[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0
[PROOFSTEP]
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr :=
  by
  intro f
  ext
  · aesop_cat
  · simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
[PROOFSTEP]
intro f
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f = f ≫ biprod.inr
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst
[PROOFSTEP]
aesop_cat
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd
[PROOFSTEP]
simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0
[PROOFSTEP]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl :=
  by
  intro f
  ext
  · aesop_cat
  · simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
⊢ ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
[PROOFSTEP]
intro f
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 = f ≫ biprod.inl
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 ≫ biprod.fst = (f ≫ biprod.inl) ≫ biprod.fst
[PROOFSTEP]
aesop_cat
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 ≫ biprod.snd = (f ≫ biprod.inl) ≫ biprod.snd
[PROOFSTEP]
simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0
[PROOFSTEP]
exact
  ⟨⟨fun f => by simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]⟩,
    ⟨fun f => by simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]⟩⟩
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
f : X ⟶ Y
⊢ leftAdd X Y 0 f = f
[PROOFSTEP]
simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
f : X ⟶ Y
⊢ leftAdd X Y f 0 = f
[PROOFSTEP]
simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0
[PROOFSTEP]
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f :=
  by
  intro f
  ext
  · aesop_cat
  · simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
[PROOFSTEP]
intro f
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.desc 0 f = biprod.snd ≫ f
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.inl ≫ biprod.desc 0 f = biprod.inl ≫ biprod.snd ≫ f
[PROOFSTEP]
aesop_cat
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.inr ≫ biprod.desc 0 f = biprod.inr ≫ biprod.snd ≫ f
[PROOFSTEP]
simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0
[PROOFSTEP]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f :=
  by
  intro f
  ext
  · aesop_cat
  · simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
⊢ ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
[PROOFSTEP]
intro f
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.desc f 0 = biprod.fst ≫ f
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.inl ≫ biprod.desc f 0 = biprod.inl ≫ biprod.fst ≫ f
[PROOFSTEP]
aesop_cat
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.inr ≫ biprod.desc f 0 = biprod.inr ≫ biprod.fst ≫ f
[PROOFSTEP]
simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0
[PROOFSTEP]
exact
  ⟨⟨fun f => by simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]⟩,
    ⟨fun f => by simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]⟩⟩
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
f : X ⟶ Y
⊢ rightAdd X Y 0 f = f
[PROOFSTEP]
simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
f : X ⟶ Y
⊢ rightAdd X Y f 0 = f
[PROOFSTEP]
simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ biprod.inl ≫ diag = biprod.lift f h
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ (biprod.inl ≫ diag) ≫ biprod.fst = biprod.lift f h ≫ biprod.fst
[PROOFSTEP]
simp
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ (biprod.inl ≫ diag) ≫ biprod.snd = biprod.lift f h ≫ biprod.snd
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ biprod.inr ≫ diag = biprod.lift g k
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ (biprod.inr ≫ diag) ≫ biprod.fst = biprod.lift g k ≫ biprod.fst
[PROOFSTEP]
simp
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ (biprod.inr ≫ diag) ≫ biprod.snd = biprod.lift g k ≫ biprod.snd
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
have h₁ : biprod.lift (f +ᵣ g) (h +ᵣ k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag := by ext <;> aesop_cat
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) ≫ biprod.fst = (biprod.lift (𝟙 X) (𝟙 X) ≫ diag) ≫ biprod.fst
[PROOFSTEP]
aesop_cat
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) ≫ biprod.snd = (biprod.lift (𝟙 X) (𝟙 X) ≫ diag) ≫ biprod.snd
[PROOFSTEP]
aesop_cat
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
have h₂ : diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.desc (f +ₗ h) (g +ₗ k) := by
  ext <;> simp [reassoc_of% hd₁, reassoc_of% hd₂]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
ext
[GOAL]
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ biprod.inl ≫ diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.inl ≫ biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
simp [reassoc_of% hd₁, reassoc_of% hd₂]
[GOAL]
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
⊢ biprod.inr ≫ diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.inr ≫ biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
simp [reassoc_of% hd₁, reassoc_of% hd₂]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAdd X Y h k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag
h₂ : diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.desc (leftAdd X Y f h) (leftAdd X Y g k)
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k)
[PROOFSTEP]
rw [leftAdd, h₁, Category.assoc, h₂, rightAdd]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ (f + g) ≫ h = f ≫ h + g ≫ h
[PROOFSTEP]
simp only [add_eq_right_addition, Category.assoc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc f g ≫ h = biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc (f ≫ h) (g ≫ h)
[PROOFSTEP]
congr
[GOAL]
case e_a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ biprod.desc f g ≫ h = biprod.desc (f ≫ h) (g ≫ h)
[PROOFSTEP]
ext
[GOAL]
case e_a.h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ biprod.inl ≫ biprod.desc f g ≫ h = biprod.inl ≫ biprod.desc (f ≫ h) (g ≫ h)
[PROOFSTEP]
simp
[GOAL]
case e_a.h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f g : X ⟶ Y
h : Y ⟶ Z
⊢ biprod.inr ≫ biprod.desc f g ≫ h = biprod.inr ≫ biprod.desc (f ≫ h) (g ≫ h)
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ f ≫ (g + h) = f ≫ g + f ≫ h
[PROOFSTEP]
simp only [add_eq_left_addition, ← Category.assoc]
[GOAL]
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ (f ≫ biprod.lift g h) ≫ biprod.desc (𝟙 Z) (𝟙 Z) = biprod.lift (f ≫ g) (f ≫ h) ≫ biprod.desc (𝟙 Z) (𝟙 Z)
[PROOFSTEP]
congr
[GOAL]
case e_a
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ f ≫ biprod.lift g h = biprod.lift (f ≫ g) (f ≫ h)
[PROOFSTEP]
ext
[GOAL]
case e_a.h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ (f ≫ biprod.lift g h) ≫ biprod.fst = biprod.lift (f ≫ g) (f ≫ h) ≫ biprod.fst
[PROOFSTEP]
simp
[GOAL]
case e_a.h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y Z : C
f : X ⟶ Y
g h : Y ⟶ Z
⊢ (f ≫ biprod.lift g h) ≫ biprod.snd = biprod.lift (f ≫ g) (f ≫ h) ≫ biprod.snd
[PROOFSTEP]
simp
