Theorem qexpcl

Description: Closure of exponentiation of rationals. For integer exponents, see qexpclz . (Contributed by NM, 16-Dec-2005)

	Ref	Expression
Assertion	qexpcl	$$⊢\left(A\in \mathbb{Q}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{Q}$$

Proof

Step	Hyp	Ref	Expression
1		qsscn	$$⊢\mathbb{Q}\subseteq ℂ$$
2		qmulcl	$$⊢\left(x\in \mathbb{Q}\wedge y\in \mathbb{Q}\right)\to xy\in \mathbb{Q}$$
3		1z	$$⊢1\in \mathbb{Z}$$
4		zq	$$⊢1\in \mathbb{Z}\to 1\in \mathbb{Q}$$
5	3 4	ax-mp	$$⊢1\in \mathbb{Q}$$
6	1 2 5	expcllem	$$⊢\left(A\in \mathbb{Q}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{Q}$$