l3lab/ntp-mathlib huggingface dataset:
theorem Biprod.column_nonzero_of_iso {W X Y Z : C} (f : W ⊞ X ⟶ Y ⊞ Z) [IsIso f] :
    𝟙 W = 0 ∨ biprod.inl ≫ f ≫ biprod.fst ≠ 0 ∨ biprod.inl ≫ f ≫ biprod.snd ≠ 0 := by
  by_contra! h
  rcases h with ⟨nz, a₁, a₂⟩
  set x := biprod.inl ≫ f ≫ inv f ≫ biprod.fst
  have h₁ : x = 𝟙 W := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biprod.total]
    conv_lhs =>
      slice 2 3
      rw [comp_add]
    simp only [Category.assoc]
    rw [comp_add_assoc, add_comp]
    conv_lhs =>
      congr
      next => skip
      slice 1 3
      rw [a₂]
    simp only [zero_comp, add_zero]
    conv_lhs =>
      slice 1 3
      rw [a₁]
    simp only [zero_comp]
  

Proof state:
case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁ X₂ Y₁ Y₂ : C
f₁₁ : X₁ ⟶ Y₁
f₁₂ : X₁ ⟶ Y₂
f₂₁ : X₂ ⟶ Y₁
f₂₂ : X₂ ⟶ Y₂
W X Y Z : C
f : W ⊞ X ⟶ Y ⊞ Z
inst✝ : IsIso f
nz : 𝟙 W ≠ 0
a₁ : biprod.inl ≫ f ≫ biprod.fst = 0
a₂ : biprod.inl ≫ f ≫ biprod.snd = 0
x : W ⟶ W := biprod.inl ≫ f ≫ inv f ≫ biprod.fst
h₁ : x = 𝟙 W
h₀ : x = 0
⊢ False

Next tactic:
exact nz (h₁.symm.trans h₀)
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
⊢ (∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0

Next tactic:
cases nonempty_fintype τ
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  

Proof state:
case intro
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
⊢ (∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0

Next tactic:
intro z
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  

Proof state:
case intro
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
⊢ 𝟙 (S s) = 0

Next tactic:
have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
t : τ
W : C
h : T t ⟶ W
⊢ biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h

Next tactic:
simp only [← Category.assoc]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
t : τ
W : C
h : T t ⟶ W
⊢ ((biproduct.ι S s ≫ f) ≫ biproduct.π T t) ≫ h = 0 ≫ h

Next tactic:
apply eq_whisker
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    

Proof state:
case w
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
t : τ
W : C
h : T t ⟶ W
⊢ (biproduct.ι S s ≫ f) ≫ biproduct.π T t = 0

Next tactic:
simp only [Category.assoc]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    

Proof state:
case w
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
t : τ
W : C
h : T t ⟶ W
⊢ biproduct.ι S s ≫ f ≫ biproduct.π T t = 0

Next tactic:
apply z
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  

Proof state:
case intro
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
⊢ 𝟙 (S s) = 0

Next tactic:
set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  

Proof state:
case intro
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
⊢ 𝟙 (S s) = 0

Next tactic:
have h₁ : x = 𝟙 (S s) := by simp
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by 

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
⊢ x = 𝟙 (S s)

Next tactic:
simp
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  

Proof state:
case intro
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
⊢ 𝟙 (S s) = 0

Next tactic:
have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      congr
      congr
      next => skip
      intro j; simp only [reassoced]
    simp
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
⊢ x = 0

Next tactic:
dsimp
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
⊢ biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s = 0

Next tactic:
rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
⊢ biproduct.ι S s ≫ f ≫ (∑ j : τ, biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0

Next tactic:
simp only [comp_sum_assoc]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
⊢ (∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0

Next tactic:
conv_lhs =>
      congr
      congr
      next => skip
      intro j; simp only [reassoced]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| (∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s

Next tactic:
congr
      congr
      next => skip
      intro j; simp only [reassoced]
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| (∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s

Next tactic:
congr
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      congr
      

Proof state:
case a
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| ∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j
case a
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| inv f ≫ biproduct.π S s

Next tactic:
congr
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      congr
      congr
      

Proof state:
case a.s
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| Finset.univ
case a.f
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| fun j => biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j
case a
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| inv f ≫ biproduct.π S s

Next tactic:
next => skip
theorem Biproduct.column_nonzero_of_iso' {σ τ : Type} [Finite τ] {S : σ → C} [HasBiproduct S]
    {T : τ → C} [HasBiproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [IsIso f] :
    (∀ t : τ, biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0 := by
  cases nonempty_fintype τ
  intro z
  have reassoced {t : τ} {W : C} (h : _ ⟶ W) :
    biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by
    simp only [← Category.assoc]
    apply eq_whisker
    simp only [Category.assoc]
    apply z
  set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
  have h₁ : x = 𝟙 (S s) := by simp
  have h₀ : x = 0 := by
    dsimp
    rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]
    simp only [comp_sum_assoc]
    conv_lhs =>
      congr
      congr
      next => 

Proof state:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
σ τ : Type
inst✝³ : Finite τ
S : σ → C
inst✝² : HasBiproduct S
T : τ → C
inst✝¹ : HasBiproduct T
s : σ
f : ⨁ S ⟶ ⨁ T
inst✝ : IsIso f
val✝ : Fintype τ
z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0
reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h
x : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s
h₁ : x = 𝟙 (S s)
| Finset.univ

Next tactic:
skip