Step |
Hyp |
Ref |
Expression |
1 |
|
cxp112d.c |
|- ( ph -> C e. CC ) |
2 |
|
cxp112d.a |
|- ( ph -> A e. CC ) |
3 |
|
cxp112d.b |
|- ( ph -> B e. CC ) |
4 |
|
cxp112d.0 |
|- ( ph -> C =/= 0 ) |
5 |
|
cxp112d.1 |
|- ( ph -> C =/= 1 ) |
6 |
1 4 2
|
cxpefd |
|- ( ph -> ( C ^c A ) = ( exp ` ( A x. ( log ` C ) ) ) ) |
7 |
1 4 3
|
cxpefd |
|- ( ph -> ( C ^c B ) = ( exp ` ( B x. ( log ` C ) ) ) ) |
8 |
6 7
|
eqeq12d |
|- ( ph -> ( ( C ^c A ) = ( C ^c B ) <-> ( exp ` ( A x. ( log ` C ) ) ) = ( exp ` ( B x. ( log ` C ) ) ) ) ) |
9 |
1 4
|
logcld |
|- ( ph -> ( log ` C ) e. CC ) |
10 |
2 9
|
mulcld |
|- ( ph -> ( A x. ( log ` C ) ) e. CC ) |
11 |
3 9
|
mulcld |
|- ( ph -> ( B x. ( log ` C ) ) e. CC ) |
12 |
10 11
|
ef11d |
|- ( ph -> ( ( exp ` ( A x. ( log ` C ) ) ) = ( exp ` ( B x. ( log ` C ) ) ) <-> E. n e. ZZ ( A x. ( log ` C ) ) = ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) ) ) |
13 |
2
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> A e. CC ) |
14 |
9
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( log ` C ) e. CC ) |
15 |
11
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( B x. ( log ` C ) ) e. CC ) |
16 |
|
ax-icn |
|- _i e. CC |
17 |
|
2cn |
|- 2 e. CC |
18 |
|
picn |
|- _pi e. CC |
19 |
17 18
|
mulcli |
|- ( 2 x. _pi ) e. CC |
20 |
16 19
|
mulcli |
|- ( _i x. ( 2 x. _pi ) ) e. CC |
21 |
20
|
a1i |
|- ( ( ph /\ n e. ZZ ) -> ( _i x. ( 2 x. _pi ) ) e. CC ) |
22 |
|
zcn |
|- ( n e. ZZ -> n e. CC ) |
23 |
22
|
adantl |
|- ( ( ph /\ n e. ZZ ) -> n e. CC ) |
24 |
21 23
|
mulcld |
|- ( ( ph /\ n e. ZZ ) -> ( ( _i x. ( 2 x. _pi ) ) x. n ) e. CC ) |
25 |
15 24
|
addcld |
|- ( ( ph /\ n e. ZZ ) -> ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) e. CC ) |
26 |
1 4 5
|
logccne0d |
|- ( ph -> ( log ` C ) =/= 0 ) |
27 |
26
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( log ` C ) =/= 0 ) |
28 |
13 14 25 27
|
ldiv |
|- ( ( ph /\ n e. ZZ ) -> ( ( A x. ( log ` C ) ) = ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) <-> A = ( ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / ( log ` C ) ) ) ) |
29 |
15 24 14 27
|
divdird |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / ( log ` C ) ) = ( ( ( B x. ( log ` C ) ) / ( log ` C ) ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) |
30 |
3 9 26
|
divcan4d |
|- ( ph -> ( ( B x. ( log ` C ) ) / ( log ` C ) ) = B ) |
31 |
30
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( ( B x. ( log ` C ) ) / ( log ` C ) ) = B ) |
32 |
31
|
oveq1d |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( B x. ( log ` C ) ) / ( log ` C ) ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) |
33 |
29 32
|
eqtrd |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / ( log ` C ) ) = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) |
34 |
33
|
eqeq2d |
|- ( ( ph /\ n e. ZZ ) -> ( A = ( ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / ( log ` C ) ) <-> A = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) ) |
35 |
28 34
|
bitrd |
|- ( ( ph /\ n e. ZZ ) -> ( ( A x. ( log ` C ) ) = ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) <-> A = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) ) |
36 |
35
|
rexbidva |
|- ( ph -> ( E. n e. ZZ ( A x. ( log ` C ) ) = ( ( B x. ( log ` C ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) <-> E. n e. ZZ A = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) ) |
37 |
8 12 36
|
3bitrd |
|- ( ph -> ( ( C ^c A ) = ( C ^c B ) <-> E. n e. ZZ A = ( B + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / ( log ` C ) ) ) ) ) |