Metamath Proof Explorer

Theorem sinpim

Description: Sine of a number subtracted from _pi . (Contributed by SN, 19-Nov-2025)

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

Proof

Step Hyp Ref Expression
1 id 
 |-  ( A e. CC -> A e. CC )
2 picn 
 |-  _pi e. CC
3 2 a1i 
 |-  ( A e. CC -> _pi e. CC )
4 1 3 subcld 
 |-  ( A e. CC -> ( A - _pi ) e. CC )
5 sinneg 
 |-  ( ( A - _pi ) e. CC -> ( sin ` -u ( A - _pi ) ) = -u ( sin ` ( A - _pi ) ) )
6 4 5 syl 
 |-  ( A e. CC -> ( sin ` -u ( A - _pi ) ) = -u ( sin ` ( A - _pi ) ) )
7 1 3 negsubdi2d 
 |-  ( A e. CC -> -u ( A - _pi ) = ( _pi - A ) )
8 7 fveq2d 
 |-  ( A e. CC -> ( sin ` -u ( A - _pi ) ) = ( sin ` ( _pi - A ) ) )
9 sincl 
 |-  ( A e. CC -> ( sin ` A ) e. CC )
10 sinmpi 
 |-  ( A e. CC -> ( sin ` ( A - _pi ) ) = -u ( sin ` A ) )
11 10 eqcomd 
 |-  ( A e. CC -> -u ( sin ` A ) = ( sin ` ( A - _pi ) ) )
12 9 11 negcon1ad 
 |-  ( A e. CC -> -u ( sin ` ( A - _pi ) ) = ( sin ` A ) )
13 6 8 12 3eqtr3d 
 |-  ( A e. CC -> ( sin ` ( _pi - A ) ) = ( sin ` A ) )