Theorem rpcxpcl

Description: Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	rpcxpcl	|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		rprege0	|- ( A e. RR+ -> ( A e. RR /\ 0 <_ A ) )
2		recxpcl	|- ( ( A e. RR /\ 0 <_ A /\ B e. RR ) -> ( A ^c B ) e. RR )
3	2	3expa	|- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) -> ( A ^c B ) e. RR )
4	1 3	sylan	|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR )
5		id	|- ( B e. RR -> B e. RR )
6		relogcl	|- ( A e. RR+ -> ( log ` A ) e. RR )
7		remulcl	|- ( ( B e. RR /\ ( log ` A ) e. RR ) -> ( B x. ( log ` A ) ) e. RR )
8	5 6 7	syl2anr	|- ( ( A e. RR+ /\ B e. RR ) -> ( B x. ( log ` A ) ) e. RR )
9		efgt0	|- ( ( B x. ( log ` A ) ) e. RR -> 0 < ( exp ` ( B x. ( log ` A ) ) ) )
10	8 9	syl	|- ( ( A e. RR+ /\ B e. RR ) -> 0 < ( exp ` ( B x. ( log ` A ) ) ) )
11		rpcnne0	|- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) )
12		recn	|- ( B e. RR -> B e. CC )
13		cxpef	|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
14	13	3expa	|- ( ( ( A e. CC /\ A =/= 0 ) /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
15	11 12 14	syl2an	|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
16	10 15	breqtrrd	|- ( ( A e. RR+ /\ B e. RR ) -> 0 < ( A ^c B ) )
17	4 16	elrpd	|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )