Metamath Proof Explorer


Theorem cxpltd

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 cxpltd
|- ( ph -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )

Proof

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 cxplt
 |-  ( ( ( 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 ) ) )