Metamath Proof Explorer

Theorem cospim

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

Ref Expression
Assertion cospim 
|- ( A e. CC -> ( cos ` ( _pi - A ) ) = -u ( cos ` 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 cosneg 
 |-  ( ( A - _pi ) e. CC -> ( cos ` -u ( A - _pi ) ) = ( cos ` ( A - _pi ) ) )
6 4 5 syl 
 |-  ( A e. CC -> ( cos ` -u ( A - _pi ) ) = ( cos ` ( A - _pi ) ) )
7 1 3 negsubdi2d 
 |-  ( A e. CC -> -u ( A - _pi ) = ( _pi - A ) )
8 7 fveq2d 
 |-  ( A e. CC -> ( cos ` -u ( A - _pi ) ) = ( cos ` ( _pi - A ) ) )
9 cosmpi 
 |-  ( A e. CC -> ( cos ` ( A - _pi ) ) = -u ( cos ` A ) )
10 6 8 9 3eqtr3d 
 |-  ( A e. CC -> ( cos ` ( _pi - A ) ) = -u ( cos ` A ) )