Theorem mul0ord

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	msq0d.1	|- ( ph -> A e. CC )
	mul0ord.2	|- ( ph -> B e. CC )
Assertion	mul0ord	|- ( ph -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	|- ( ph -> A e. CC )
2		mul0ord.2	|- ( ph -> B e. CC )
3		mul0or	|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )