Theorem cxpled

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

	Ref	Expression
Hypotheses	recxpcld.1	|- ( ph -> A e. RR )
	cxpltd.2	|- ( ph -> 1 < A )
	cxpltd.3	|- ( ph -> B e. RR )
	cxpltd.4	|- ( ph -> C e. RR )
Assertion	cxpled	|- ( ph -> ( B <_ C <-> ( A ^c B ) <_ ( A ^c C ) ) )

Proof

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