Step |
Hyp |
Ref |
Expression |
1 |
|
picn |
|- _pi e. CC |
2 |
|
cossub |
|- ( ( 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
|
syl5eq |
|- ( 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
|
syl5eq |
|- ( 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
|
addid1d |
|- ( 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 ) ) |