Step |
Hyp |
Ref |
Expression |
1 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
2 |
|
1z |
|- 1 e. ZZ |
3 |
|
sinper |
|- ( ( -u A e. CC /\ 1 e. ZZ ) -> ( sin ` ( -u A + ( 1 x. ( 2 x. _pi ) ) ) ) = ( sin ` -u A ) ) |
4 |
1 2 3
|
sylancl |
|- ( A e. CC -> ( sin ` ( -u A + ( 1 x. ( 2 x. _pi ) ) ) ) = ( sin ` -u A ) ) |
5 |
|
2cn |
|- 2 e. CC |
6 |
|
picn |
|- _pi e. CC |
7 |
5 6
|
mulcli |
|- ( 2 x. _pi ) e. CC |
8 |
7
|
mulid2i |
|- ( 1 x. ( 2 x. _pi ) ) = ( 2 x. _pi ) |
9 |
8
|
oveq2i |
|- ( -u A + ( 1 x. ( 2 x. _pi ) ) ) = ( -u A + ( 2 x. _pi ) ) |
10 |
|
negsubdi |
|- ( ( A e. CC /\ ( 2 x. _pi ) e. CC ) -> -u ( A - ( 2 x. _pi ) ) = ( -u A + ( 2 x. _pi ) ) ) |
11 |
|
negsubdi2 |
|- ( ( A e. CC /\ ( 2 x. _pi ) e. CC ) -> -u ( A - ( 2 x. _pi ) ) = ( ( 2 x. _pi ) - A ) ) |
12 |
10 11
|
eqtr3d |
|- ( ( A e. CC /\ ( 2 x. _pi ) e. CC ) -> ( -u A + ( 2 x. _pi ) ) = ( ( 2 x. _pi ) - A ) ) |
13 |
7 12
|
mpan2 |
|- ( A e. CC -> ( -u A + ( 2 x. _pi ) ) = ( ( 2 x. _pi ) - A ) ) |
14 |
9 13
|
eqtrid |
|- ( A e. CC -> ( -u A + ( 1 x. ( 2 x. _pi ) ) ) = ( ( 2 x. _pi ) - A ) ) |
15 |
14
|
fveq2d |
|- ( A e. CC -> ( sin ` ( -u A + ( 1 x. ( 2 x. _pi ) ) ) ) = ( sin ` ( ( 2 x. _pi ) - A ) ) ) |
16 |
4 15
|
eqtr3d |
|- ( A e. CC -> ( sin ` -u A ) = ( sin ` ( ( 2 x. _pi ) - A ) ) ) |
17 |
|
sinneg |
|- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) ) |
18 |
16 17
|
eqtr3d |
|- ( A e. CC -> ( sin ` ( ( 2 x. _pi ) - A ) ) = -u ( sin ` A ) ) |