Theorem mulcan2d

Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulcand.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulcand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mulcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	mulcand.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	mulcan2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mulcand.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulcand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		mulcand.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5	1 3	mulcomd	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6	2 3	mulcomd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
7	5 6	eqeq12d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ) )
8	1 2 3 4	mulcand	⊢ ( 𝜑 → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
9	7 8	bitrd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) )