Theorem: CategoryTheory.Idempotents.isIdempotentComplete_iff_idempotents_have_kernels
File: Mathlib/CategoryTheory/Idempotents/Basic.lean

tactic: rw [isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p

tactic: constructor
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
state_after: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p

case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p) → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p

tactic: intro h X p hp
state_before: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p) → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
state_after: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
⊢ HasKernel p

tactic: haveI : HasEqualizer (𝟙 X) (𝟙 X - p) := h X (𝟙 _ - p) (idem_of_id_sub_idem p hp)
state_before: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
⊢ HasKernel p
state_after: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) (𝟙 X - p)
⊢ HasKernel p

tactic: convert hasKernel_of_hasEqualizer (𝟙 X) (𝟙 X - p)
state_before: case mp
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) (𝟙 X - p)
⊢ HasKernel p
state_after: case h.e'_6
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) (𝟙 X - p)
⊢ p = 𝟙 X - (𝟙 X - p)

tactic: rw [sub_sub_cancel]
state_before: case h.e'_6
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) (𝟙 X - p)
⊢ p = 𝟙 X - (𝟙 X - p)
state_after: no goals

tactic: intro h X p hp
state_before: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
⊢ (∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p) → ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
state_after: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
X : C
p : X ⟶ X
hp : p ≫ p = p
⊢ HasEqualizer (𝟙 X) p

tactic: haveI : HasKernel (𝟙 _ - p) := h X (𝟙 _ - p) (idem_of_id_sub_idem p hp)
state_before: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
X : C
p : X ⟶ X
hp : p ≫ p = p
⊢ HasEqualizer (𝟙 X) p
state_after: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasKernel (𝟙 X - p)
⊢ HasEqualizer (𝟙 X) p

tactic: apply Preadditive.hasEqualizer_of_hasKernel
state_before: case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasKernel p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasKernel (𝟙 X - p)
⊢ HasEqualizer (𝟙 X) p
state_after: no goals