Theorem: CancelDenoms.cancel_factors_eq
File: Mathlib/Tactic/CancelDenoms/Core.lean

tactic: rw [← ha, ← hb, ← mul_assoc bd, ← mul_assoc ad, mul_comm bd]
state_before: α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ (a = b) = (1 / gcd * (bd * a') = 1 / gcd * (ad * b'))
state_after: α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ (a = b) = (1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b))

tactic: ext
state_before: α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ (a = b) = (1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b))
state_after: case a
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ a = b ↔ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)

tactic: constructor
state_before: case a
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ a = b ↔ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)
state_after: case a.mp
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ a = b → 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)

case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b) → a = b

tactic: rintro rfl
state_before: case a.mp
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ a = b → 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)
state_after: case a.mp
α : Type u_1
inst✝ : Field α
a ad bd a' b' gcd : α
ha : ad * a = a'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
hb : bd * a = b'
⊢ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * a)

tactic: rfl
state_before: case a.mp
α : Type u_1
inst✝ : Field α
a ad bd a' b' gcd : α
ha : ad * a = a'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
hb : bd * a = b'
⊢ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * a)
state_after: no goals

tactic: intro h
state_before: case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
⊢ 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b) → a = b
state_after: case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)
⊢ a = b

tactic: simp only [← mul_assoc] at h
state_before: case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)
⊢ a = b
state_after: case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ a = b

tactic: refine mul_left_cancel₀ (mul_ne_zero ?_ ?_) h
state_before: case a.mpr
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ a = b
state_after: case a.mpr.refine_1
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd * ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0

tactic: on_goal 1 => apply mul_ne_zero
state_before: case a.mpr.refine_1
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd * ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0
state_after: case a.mpr.refine_1.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd ≠ 0

case a.mpr.refine_1.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0

tactic: on_goal 1 => apply div_ne_zero
state_before: case a.mpr.refine_1.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd ≠ 0

case a.mpr.refine_1.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0
state_after: case a.mpr.refine_1.ha.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 ≠ 0

case a.mpr.refine_1.ha.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ gcd ≠ 0

case a.mpr.refine_1.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0

tactic: all_goals assumption
state_before: case a.mpr.refine_1.ha.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ gcd ≠ 0

case a.mpr.refine_1.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ ad ≠ 0

case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0
state_after: no goals

tactic: apply mul_ne_zero
state_before: case a.mpr.refine_1
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd * ad ≠ 0
state_after: case a.mpr.refine_1.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd ≠ 0

case a.mpr.refine_1.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ ad ≠ 0

tactic: apply div_ne_zero
state_before: case a.mpr.refine_1.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 / gcd ≠ 0
state_after: case a.mpr.refine_1.ha.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 ≠ 0

case a.mpr.refine_1.ha.hb
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ gcd ≠ 0

tactic: exact one_ne_zero
state_before: case a.mpr.refine_1.ha.ha
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ 1 ≠ 0
state_after: no goals

tactic: assumption
state_before: case a.mpr.refine_2
α : Type u_1
inst✝ : Field α
a b ad bd a' b' gcd : α
ha : ad * a = a'
hb : bd * b = b'
had : ad ≠ 0
hbd : bd ≠ 0
hgcd : gcd ≠ 0
h : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b
⊢ bd ≠ 0
state_after: no goals