Metamath Proof Explorer

Theorem sinmpi

Description: Sine of a number less _pi . (Contributed by Paul Chapman, 15-Mar-2008)

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

Proof

Step Hyp Ref Expression
1 picn 
 |-  _pi e. CC
2 sinsub 
 |-  ( ( 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 eqtrid 
 |-  ( 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 eqtrid 
 |-  ( 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 subid1d 
 |-  ( 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 ) )