Theorem cxple2d

Description: Ordering property for complex exponentiation. (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 )
	mulcxpd.4	|- ( ph -> 0 <_ B )
	cxple2d.5	|- ( ph -> C e. RR+ )
Assertion	cxple2d	|- ( ph -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )

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		mulcxpd.4	|- ( ph -> 0 <_ B )
5		cxple2d.5	|- ( ph -> C e. RR+ )
6		cxple2	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
7	1 2 3 4 5 6	syl221anc	|- ( ph -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )