Metamath Proof Explorer

Theorem cxplt

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014)

Ref Expression
Assertion cxplt 
|- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )

Proof

Step Hyp Ref Expression
1 simprl 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
2 rplogcl 
 |-  ( ( A e. RR /\ 1 < A ) -> ( log ` A ) e. RR+ )
3 2 adantr 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR+ )
4 3 rpred 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR )
5 1 4 remulcld 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B x. ( log ` A ) ) e. RR )
6 simprr 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
7 6 4 remulcld 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( C x. ( log ` A ) ) e. RR )
8 eflt 
 |-  ( ( ( B x. ( log ` A ) ) e. RR /\ ( C x. ( log ` A ) ) e. RR ) -> ( ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
9 5 7 8 syl2anc 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
10 1 6 3 ltmul1d 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( B x. ( log ` A ) ) < ( C x. ( log ` A ) ) ) )
11 simpll 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR )
12 11 recnd 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> A e. CC )
13 0red 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 e. RR )
14 1red 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 1 e. RR )
15 0lt1 
 |-  0 < 1
16 15 a1i 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 < 1 )
17 simplr 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 1 < A )
18 13 14 11 16 17 lttrd 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> 0 < A )
19 18 gt0ne0d 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> A =/= 0 )
20 1 recnd 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
21 cxpef 
 |-  ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
22 12 19 20 21 syl3anc 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
23 6 recnd 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
24 cxpef 
 |-  ( ( A e. CC /\ A =/= 0 /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
25 12 19 23 24 syl3anc 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
26 22 25 breq12d 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( A ^c B ) < ( A ^c C ) <-> ( exp ` ( B x. ( log ` A ) ) ) < ( exp ` ( C x. ( log ` A ) ) ) ) )
27 9 10 26 3bitr4d 
 |-  ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )