Theorem: CategoryTheory.Limits.kernelForkBiproductToSubtype_cone
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 pᶜ ≫ biproduct.toSubtype f p = 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 p
k : Subtype pᶜ
⊢ biproduct.ι (Subtype.restrict pᶜ f) k ≫
      (biproduct.fromSubtype f pᶜ ≫ biproduct.toSubtype f p) ≫ biproduct.π (Subtype.restrict p f) j =
    biproduct.ι (Subtype.restrict pᶜ f) k ≫ 0 ≫ biproduct.π (Subtype.restrict p 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 p
k : Subtype pᶜ
⊢ biproduct.ι (Subtype.restrict pᶜ f) k ≫
      (biproduct.fromSubtype f pᶜ ≫ biproduct.toSubtype f p) ≫ biproduct.π (Subtype.restrict p f) j =
    biproduct.ι (Subtype.restrict pᶜ f) k ≫ 0 ≫ biproduct.π (Subtype.restrict p 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 p
k : Subtype pᶜ
⊢ (if h : p ↑k then biproduct.ι (Subtype.restrict p f) ⟨↑k, h⟩ else 0) ≫ biproduct.π (Subtype.restrict p f) j = 0
tactic: simp only [Category.assoc, 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 p
k : Subtype pᶜ
⊢ (if h : p ↑k then biproduct.ι (Subtype.restrict p f) ⟨↑k, h⟩ else 0) ≫ biproduct.π (Subtype.restrict p 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 p
k : Subtype pᶜ
⊢ 0 ≫ biproduct.π (Subtype.restrict p f) j = 0
tactic: erw [dif_neg k.2]
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 p
k : Subtype pᶜ
⊢ 0 ≫ biproduct.π (Subtype.restrict p f) j = 0
state_after: no goals
tactic: simp only [zero_comp]
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' : W' ⟶ ⨁ f) (eq' : g' ≫ biproduct.toSubtype f p = 0),
    (fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' eq' ≫ biproduct.fromSubtype f pᶜ = 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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
⊢ (fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' w ≫ biproduct.fromSubtype f pᶜ = 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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
⊢ (fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' w ≫ biproduct.fromSubtype f pᶜ = 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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
⊢ ((fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' w ≫ biproduct.fromSubtype f pᶜ) ≫ biproduct.π f j =
    g' ≫ biproduct.π f j
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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
⊢ ((fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' w ≫ biproduct.fromSubtype f pᶜ) ≫ biproduct.π f j =
    g' ≫ biproduct.π f j
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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
⊢ (g' ≫ biproduct.π f j ≫ if (p j)ᶜ then 𝟙 (f j) else 0) = g' ≫ biproduct.π f j
tactic: simp only [Category.assoc, biproduct.toSubtype_fromSubtype, Pi.compl_apply,
          biproduct.map_π]
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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
⊢ (g' ≫ biproduct.π f j ≫ if (p j)ᶜ then 𝟙 (f j) else 0) = g' ≫ biproduct.π f j
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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
h : (p j)ᶜ
⊢ g' ≫ biproduct.π f j ≫ 𝟙 (f j) = g' ≫ biproduct.π f j

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' : W' ⟶ ⨁ f
w : g' ≫ biproduct.toSubtype f p = 0
j : K
h : ¬(p j)ᶜ
⊢ g' ≫ biproduct.π f j ≫ 0 = g' ≫ biproduct.π f j
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
⊢ ∀ {W' : C} (g' : W' ⟶ ⨁ f) (eq' : g' ≫ biproduct.toSubtype f p = 0) (m : W' ⟶ ⨁ Subtype.restrict pᶜ f),
    m ≫ biproduct.fromSubtype f pᶜ = g' → m = (fun {W} g x => g ≫ biproduct.toSubtype f pᶜ) g' eq'
state_after: no goals
tactic: aesop_cat