Theorem nnexpcld

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

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