Theorem: CategoryTheory.MorphismProperty.LeftFraction.Localization.map_eq_iff
File: Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean

tactic: simp only [← Hom.map_mk _ (Q W)]
state_before: C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ f.map (Q W) ⋯ = g.map (Q W) ⋯ ↔ LeftFractionRel f g
state_after: C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯ ↔ LeftFractionRel f g

tactic: constructor
state_before: C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯ ↔ LeftFractionRel f g
state_after: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯ → LeftFractionRel f g

case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ LeftFractionRel f g → (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯

tactic: intro h
state_before: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯ → LeftFractionRel f g
state_after: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
⊢ LeftFractionRel f g

tactic: rw [← homMk_eq_iff_leftFractionRel, homMk_eq, homMk_eq]
state_before: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
⊢ LeftFractionRel f g
state_after: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
⊢ f.map (Q W) ⋯ = g.map (Q W) ⋯

tactic: exact h
state_before: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
⊢ f.map (Q W) ⋯ = g.map (Q W) ⋯
state_after: no goals

tactic: intro h
state_before: case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ LeftFractionRel f g → (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
state_after: case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : LeftFractionRel f g
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯

tactic: congr 1
state_before: case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : LeftFractionRel f g
⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯
state_after: case mpr.e_f
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : LeftFractionRel f g
⊢ Hom.mk f = Hom.mk g

tactic: exact Quot.sound h
state_before: case mpr.e_f
C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.97896, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
h : LeftFractionRel f g
⊢ Hom.mk f = Hom.mk g
state_after: no goals