Metamath Proof Explorer


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 ) ) )