Theorem cxple3d

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	|- ( ph -> A e. RR+ )
	rpcxpcld.2	|- ( ph -> B e. RR )
	cxplt3d.3	|- ( ph -> A < 1 )
	cxplt3d.4	|- ( ph -> C e. RR )
Assertion	cxple3d	|- ( ph -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	|- ( ph -> A e. RR+ )
2		rpcxpcld.2	|- ( ph -> B e. RR )
3		cxplt3d.3	|- ( ph -> A < 1 )
4		cxplt3d.4	|- ( ph -> C e. RR )
5		cxple3	|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
6	1 3 2 4 5	syl22anc	|- ( ph -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )