Theorem divne0i

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divneq0.3	$$⊢A\ne 0$$
	divneq0.4	$$⊢B\ne 0$$
Assertion	divne0i	$$⊢\frac{A}{B}\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divneq0.3	$$⊢A\ne 0$$
4		divneq0.4	$$⊢B\ne 0$$
5		divne0	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\ne 0$$
6	1 3 2 4 5	mp4an	$$⊢\frac{A}{B}\ne 0$$