Theorem mulne0d

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	$$⊢\phi \to B\in ℂ$$
	mulne0d.3	$$⊢\phi \to A\ne 0$$
	mulne0d.4	$$⊢\phi \to B\ne 0$$
Assertion	mulne0d	$$⊢\phi \to AB\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	$$⊢\phi \to A\in ℂ$$
2		mul0ord.2	$$⊢\phi \to B\in ℂ$$
3		mulne0d.3	$$⊢\phi \to A\ne 0$$
4		mulne0d.4	$$⊢\phi \to B\ne 0$$
5	1 2	mulne0bd	$$⊢\phi \to \left(\left(A\ne 0\wedge B\ne 0\right)\leftrightarrow AB\ne 0\right)$$
6	3 4 5	mpbi2and	$$⊢\phi \to AB\ne 0$$