Theorem: CategoryTheory.pullbackShiftFunctorZero_inv_app
File: Mathlib/CategoryTheory/Shift/Pullback.lean

Proof steps:
state_before: C : Type u_1
inst✝³ : Category.{?u.3896, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ 0 = φ 0
state_after: no goals
tactic: simp
state_before: C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorZero (PullbackShift C φ) A).inv.app X =
    (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X
state_after: C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorZero C B).inv.app X ≫ (HasShift.shift.map (Discrete.addMonoidalFunctor φ).ε).app X =
    (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X
tactic: change (shiftFunctorZero C B).inv.app X ≫ _ = _
state_before: C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorZero C B).inv.app X ≫ (HasShift.shift.map (Discrete.addMonoidalFunctor φ).ε).app X =
    (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X
state_after: C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorZero C B).inv.app X ≫ (HasShift.shift.map (eqToHom ⋯)).app X =
    (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X
tactic: dsimp [Discrete.eqToHom]
state_before: C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ (shiftFunctorZero C B).inv.app X ≫ (HasShift.shift.map (eqToHom ⋯)).app X =
    (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X
state_after: case e_a.e_self
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ HasShift.shift.map (eqToHom ⋯) = (pullbackShiftIso C φ 0 0 ⋯).inv
tactic: congr 2
state_before: case e_a.e_self
C : Type u_1
inst✝³ : Category.{u_4, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
φ : A →+ B
inst✝ : HasShift C B
X : PullbackShift C φ
a₁ a₂ a₃ : A
h : a₁ + a₂ = a₃
b₁ b₂ b₃ : B
h₁ : b₁ = φ a₁
h₂ : b₂ = φ a₂
h₃ : b₃ = φ a₃
⊢ HasShift.shift.map (eqToHom ⋯) = (pullbackShiftIso C φ 0 0 ⋯).inv
state_after: no goals
tactic: apply eqToHom_map