Theorem zexpcld

Description: Closure of exponentiation of integers, deductive form. (Contributed by SN, 15-Sep-2024)

	Ref	Expression
Hypotheses	zexpcld.1	$$⊢\phi \to A\in \mathbb{Z}$$
	zexpcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	zexpcld	$$⊢\phi \to A^N\in \mathbb{Z}$$

Proof

Step	Hyp	Ref	Expression
1		zexpcld.1	$$⊢\phi \to A\in \mathbb{Z}$$
2		zexpcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		zexpcl	$$⊢\left(A\in \mathbb{Z}\wedge N\in {\mathbb{N}}_0\right)\to A^N\in \mathbb{Z}$$
4	1 2 3	syl2anc	$$⊢\phi \to A^N\in \mathbb{Z}$$