Theorem: CategoryTheory.Limits.cokernelCoforkBiproductFromSubtype_cocone
File: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean

Proof steps:
state_before: J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
⊢ biproduct.fromSubtype f p ≫ biproduct.toSubtype f pᶜ = 0
state_after: case w.w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ biproduct.ι (Subtype.restrict p f) k ≫
      (biproduct.fromSubtype f p ≫ biproduct.toSubtype f pᶜ) ≫ biproduct.π (Subtype.restrict pᶜ f) j =
    biproduct.ι (Subtype.restrict p f) k ≫ 0 ≫ biproduct.π (Subtype.restrict pᶜ f) j
tactic: ext j k
state_before: case w.w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ biproduct.ι (Subtype.restrict p f) k ≫
      (biproduct.fromSubtype f p ≫ biproduct.toSubtype f pᶜ) ≫ biproduct.π (Subtype.restrict pᶜ f) j =
    biproduct.ι (Subtype.restrict p f) k ≫ 0 ≫ biproduct.π (Subtype.restrict pᶜ f) j
state_after: case w.w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ (if h : (p ↑k)ᶜ then biproduct.ι (Subtype.restrict pᶜ f) ⟨↑k, ⋯⟩ else 0) ≫ biproduct.π (Subtype.restrict pᶜ f) j = 0
tactic: simp only [Category.assoc, Pi.compl_apply, biproduct.ι_fromSubtype_assoc,
          biproduct.ι_toSubtype_assoc, comp_zero, zero_comp]
state_before: case w.w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ (if h : (p ↑k)ᶜ then biproduct.ι (Subtype.restrict pᶜ f) ⟨↑k, ⋯⟩ else 0) ≫ biproduct.π (Subtype.restrict pᶜ f) j = 0
state_after: case w.w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ 0 ≫ biproduct.π (Subtype.restrict pᶜ f) j = 0

case w.w.hnc
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
j : Subtype pᶜ
k : Subtype p
⊢ ¬(p ↑k)ᶜ
tactic: rw [dif_neg]
state_before: J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
⊢ ∀ {Z' : C} (g' : ⨁ f ⟶ Z') (eq' : biproduct.fromSubtype f p ≫ g' = 0),
    biproduct.toSubtype f pᶜ ≫ (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' eq' = g'
state_after: J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
⊢ biproduct.toSubtype f pᶜ ≫ (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' w = g'
tactic: intro W g' w
state_before: J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
⊢ biproduct.toSubtype f pᶜ ≫ (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' w = g'
state_after: case w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
⊢ biproduct.ι f j ≫ biproduct.toSubtype f pᶜ ≫ (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' w =
    biproduct.ι f j ≫ g'
tactic: ext j
state_before: case w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
⊢ biproduct.ι f j ≫ biproduct.toSubtype f pᶜ ≫ (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' w =
    biproduct.ι f j ≫ g'
state_after: case w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
⊢ (if (p j)ᶜ then 𝟙 (f j) else 0) ≫ biproduct.ι f j ≫ g' = biproduct.ι f j ≫ g'
tactic: simp only [biproduct.toSubtype_fromSubtype_assoc, Pi.compl_apply, biproduct.ι_map_assoc]
state_before: case w
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
⊢ (if (p j)ᶜ then 𝟙 (f j) else 0) ≫ biproduct.ι f j ≫ g' = biproduct.ι f j ≫ g'
state_after: case pos
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
h : (p j)ᶜ
⊢ 𝟙 (f j) ≫ biproduct.ι f j ≫ g' = biproduct.ι f j ≫ g'

case neg
J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
W : C
g' : ⨁ f ⟶ W
w : biproduct.fromSubtype f p ≫ g' = 0
j : K
h : ¬(p j)ᶜ
⊢ 0 ≫ biproduct.ι f j ≫ g' = biproduct.ι f j ≫ g'
tactic: split_ifs with h
state_before: J : Type w
K✝ : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
K : Type
inst✝¹ : Finite K
inst✝ : HasFiniteBiproducts C
f : K → C
p : Set K
⊢ ∀ {Z' : C} (g' : ⨁ f ⟶ Z') (eq' : biproduct.fromSubtype f p ≫ g' = 0) (m : ⨁ Subtype.restrict pᶜ f ⟶ Z'),
    biproduct.toSubtype f pᶜ ≫ m = g' → m = (fun {W} g x => biproduct.fromSubtype f pᶜ ≫ g) g' eq'
state_after: no goals
tactic: aesop_cat