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

tactic: trans l₁' ≫ pullback.fst
state_before: C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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).inv ≫ pullback.fst ≫ pullback.fst = pullback.fst

tactic: rw [← Category.assoc]
state_before: C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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).inv ≫ pullback.fst ≫ pullback.fst =
    pullback.lift pullback.fst (pullback.snd ≫ pullback.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst

C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst =
    pullback.fst

tactic: congr 1
state_before: C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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).inv ≫ pullback.fst) ≫ pullback.fst =
    pullback.lift pullback.fst (pullback.snd ≫ pullback.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst

C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst =
    pullback.fst

tactic: exact IsLimit.conePointUniqueUpToIso_inv_comp _ _ WalkingCospan.left
state_before: case e_a
C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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).inv ≫ pullback.fst =
    pullback.lift pullback.fst (pullback.snd ≫ pullback.fst)
      (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂)

C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst =
    pullback.fst

tactic: exact pullback.lift_fst _ _ _
state_before: C : Type u
inst✝⁷ : Category C
D : Type u₂
inst✝⁶ : Category 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.fst)
        (_ : pullback.fst ≫ f₁ = (pullback.snd ≫ pullback.fst) ≫ f₂) ≫
      pullback.fst =
    pullback.fst