Theorem mulne0bd

Description: The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	msq0d.1	|- ( ph -> A e. CC )
	mul0ord.2	|- ( ph -> B e. CC )
Assertion	mulne0bd	|- ( ph -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	|- ( ph -> A e. CC )
2		mul0ord.2	|- ( ph -> B e. CC )
3		mulne0b	|- ( ( A e. CC /\ B e. CC ) -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) )
4	1 2 3	syl2anc	|- ( ph -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) )