Theorem recne0

Description: The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	recne0	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{1}{A}\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		ax-1ne0	$$⊢1\ne 0$$
3		divne0	$$⊢\left(\left(1\in ℂ\wedge 1\ne 0\right)\wedge \left(A\in ℂ\wedge A\ne 0\right)\right)\to \frac{1}{A}\ne 0$$
4	1 2 3	mpanl12	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{1}{A}\ne 0$$