true ⟶ Y₂) true) f₄
inst✝² : HasPullback f₁ (pullback.fst ≫ f₂)
inst✝¹ : HasPullback (pullback.snd ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst ≫ f₂)
⊢ pullback.lift (pullback.fst ≫ pullback.snd) pullback.snd ⋯ ≫ pullback.fst = pullback.fst ≫ pullback.snd
state_after: no goals