Metamath Proof Explorer


Theorem cosppi

Description: Cosine of a number plus _pi . (Contributed by NM, 18-Aug-2008)

Ref Expression
Assertion cosppi ( 𝐴 ∈ ℂ → ( cos ‘ ( 𝐴 + π ) ) = - ( cos ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 picn π ∈ ℂ
2 cosadd ( ( 𝐴 ∈ ℂ ∧ π ∈ ℂ ) → ( cos ‘ ( 𝐴 + π ) ) = ( ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) − ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) ) )
3 1 2 mpan2 ( 𝐴 ∈ ℂ → ( cos ‘ ( 𝐴 + π ) ) = ( ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) − ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) ) )
4 cospi ( cos ‘ π ) = - 1
5 4 oveq2i ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) = ( ( cos ‘ 𝐴 ) · - 1 )
6 coscl ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ )
7 neg1cn - 1 ∈ ℂ
8 mulcom ( ( ( cos ‘ 𝐴 ) ∈ ℂ ∧ - 1 ∈ ℂ ) → ( ( cos ‘ 𝐴 ) · - 1 ) = ( - 1 · ( cos ‘ 𝐴 ) ) )
9 7 8 mpan2 ( ( cos ‘ 𝐴 ) ∈ ℂ → ( ( cos ‘ 𝐴 ) · - 1 ) = ( - 1 · ( cos ‘ 𝐴 ) ) )
10 mulm1 ( ( cos ‘ 𝐴 ) ∈ ℂ → ( - 1 · ( cos ‘ 𝐴 ) ) = - ( cos ‘ 𝐴 ) )
11 9 10 eqtrd ( ( cos ‘ 𝐴 ) ∈ ℂ → ( ( cos ‘ 𝐴 ) · - 1 ) = - ( cos ‘ 𝐴 ) )
12 6 11 syl ( 𝐴 ∈ ℂ → ( ( cos ‘ 𝐴 ) · - 1 ) = - ( cos ‘ 𝐴 ) )
13 5 12 syl5eq ( 𝐴 ∈ ℂ → ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) = - ( cos ‘ 𝐴 ) )
14 sinpi ( sin ‘ π ) = 0
15 14 oveq2i ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) = ( ( sin ‘ 𝐴 ) · 0 )
16 sincl ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ )
17 16 mul01d ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) · 0 ) = 0 )
18 15 17 syl5eq ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) = 0 )
19 13 18 oveq12d ( 𝐴 ∈ ℂ → ( ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) − ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) ) = ( - ( cos ‘ 𝐴 ) − 0 ) )
20 6 negcld ( 𝐴 ∈ ℂ → - ( cos ‘ 𝐴 ) ∈ ℂ )
21 20 subid1d ( 𝐴 ∈ ℂ → ( - ( cos ‘ 𝐴 ) − 0 ) = - ( cos ‘ 𝐴 ) )
22 19 21 eqtrd ( 𝐴 ∈ ℂ → ( ( ( cos ‘ 𝐴 ) · ( cos ‘ π ) ) − ( ( sin ‘ 𝐴 ) · ( sin ‘ π ) ) ) = - ( cos ‘ 𝐴 ) )
23 3 22 eqtrd ( 𝐴 ∈ ℂ → ( cos ‘ ( 𝐴 + π ) ) = - ( cos ‘ 𝐴 ) )