Theorem: CategoryTheory.Limits.biproduct.ι_toSubtype_subtype
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
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j : Subtype p
⊢ ι f ↑j ≫ toSubtype f p = ι (Subtype.restrict p 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
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
⊢ (ι f ↑j ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j✝ = ι (Subtype.restrict p f) j ≫ π (Subtype.restrict p f) j✝
tactic: ext
state_before: case w
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
⊢ (ι f ↑j ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j✝ = ι (Subtype.restrict p f) j ≫ π (Subtype.restrict p 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
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
⊢ (if h : ↑j = ↑j✝ then eqToHom ⋯ else 0) = if h : j = j✝ then eqToHom ⋯ else 0
tactic: rw [biproduct.toSubtype, Category.assoc, biproduct.lift_π, biproduct.ι_π, biproduct.ι_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
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
⊢ (if h : ↑j = ↑j✝ then eqToHom ⋯ else 0) = if h : j = j✝ then eqToHom ⋯ else 0
state_after: case pos
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ↑j = ↑j✝
h₂ : j = j✝
⊢ eqToHom ⋯ = eqToHom ⋯

case neg
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ↑j = ↑j✝
h₂ : ¬j = j✝
⊢ eqToHom ⋯ = 0

case pos
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ¬↑j = ↑j✝
h₂ : j = j✝
⊢ 0 = eqToHom ⋯

case neg
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ¬↑j = ↑j✝
h₂ : ¬j = j✝
⊢ 0 = 0
tactic: split_ifs with h₁ h₂ h₂
state_before: case pos
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ↑j = ↑j✝
h₂ : j = j✝
⊢ eqToHom ⋯ = eqToHom ⋯

case neg
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ↑j = ↑j✝
h₂ : ¬j = j✝
⊢ eqToHom ⋯ = 0

case pos
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ¬↑j = ↑j✝
h₂ : j = j✝
⊢ 0 = eqToHom ⋯

case neg
J : Type w
K : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f : J → C
inst✝¹ : HasBiproduct f
p : J → Prop
inst✝ : HasBiproduct (Subtype.restrict p f)
j j✝ : Subtype p
h₁ : ¬↑j = ↑j✝
h₂ : ¬j = j✝
⊢ 0 = 0
state_after: no goals
tactic: exacts [rfl, False.elim (h₂ (Subtype.ext h₁)), False.elim (h₁ (congr_arg Subtype.val h₂)), rfl]