Theorem nn0expcli

Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015)

	Ref	Expression
Hypotheses	nn0expcli.1	$$⊢A\in {\mathbb{N}}_0$$
	nn0expcli.2	$$⊢N\in {\mathbb{N}}_0$$
Assertion	nn0expcli	$$⊢A^N\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		nn0expcli.1	$$⊢A\in {\mathbb{N}}_0$$
2		nn0expcli.2	$$⊢N\in {\mathbb{N}}_0$$
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	mp2an	$$⊢A^N\in {\mathbb{N}}_0$$