Theorem: CategoryTheory.Localization.Construction.natTransExtension_app
File: Mathlib/CategoryTheory/Localization/Construction.lean

Proof steps:
state_before: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f
state_after: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
tactic: suffices MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤ by
      intro X Y f
      simpa only [← this] using MorphismProperty.top_apply f
state_before: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
state_after: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)
tactic: refine morphismProperty_is_top'
      (MorphismProperty.naturalityProperty (NatTransExtension.app τ))
      ?_ (MorphismProperty.naturalityProperty.stableUnderInverse _)
state_before: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)
state_after: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
X Y : C
f : X ⟶ Y
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)
tactic: intros X Y f
state_before: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
X Y : C
f : X ⟶ Y
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) (W.Q.map f)
state_after: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
X Y : C
f : X ⟶ Y
⊢ F₁.map (W.Q.map f) ≫ NatTransExtension.app τ (W.Q.obj Y) = NatTransExtension.app τ (W.Q.obj X) ≫ F₂.map (W.Q.map f)
tactic: dsimp
state_before: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
X Y : C
f : X ⟶ Y
⊢ F₁.map (W.Q.map f) ≫ NatTransExtension.app τ (W.Q.obj Y) = NatTransExtension.app τ (W.Q.obj X) ≫ F₂.map (W.Q.map f)
state_after: no goals
tactic: simpa only [NatTransExtension.app_eq] using τ.naturality f