Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g✝ : X✝ ⟶ Y✝
inst✝¹ : HasEqualizers C
inst✝ : HasPullbacks C
X Y S T : C
f g : X ⟶ Y
s : X ⟶ S
t : Y ⟶ S
hf : f ≫ t = s
hg : g ≫ t = s
v : T ⟶ S
lhs : pullback s v ⟶ pullback t v := pullback.map s v t v f (𝟙 T) (𝟙 S) ⋯ ⋯
rhs : pullback s v ⟶ pullback t v := pullback.map s v t v g (𝟙 T) (𝟙 S) ⋯ ⋯
hl : pullback.fst s v ≫ f = lhs ≫ pullback.fst t v
hr : pullback.fst s v ≫ g = rhs ≫ pullback.fst t v
⊢ equalizer.lift (pullback.map (equalizer.ι f g) (pullback.fst s v) s v (equalizer.ι f g) (pullback.snd s v) s ⋯ ⋯) ⋯ ≫
      pullback.lift (equalizer.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) ⋯)
        (pullback.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) (equalizer.ι lhs rhs ≫ pullback.snd s v) ⋯) ⋯ =
    𝟙 (pullback (equalizer.ι f g) (pullback.fst s v))

Tactic:
ext

Goal state after:
case h₀
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g✝ : X✝ ⟶ Y✝
inst✝¹ : HasEqualizers C
inst✝ : HasPullbacks C
X Y S T : C
f g : X ⟶ Y
s : X ⟶ S
t : Y ⟶ S
hf : f ≫ t = s
hg : g ≫ t = s
v : T ⟶ S
lhs : pullback s v ⟶ pullback t v := ⋯
rhs : pullback s v ⟶ pullback t v := ⋯
hl : pullback.fst s v ≫ f = lhs ≫ pullback.fst t v
hr : pullback.fst s v ≫ g = rhs ≫ pullback.fst t v
⊢ ((equalizer.lift (pullback.map (equalizer.ι f g) (pullback.fst s v) s v (equalizer.ι f g) (pullback.snd s v) s ⋯ ⋯)
            ⋯ ≫
          pullback.lift (equalizer.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) ⋯)
            (pullback.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) (equalizer.ι lhs rhs ≫ pullback.snd s v) ⋯) ⋯) ≫
        pullback.fst (equalizer.ι f g) (pullback.fst s v)) ≫
      equalizer.ι f g =
    (𝟙 (pullback (equalizer.ι f g) (pullback.fst s v)) ≫ pullback.fst (equalizer.ι f g) (pullback.fst s v)) ≫
      equalizer.ι f g

case h₁.h₀
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g✝ : X✝ ⟶ Y✝
inst✝¹ : HasEqualizers C
inst✝ : HasPullbacks C
X Y S T : C
f g : X ⟶ Y
s : X ⟶ S
t : Y ⟶ S
hf : f ≫ t = s
hg : g ≫ t = s
v : T ⟶ S
lhs : pullback s v ⟶ pullback t v := ⋯
rhs : pullback s v ⟶ pullback t v := ⋯
hl : pullback.fst s v ≫ f = lhs ≫ pullback.fst t v
hr : pullback.fst s v ≫ g = rhs ≫ pullback.fst t v
⊢ ((equalizer.lift (pullback.map (equalizer.ι f g) (pullback.fst s v) s v (equalizer.ι f g) (pullback.snd s v) s ⋯ ⋯)
            ⋯ ≫
          pullback.lift (equalizer.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) ⋯)
            (pullback.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) (equalizer.ι lhs rhs ≫ pullback.snd s v) ⋯) ⋯) ≫
        pullback.snd (equalizer.ι f g) (pullback.fst s v)) ≫
      pullback.fst s v =
    (𝟙 (pullback (equalizer.ι f g) (pullback.fst s v)) ≫ pullback.snd (equalizer.ι f g) (pullback.fst s v)) ≫
      pullback.fst s v

case h₁
C : Type u
inst✝² : Category.{v, u} C
X✝ Y✝ : C
f✝ g✝ : X✝ ⟶ Y✝
inst✝¹ : HasEqualizers C
inst✝ : HasPullbacks C
X Y S T : C
f g : X ⟶ Y
s : X ⟶ S
t : Y ⟶ S
hf : f ≫ t = s
hg : g ≫ t = s
v : T ⟶ S
lhs : pullback s v ⟶ pullback t v := ⋯
rhs : pullback s v ⟶ pullback t v := ⋯
hl : pullback.fst s v ≫ f = lhs ≫ pullback.fst t v
hr : pullback.fst s v ≫ g = rhs ≫ pullback.fst t v
⊢ ((equalizer.lift (pullback.map (equalizer.ι f g) (pullback.fst s v) s v (equalizer.ι f g) (pullback.snd s v) s ⋯ ⋯)
            ⋯ ≫
          pullback.lift (equalizer.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) ⋯)
            (pullback.lift (equalizer.ι lhs rhs ≫ pullback.fst s v) (equalizer.ι lhs rhs ≫ pullback.snd s v) ⋯) ⋯) ≫
        pullback.snd (equalizer.ι f g) (pullback.fst s v)) ≫
      pullback.snd s v =
    (𝟙 (pullback (equalizer.ι f g) (pullback.fst s v)) ≫ pullback.snd (equalizer.ι f g) (pullback.fst s v)) ≫
      pullback.snd s v