Theorem: CategoryTheory.Functor.IsLocalization.of_iso
File: Mathlib/CategoryTheory/Localization/Predicate.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
⊢ L₂.IsLocalization W
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₁
⊢ L₂.IsLocalization W
tactic: have h := Localization.inverts L₁ W
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₁
⊢ L₂.IsLocalization W
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
⊢ L₂.IsLocalization W
tactic: rw [MorphismProperty.IsInvertedBy.iff_of_iso W e] at h
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
⊢ L₂.IsLocalization W
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
F₁ : W.Localization ⥤ D := Construction.lift L₁ ⋯
⊢ L₂.IsLocalization W
tactic: let F₁ := Localization.Construction.lift L₁ (Localization.inverts L₁ W)
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
F₁ : W.Localization ⥤ D := Construction.lift L₁ ⋯
⊢ L₂.IsLocalization W
state_after: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
F₁ : W.Localization ⥤ D := Construction.lift L₁ ⋯
F₂ : W.Localization ⥤ D := Construction.lift L₂ h
⊢ L₂.IsLocalization W
tactic: let F₂ := Localization.Construction.lift L₂ h
state_before: C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_5, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝¹ : Category.{?u.61043, u_3} E
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
h : W.IsInvertedBy L₂
F₁ : W.Localization ⥤ D := Construction.lift L₁ ⋯
F₂ : W.Localization ⥤ D := Construction.lift L₂ h
⊢ L₂.IsLocalization W
state_after: no goals
tactic: exact
    { inverts := h
      isEquivalence := Functor.isEquivalence_of_iso (liftNatIso W.Q W L₁ L₂ F₁ F₂ e) }