Theorem zexpcl

Description: Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005)

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

Proof

Step	Hyp	Ref	Expression
1		zsscn	$$⊢\mathbb{Z}\subseteq ℂ$$
2		zmulcl	$$⊢\left(x\in \mathbb{Z}\wedge y\in \mathbb{Z}\right)\to xy\in \mathbb{Z}$$
3		1z	$$⊢1\in \mathbb{Z}$$
4	1 2 3	expcllem	$$⊢\left(A\in \mathbb{Z}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{Z}$$