Metamath Proof Explorer


Theorem sinppi

Description: Sine of a number plus _pi . (Contributed by NM, 10-Aug-2008)

Ref Expression
Assertion sinppi
|- ( A e. CC -> ( sin ` ( A + _pi ) ) = -u ( sin ` A ) )

Proof

Step Hyp Ref Expression
1 picn
 |-  _pi e. CC
2 sinadd
 |-  ( ( A e. CC /\ _pi e. CC ) -> ( sin ` ( A + _pi ) ) = ( ( ( sin ` A ) x. ( cos ` _pi ) ) + ( ( cos ` A ) x. ( sin ` _pi ) ) ) )
3 1 2 mpan2
 |-  ( A e. CC -> ( sin ` ( A + _pi ) ) = ( ( ( sin ` A ) x. ( cos ` _pi ) ) + ( ( cos ` A ) x. ( sin ` _pi ) ) ) )
4 cospi
 |-  ( cos ` _pi ) = -u 1
5 4 oveq2i
 |-  ( ( sin ` A ) x. ( cos ` _pi ) ) = ( ( sin ` A ) x. -u 1 )
6 sincl
 |-  ( A e. CC -> ( sin ` A ) e. CC )
7 neg1cn
 |-  -u 1 e. CC
8 mulcom
 |-  ( ( ( sin ` A ) e. CC /\ -u 1 e. CC ) -> ( ( sin ` A ) x. -u 1 ) = ( -u 1 x. ( sin ` A ) ) )
9 7 8 mpan2
 |-  ( ( sin ` A ) e. CC -> ( ( sin ` A ) x. -u 1 ) = ( -u 1 x. ( sin ` A ) ) )
10 mulm1
 |-  ( ( sin ` A ) e. CC -> ( -u 1 x. ( sin ` A ) ) = -u ( sin ` A ) )
11 9 10 eqtrd
 |-  ( ( sin ` A ) e. CC -> ( ( sin ` A ) x. -u 1 ) = -u ( sin ` A ) )
12 6 11 syl
 |-  ( A e. CC -> ( ( sin ` A ) x. -u 1 ) = -u ( sin ` A ) )
13 5 12 syl5eq
 |-  ( A e. CC -> ( ( sin ` A ) x. ( cos ` _pi ) ) = -u ( sin ` A ) )
14 sinpi
 |-  ( sin ` _pi ) = 0
15 14 oveq2i
 |-  ( ( cos ` A ) x. ( sin ` _pi ) ) = ( ( cos ` A ) x. 0 )
16 coscl
 |-  ( A e. CC -> ( cos ` A ) e. CC )
17 16 mul01d
 |-  ( A e. CC -> ( ( cos ` A ) x. 0 ) = 0 )
18 15 17 syl5eq
 |-  ( A e. CC -> ( ( cos ` A ) x. ( sin ` _pi ) ) = 0 )
19 13 18 oveq12d
 |-  ( A e. CC -> ( ( ( sin ` A ) x. ( cos ` _pi ) ) + ( ( cos ` A ) x. ( sin ` _pi ) ) ) = ( -u ( sin ` A ) + 0 ) )
20 6 negcld
 |-  ( A e. CC -> -u ( sin ` A ) e. CC )
21 20 addid1d
 |-  ( A e. CC -> ( -u ( sin ` A ) + 0 ) = -u ( sin ` A ) )
22 19 21 eqtrd
 |-  ( A e. CC -> ( ( ( sin ` A ) x. ( cos ` _pi ) ) + ( ( cos ` A ) x. ( sin ` _pi ) ) ) = -u ( sin ` A ) )
23 3 22 eqtrd
 |-  ( A e. CC -> ( sin ` ( A + _pi ) ) = -u ( sin ` A ) )