Theorem: CategoryTheory.GlueData.t'_iij
File: Mathlib/CategoryTheory/GlueData.lean

tactic: have eq₁ := D.t_fac i i j
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = pullback.fst ≫ D.t i i
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom

tactic: have eq₂ := (IsIso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = pullback.fst ≫ D.t i i
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = pullback.fst ≫ D.t i i
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom

tactic: rw [D.t_id, Category.comp_id, eq₂] at eq₁
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = pullback.fst ≫ D.t i i
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom

tactic: have eq₃ := (IsIso.eq_comp_inv (D.f i i)).mp eq₁
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₃ : (D.t' i i j ≫ pullback.snd) ≫ D.f i i = pullback.snd ≫ D.f i j
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom

tactic: rw [Category.assoc, ← pullback.condition, ← Category.assoc] at eq₃
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₃ : (D.t' i i j ≫ pullback.snd) ≫ D.f i i = pullback.snd ≫ D.f i j
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₃ : (D.t' i i j ≫ pullback.fst) ≫ D.f i j = pullback.snd ≫ D.f i j
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom

tactic: exact
  Mono.right_cancellation _ _
    ((Mono.right_cancellation _ _ eq₃).trans (pullbackSymmetry_hom_comp_fst _ _).symm)
state_before: C : Type u₁
inst✝¹ : Category.{v, u₁} C
C' : Type u₂
inst✝ : Category.{v, u₂} C'
D : GlueData C
i j : D.J
eq₁ : D.t' i i j ≫ pullback.snd = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₂ : pullback.fst = (pullback.snd ≫ D.f i j) ≫ inv (D.f i i)
eq₃ : (D.t' i i j ≫ pullback.fst) ≫ D.f i j = pullback.snd ≫ D.f i j
⊢ D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom
state_after: no goals