Step |
Hyp |
Ref |
Expression |
1 |
|
cxp111d.a |
|- ( ph -> A e. CC ) |
2 |
|
cxp111d.b |
|- ( ph -> B e. CC ) |
3 |
|
cxp111d.c |
|- ( ph -> C e. CC ) |
4 |
|
cxp111d.1 |
|- ( ph -> A =/= 0 ) |
5 |
|
cxp111d.2 |
|- ( ph -> B =/= 0 ) |
6 |
|
cxp111d.3 |
|- ( ph -> C =/= 0 ) |
7 |
1 4 3
|
cxpefd |
|- ( ph -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) ) |
8 |
2 5 3
|
cxpefd |
|- ( ph -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) ) |
9 |
7 8
|
eqeq12d |
|- ( ph -> ( ( A ^c C ) = ( B ^c C ) <-> ( exp ` ( C x. ( log ` A ) ) ) = ( exp ` ( C x. ( log ` B ) ) ) ) ) |
10 |
1 4
|
logcld |
|- ( ph -> ( log ` A ) e. CC ) |
11 |
3 10
|
mulcld |
|- ( ph -> ( C x. ( log ` A ) ) e. CC ) |
12 |
2 5
|
logcld |
|- ( ph -> ( log ` B ) e. CC ) |
13 |
3 12
|
mulcld |
|- ( ph -> ( C x. ( log ` B ) ) e. CC ) |
14 |
11 13
|
ef11d |
|- ( ph -> ( ( exp ` ( C x. ( log ` A ) ) ) = ( exp ` ( C x. ( log ` B ) ) ) <-> E. n e. ZZ ( C x. ( log ` A ) ) = ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) ) ) |
15 |
11
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( C x. ( log ` A ) ) e. CC ) |
16 |
13
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( C x. ( log ` B ) ) e. CC ) |
17 |
|
ax-icn |
|- _i e. CC |
18 |
|
2cn |
|- 2 e. CC |
19 |
|
picn |
|- _pi e. CC |
20 |
18 19
|
mulcli |
|- ( 2 x. _pi ) e. CC |
21 |
17 20
|
mulcli |
|- ( _i x. ( 2 x. _pi ) ) e. CC |
22 |
21
|
a1i |
|- ( ( ph /\ n e. ZZ ) -> ( _i x. ( 2 x. _pi ) ) e. CC ) |
23 |
|
zcn |
|- ( n e. ZZ -> n e. CC ) |
24 |
23
|
adantl |
|- ( ( ph /\ n e. ZZ ) -> n e. CC ) |
25 |
22 24
|
mulcld |
|- ( ( ph /\ n e. ZZ ) -> ( ( _i x. ( 2 x. _pi ) ) x. n ) e. CC ) |
26 |
16 25
|
addcld |
|- ( ( ph /\ n e. ZZ ) -> ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) e. CC ) |
27 |
3
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> C e. CC ) |
28 |
6
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> C =/= 0 ) |
29 |
|
div11 |
|- ( ( ( C x. ( log ` A ) ) e. CC /\ ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( ( C x. ( log ` A ) ) / C ) = ( ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / C ) <-> ( C x. ( log ` A ) ) = ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) ) ) |
30 |
15 26 27 28 29
|
syl112anc |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( C x. ( log ` A ) ) / C ) = ( ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / C ) <-> ( C x. ( log ` A ) ) = ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) ) ) |
31 |
10 3 6
|
divcan3d |
|- ( ph -> ( ( C x. ( log ` A ) ) / C ) = ( log ` A ) ) |
32 |
31
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( ( C x. ( log ` A ) ) / C ) = ( log ` A ) ) |
33 |
16 25 27 28
|
divdird |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / C ) = ( ( ( C x. ( log ` B ) ) / C ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) |
34 |
12 3 6
|
divcan3d |
|- ( ph -> ( ( C x. ( log ` B ) ) / C ) = ( log ` B ) ) |
35 |
34
|
adantr |
|- ( ( ph /\ n e. ZZ ) -> ( ( C x. ( log ` B ) ) / C ) = ( log ` B ) ) |
36 |
35
|
oveq1d |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( C x. ( log ` B ) ) / C ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) |
37 |
33 36
|
eqtrd |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / C ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) |
38 |
32 37
|
eqeq12d |
|- ( ( ph /\ n e. ZZ ) -> ( ( ( C x. ( log ` A ) ) / C ) = ( ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) / C ) <-> ( log ` A ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) ) |
39 |
30 38
|
bitr3d |
|- ( ( ph /\ n e. ZZ ) -> ( ( C x. ( log ` A ) ) = ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) <-> ( log ` A ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) ) |
40 |
39
|
rexbidva |
|- ( ph -> ( E. n e. ZZ ( C x. ( log ` A ) ) = ( ( C x. ( log ` B ) ) + ( ( _i x. ( 2 x. _pi ) ) x. n ) ) <-> E. n e. ZZ ( log ` A ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) ) |
41 |
9 14 40
|
3bitrd |
|- ( ph -> ( ( A ^c C ) = ( B ^c C ) <-> E. n e. ZZ ( log ` A ) = ( ( log ` B ) + ( ( ( _i x. ( 2 x. _pi ) ) x. n ) / C ) ) ) ) |