Theorem mulne0i

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995)

	Ref	Expression
Hypotheses	muln0.1	$$⊢A\in ℂ$$
	muln0.2	$$⊢B\in ℂ$$
	muln0.3	$$⊢A\ne 0$$
	muln0.4	$$⊢B\ne 0$$
Assertion	mulne0i	$$⊢AB\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		muln0.1	$$⊢A\in ℂ$$
2		muln0.2	$$⊢B\in ℂ$$
3		muln0.3	$$⊢A\ne 0$$
4		muln0.4	$$⊢B\ne 0$$
5		mulne0	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to AB\ne 0$$
6	1 3 2 4 5	mp4an	$$⊢AB\ne 0$$