Theorem cxplead

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

	Ref	Expression
Hypotheses	recxpcld.1	|- ( ph -> A e. RR )
	cxplead.2	|- ( ph -> 1 <_ A )
	cxplead.3	|- ( ph -> B e. RR )
	cxplead.4	|- ( ph -> C e. RR )
	cxplead.5	|- ( ph -> B <_ C )
Assertion	cxplead	|- ( ph -> ( A ^c B ) <_ ( A ^c C ) )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	|- ( ph -> A e. RR )
2		cxplead.2	|- ( ph -> 1 <_ A )
3		cxplead.3	|- ( ph -> B e. RR )
4		cxplead.4	|- ( ph -> C e. RR )
5		cxplead.5	|- ( ph -> B <_ C )
6		cxplea	|- ( ( ( A e. RR /\ 1 <_ A ) /\ ( B e. RR /\ C e. RR ) /\ B <_ C ) -> ( A ^c B ) <_ ( A ^c C ) )
7	1 2 3 4 5 6	syl221anc	|- ( ph -> ( A ^c B ) <_ ( A ^c C ) )