Theorem rpcxpcld

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

	Ref	Expression
Hypotheses	rpcxpcld.1	|- ( ph -> A e. RR+ )
	rpcxpcld.2	|- ( ph -> B e. RR )
Assertion	rpcxpcld	|- ( ph -> ( A ^c B ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	|- ( ph -> A e. RR+ )
2		rpcxpcld.2	|- ( ph -> B e. RR )
3		rpcxpcl	|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
4	1 2 3	syl2anc	|- ( ph -> ( A ^c B ) e. RR+ )