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 ) ) ) |
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 ) ) ) |