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 ) ) ) |
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 ) ) ) |