Theorem expge0d

Description: A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	reexpcld.1	|- ( ph -> A e. RR )
	reexpcld.2	|- ( ph -> N e. NN0 )
	expge0d.3	|- ( ph -> 0 <_ A )
Assertion	expge0d	|- ( ph -> 0 <_ ( A ^ N ) )

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	|- ( ph -> A e. RR )
2		reexpcld.2	|- ( ph -> N e. NN0 )
3		expge0d.3	|- ( ph -> 0 <_ A )
4		expge0	|- ( ( A e. RR /\ N e. NN0 /\ 0 <_ A ) -> 0 <_ ( A ^ N ) )
5	1 2 3 4	syl3anc	|- ( ph -> 0 <_ ( A ^ N ) )