| Step |
Hyp |
Ref |
Expression |
| 1 |
|
picn |
|- _pi e. CC |
| 2 |
|
cosadd |
|- ( ( A e. CC /\ _pi e. CC ) -> ( cos ` ( A + _pi ) ) = ( ( ( cos ` A ) x. ( cos ` _pi ) ) - ( ( sin ` A ) x. ( sin ` _pi ) ) ) ) |
| 3 |
1 2
|
mpan2 |
|- ( A e. CC -> ( cos ` ( A + _pi ) ) = ( ( ( cos ` A ) x. ( cos ` _pi ) ) - ( ( sin ` A ) x. ( sin ` _pi ) ) ) ) |
| 4 |
|
cospi |
|- ( cos ` _pi ) = -u 1 |
| 5 |
4
|
oveq2i |
|- ( ( cos ` A ) x. ( cos ` _pi ) ) = ( ( cos ` A ) x. -u 1 ) |
| 6 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
| 7 |
|
neg1cn |
|- -u 1 e. CC |
| 8 |
|
mulcom |
|- ( ( ( cos ` A ) e. CC /\ -u 1 e. CC ) -> ( ( cos ` A ) x. -u 1 ) = ( -u 1 x. ( cos ` A ) ) ) |
| 9 |
7 8
|
mpan2 |
|- ( ( cos ` A ) e. CC -> ( ( cos ` A ) x. -u 1 ) = ( -u 1 x. ( cos ` A ) ) ) |
| 10 |
|
mulm1 |
|- ( ( cos ` A ) e. CC -> ( -u 1 x. ( cos ` A ) ) = -u ( cos ` A ) ) |
| 11 |
9 10
|
eqtrd |
|- ( ( cos ` A ) e. CC -> ( ( cos ` A ) x. -u 1 ) = -u ( cos ` A ) ) |
| 12 |
6 11
|
syl |
|- ( A e. CC -> ( ( cos ` A ) x. -u 1 ) = -u ( cos ` A ) ) |
| 13 |
5 12
|
eqtrid |
|- ( A e. CC -> ( ( cos ` A ) x. ( cos ` _pi ) ) = -u ( cos ` A ) ) |
| 14 |
|
sinpi |
|- ( sin ` _pi ) = 0 |
| 15 |
14
|
oveq2i |
|- ( ( sin ` A ) x. ( sin ` _pi ) ) = ( ( sin ` A ) x. 0 ) |
| 16 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
| 17 |
16
|
mul01d |
|- ( A e. CC -> ( ( sin ` A ) x. 0 ) = 0 ) |
| 18 |
15 17
|
eqtrid |
|- ( A e. CC -> ( ( sin ` A ) x. ( sin ` _pi ) ) = 0 ) |
| 19 |
13 18
|
oveq12d |
|- ( A e. CC -> ( ( ( cos ` A ) x. ( cos ` _pi ) ) - ( ( sin ` A ) x. ( sin ` _pi ) ) ) = ( -u ( cos ` A ) - 0 ) ) |
| 20 |
6
|
negcld |
|- ( A e. CC -> -u ( cos ` A ) e. CC ) |
| 21 |
20
|
subid1d |
|- ( A e. CC -> ( -u ( cos ` A ) - 0 ) = -u ( cos ` A ) ) |
| 22 |
19 21
|
eqtrd |
|- ( A e. CC -> ( ( ( cos ` A ) x. ( cos ` _pi ) ) - ( ( sin ` A ) x. ( sin ` _pi ) ) ) = -u ( cos ` A ) ) |
| 23 |
3 22
|
eqtrd |
|- ( A e. CC -> ( cos ` ( A + _pi ) ) = -u ( cos ` A ) ) |