Theorem nnexpcld

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

	Ref	Expression
Hypotheses	nnexpcld.1	|- ( ph -> A e. NN )
	nnexpcld.2	|- ( ph -> N e. NN0 )
Assertion	nnexpcld	|- ( ph -> ( A ^ N ) e. NN )

Proof

Step	Hyp	Ref	Expression
1		nnexpcld.1	|- ( ph -> A e. NN )
2		nnexpcld.2	|- ( ph -> N e. NN0 )
3		nnexpcl	|- ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )
4	1 2 3	syl2anc	|- ( ph -> ( A ^ N ) e. NN )