Theorem mulne0bd

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

	Ref	Expression
Hypotheses	msq0d.1	$$⊢\phi \to A\in ℂ$$
	mul0ord.2	$$⊢\phi \to B\in ℂ$$
Assertion	mulne0bd	$$⊢\phi \to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	$$⊢\phi \to A\in ℂ$$
2		mul0ord.2	$$⊢\phi \to B\in ℂ$$
3		mulne0b	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$