Theorem recxpcld

Description: Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	|- ( ph -> A e. RR )
	recxpcld.2	|- ( ph -> 0 <_ A )
	recxpcld.3	|- ( ph -> B e. RR )
Assertion	recxpcld	|- ( ph -> ( A ^c B ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	|- ( ph -> A e. RR )
2		recxpcld.2	|- ( ph -> 0 <_ A )
3		recxpcld.3	|- ( ph -> B e. RR )
4		recxpcl	|- ( ( A e. RR /\ 0 <_ A /\ B e. RR ) -> ( A ^c B ) e. RR )
5	1 2 3 4	syl3anc	|- ( ph -> ( A ^c B ) e. RR )