Theorem divclzi

Description: Closure law for division. (Contributed by NM, 7-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
Assertion	divclzi	$$⊢B\ne 0\to \frac{A}{B}\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divcl	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{A}{B}\in ℂ$$
4	1 2 3	mp3an12	$$⊢B\ne 0\to \frac{A}{B}\in ℂ$$