Theorem nn0expcld

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

	Ref	Expression
Hypotheses	nn0expcld.1	$$⊢\phi \to A\in {\mathbb{N}}_0$$
	nn0expcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
Assertion	nn0expcld	$$⊢\phi \to A^N\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		nn0expcld.1	$$⊢\phi \to A\in {\mathbb{N}}_0$$
2		nn0expcld.2	$$⊢\phi \to N\in {\mathbb{N}}_0$$
3		nn0expcl	$$⊢\left(A\in {\mathbb{N}}_0\wedge N\in {\mathbb{N}}_0\right)\to A^N\in {\mathbb{N}}_0$$
4	1 2 3	syl2anc	$$⊢\phi \to A^N\in {\mathbb{N}}_0$$