Theorem mulne0d

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 )
	mulne0d.3	|- ( ph -> A =/= 0 )
	mulne0d.4	|- ( ph -> B =/= 0 )
Assertion	mulne0d	|- ( ph -> ( A x. B ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	|- ( ph -> A e. CC )
2		mul0ord.2	|- ( ph -> B e. CC )
3		mulne0d.3	|- ( ph -> A =/= 0 )
4		mulne0d.4	|- ( ph -> B =/= 0 )
5	1 2	mulne0bd	|- ( ph -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) )
6	3 4 5	mpbi2and	|- ( ph -> ( A x. B ) =/= 0 )