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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 eqtrid ⊢ ( 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 eqtrid ⊢ ( 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 ) )