Theorem divne0

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007)

	Ref	Expression
Assertion	divne0	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		divne0b	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$
2	1	3expb	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(A\ne 0\leftrightarrow \frac{A}{B}\ne 0\right)$$
3	2	biimpa	$$⊢\left(\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\wedge A\ne 0\right)\to \frac{A}{B}\ne 0$$
4	3	an32s	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\ne 0$$