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Metamath Proof Explorer

Theorem cos2pi

Description: The cosine of 2 _pi is 1. (Contributed by Paul Chapman, 23-Jan-2008)

Ref Expression
Assertion cos2pi 
|- ( cos ` ( 2 x. _pi ) ) = 1

Proof

Step Hyp Ref Expression
1 picn 
 |-  _pi e. CC
2 cos2t 
 |-  ( _pi e. CC -> ( cos ` ( 2 x. _pi ) ) = ( ( 2 x. ( ( cos ` _pi ) ^ 2 ) ) - 1 ) )
3 1 2 ax-mp 
 |-  ( cos ` ( 2 x. _pi ) ) = ( ( 2 x. ( ( cos ` _pi ) ^ 2 ) ) - 1 )
4 cospi 
 |-  ( cos ` _pi ) = -u 1
5 4 oveq1i 
 |-  ( ( cos ` _pi ) ^ 2 ) = ( -u 1 ^ 2 )
6 ax-1cn 
 |-  1 e. CC
7 sqneg 
 |-  ( 1 e. CC -> ( -u 1 ^ 2 ) = ( 1 ^ 2 ) )
8 6 7 ax-mp 
 |-  ( -u 1 ^ 2 ) = ( 1 ^ 2 )
9 sq1 
 |-  ( 1 ^ 2 ) = 1
10 5 8 9 3eqtri 
 |-  ( ( cos ` _pi ) ^ 2 ) = 1
11 10 oveq2i 
 |-  ( 2 x. ( ( cos ` _pi ) ^ 2 ) ) = ( 2 x. 1 )
12 2t1e2 
 |-  ( 2 x. 1 ) = 2
13 11 12 eqtri 
 |-  ( 2 x. ( ( cos ` _pi ) ^ 2 ) ) = 2
14 13 oveq1i 
 |-  ( ( 2 x. ( ( cos ` _pi ) ^ 2 ) ) - 1 ) = ( 2 - 1 )
15 2m1e1 
 |-  ( 2 - 1 ) = 1
16 3 14 15 3eqtri 
 |-  ( cos ` ( 2 x. _pi ) ) = 1