Theorem mul0ori

Description: If a product is zero, one of its factors must be zero. Theorem I.11 of Apostol p. 18. (Contributed by NM, 7-Oct-1999)

	Ref	Expression
Hypotheses	mul0or.1	|- A e. CC
	mul0or.2	|- B e. CC
Assertion	mul0ori	|- ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		mul0or.1	|- A e. CC
2		mul0or.2	|- B e. CC
3		mul0or	|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )
4	1 2 3	mp2an	|- ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) )