Theorem qdivcl

Description: Closure of division of rationals. (Contributed by NM, 3-Aug-2004)

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

Proof

Step	Hyp	Ref	Expression
1		qcn	$$⊢A\in \mathbb{Q}\to A\in ℂ$$
2		qcn	$$⊢B\in \mathbb{Q}\to B\in ℂ$$
3		id	$$⊢B\ne 0\to B\ne 0$$
4		divrec	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$
5	1 2 3 4	syl3an	$$⊢\left(A\in \mathbb{Q}\wedge B\in \mathbb{Q}\wedge B\ne 0\right)\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$
6		qreccl	$$⊢\left(B\in \mathbb{Q}\wedge B\ne 0\right)\to \frac{1}{B}\in \mathbb{Q}$$
7		qmulcl	$$⊢\left(A\in \mathbb{Q}\wedge \frac{1}{B}\in \mathbb{Q}\right)\to A\left(\frac{1}{B}\right)\in \mathbb{Q}$$
8	6 7	sylan2	$$⊢\left(A\in \mathbb{Q}\wedge \left(B\in \mathbb{Q}\wedge B\ne 0\right)\right)\to A\left(\frac{1}{B}\right)\in \mathbb{Q}$$
9	8	3impb	$$⊢\left(A\in \mathbb{Q}\wedge B\in \mathbb{Q}\wedge B\ne 0\right)\to A\left(\frac{1}{B}\right)\in \mathbb{Q}$$
10	5 9	eqeltrd	$$⊢\left(A\in \mathbb{Q}\wedge B\in \mathbb{Q}\wedge B\ne 0\right)\to \frac{A}{B}\in \mathbb{Q}$$