| Step |
Hyp |
Ref |
Expression |
| 1 |
|
picn |
⊢ π ∈ ℂ |
| 2 |
|
sinadd |
⊢ ( ( 𝐴 ∈ ℂ ∧ π ∈ ℂ ) → ( sin ‘ ( 𝐴 + π ) ) = ( ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) + ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) ) ) |
| 3 |
1 2
|
mpan2 |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( 𝐴 + π ) ) = ( ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) + ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) ) ) |
| 4 |
|
cospi |
⊢ ( cos ‘ π ) = - 1 |
| 5 |
4
|
oveq2i |
⊢ ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) = ( ( sin ‘ 𝐴 ) · - 1 ) |
| 6 |
|
sincl |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ ) |
| 7 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
| 8 |
|
mulcom |
⊢ ( ( ( sin ‘ 𝐴 ) ∈ ℂ ∧ - 1 ∈ ℂ ) → ( ( sin ‘ 𝐴 ) · - 1 ) = ( - 1 · ( sin ‘ 𝐴 ) ) ) |
| 9 |
7 8
|
mpan2 |
⊢ ( ( sin ‘ 𝐴 ) ∈ ℂ → ( ( sin ‘ 𝐴 ) · - 1 ) = ( - 1 · ( sin ‘ 𝐴 ) ) ) |
| 10 |
|
mulm1 |
⊢ ( ( sin ‘ 𝐴 ) ∈ ℂ → ( - 1 · ( sin ‘ 𝐴 ) ) = - ( sin ‘ 𝐴 ) ) |
| 11 |
9 10
|
eqtrd |
⊢ ( ( sin ‘ 𝐴 ) ∈ ℂ → ( ( sin ‘ 𝐴 ) · - 1 ) = - ( sin ‘ 𝐴 ) ) |
| 12 |
6 11
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) · - 1 ) = - ( sin ‘ 𝐴 ) ) |
| 13 |
5 12
|
syl5eq |
⊢ ( 𝐴 ∈ ℂ → ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) = - ( sin ‘ 𝐴 ) ) |
| 14 |
|
sinpi |
⊢ ( sin ‘ π ) = 0 |
| 15 |
14
|
oveq2i |
⊢ ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) = ( ( cos ‘ 𝐴 ) · 0 ) |
| 16 |
|
coscl |
⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ ) |
| 17 |
16
|
mul01d |
⊢ ( 𝐴 ∈ ℂ → ( ( cos ‘ 𝐴 ) · 0 ) = 0 ) |
| 18 |
15 17
|
syl5eq |
⊢ ( 𝐴 ∈ ℂ → ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) = 0 ) |
| 19 |
13 18
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) + ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) ) = ( - ( sin ‘ 𝐴 ) + 0 ) ) |
| 20 |
6
|
negcld |
⊢ ( 𝐴 ∈ ℂ → - ( sin ‘ 𝐴 ) ∈ ℂ ) |
| 21 |
20
|
addid1d |
⊢ ( 𝐴 ∈ ℂ → ( - ( sin ‘ 𝐴 ) + 0 ) = - ( sin ‘ 𝐴 ) ) |
| 22 |
19 21
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) · ( cos ‘ π ) ) + ( ( cos ‘ 𝐴 ) · ( sin ‘ π ) ) ) = - ( sin ‘ 𝐴 ) ) |
| 23 |
3 22
|
eqtrd |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( 𝐴 + π ) ) = - ( sin ‘ 𝐴 ) ) |