Theorem: CategoryTheory.HomOrthogonal.matrixDecomposition_comp
File: Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean

tactic: ext ⟨c, ⟨⟩⟩ ⟨a, j_property⟩
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
i : ι
⊢ o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * o.matrixDecomposition z i
state_after: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property : a ∈ f ⁻¹' {h c}
⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ =
    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩

tactic: simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property
state_before: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property : a ∈ f ⁻¹' {h c}
⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ =
    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩
state_after: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ =
    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩

tactic: simp only [Matrix.mul_apply, Limits.biproduct.components,
  HomOrthogonal.matrixDecomposition_apply, Category.comp_id, Category.id_comp, Category.assoc,
  End.mul_def, eqToHom_refl, eqToHom_trans_assoc, Finset.sum_congr]
state_before: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ =
    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩
state_after: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c

tactic: conv_lhs => rw [← Category.id_comp w, ← biproduct.total]
state_before: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c
state_after: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c

tactic: simp only [Preadditive.sum_comp, Preadditive.comp_sum]
state_before: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c
state_after: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∑ j : β,
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c

tactic: apply Finset.sum_congr_set
state_before: case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∑ j : β,
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
    ∑ x : ↑(g ⁻¹' {h c}),
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c
state_after: case a.mk.refl.mk.w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ (x : β) (h_1 : x ∈ g ⁻¹' {h c}),
    eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑⟨x, h_1⟩ ≫
              biproduct.ι (fun a => s (g a)) ↑⟨x, h_1⟩ ≫ w ≫ biproduct.π (fun b => s (h b)) c

case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ x ∉ g ⁻¹' {h c},
    eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
      0

tactic: intros
state_before: case a.mk.refl.mk.w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ (x : β) (h_1 : x ∈ g ⁻¹' {h c}),
    eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
      eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            biproduct.π (fun b => s (g b)) ↑⟨x, h_1⟩ ≫
              biproduct.ι (fun a => s (g a)) ↑⟨x, h_1⟩ ≫ w ≫ biproduct.π (fun b => s (h b)) c
state_after: case a.mk.refl.mk.w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
x✝ : β
h✝ : x✝ ∈ g ⁻¹' {h c}
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) x✝ ≫ biproduct.ι (fun b => s (g b)) x✝) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          biproduct.π (fun b => s (g b)) ↑⟨x✝, h✝⟩ ≫
            biproduct.ι (fun a => s (g a)) ↑⟨x✝, h✝⟩ ≫ w ≫ biproduct.π (fun b => s (h b)) c

tactic: simp
state_before: case a.mk.refl.mk.w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
x✝ : β
h✝ : x✝ ∈ g ⁻¹' {h c}
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) x✝ ≫ biproduct.ι (fun b => s (g b)) x✝) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          biproduct.π (fun b => s (g b)) ↑⟨x✝, h✝⟩ ≫
            biproduct.ι (fun a => s (g a)) ↑⟨x✝, h✝⟩ ≫ w ≫ biproduct.π (fun b => s (h b)) c
state_after: no goals

tactic: intro b nm
state_before: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
⊢ ∀ x ∉ g ⁻¹' {h c},
    eqToHom ⋯ ≫
        biproduct.ι (fun a => s (f a)) a ≫
          z ≫
            ((biproduct.π (fun b => s (g b)) x ≫ biproduct.ι (fun b => s (g b)) x) ≫ w) ≫
              biproduct.π (fun b => s (h b)) c =
      0
state_after: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : b ∉ g ⁻¹' {h c}
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    0

tactic: simp only [Set.mem_preimage, Set.mem_singleton_iff] at nm
state_before: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : b ∉ g ⁻¹' {h c}
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    0
state_after: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    0

tactic: simp only [Category.assoc]
state_before: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫
          ((biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b) ≫ w) ≫
            biproduct.π (fun b => s (h b)) c =
    0
state_after: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫ biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    0

tactic: have : biproduct.ι (fun b ↦ s (g b)) b ≫ w ≫ biproduct.π (fun b ↦ s (h b)) c = 0 := by
  apply o.eq_zero nm
state_before: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫ biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    0
state_after: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
this : biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c = 0
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫ biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    0

tactic: simp only [this, comp_zero]
state_before: case a.mk.refl.mk.w'
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
this : biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c = 0
⊢ eqToHom ⋯ ≫
      biproduct.ι (fun a => s (f a)) a ≫
        z ≫ biproduct.π (fun b => s (g b)) b ≫ biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c =
    0
state_after: no goals

tactic: apply o.eq_zero nm
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)
c : γ
a : α
j_property✝ : a ∈ f ⁻¹' {h c}
j_property : f a = h c
b : β
nm : ¬g b = h c
⊢ biproduct.ι (fun b => s (g b)) b ≫ w ≫ biproduct.π (fun b => s (h b)) c = 0
state_after: no goals