Step |
Hyp |
Ref |
Expression |
1 |
|
negcl |
⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ ) |
2 |
|
1z |
⊢ 1 ∈ ℤ |
3 |
|
sinper |
⊢ ( ( - 𝐴 ∈ ℂ ∧ 1 ∈ ℤ ) → ( sin ‘ ( - 𝐴 + ( 1 · ( 2 · π ) ) ) ) = ( sin ‘ - 𝐴 ) ) |
4 |
1 2 3
|
sylancl |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( - 𝐴 + ( 1 · ( 2 · π ) ) ) ) = ( sin ‘ - 𝐴 ) ) |
5 |
|
2cn |
⊢ 2 ∈ ℂ |
6 |
|
picn |
⊢ π ∈ ℂ |
7 |
5 6
|
mulcli |
⊢ ( 2 · π ) ∈ ℂ |
8 |
7
|
mulid2i |
⊢ ( 1 · ( 2 · π ) ) = ( 2 · π ) |
9 |
8
|
oveq2i |
⊢ ( - 𝐴 + ( 1 · ( 2 · π ) ) ) = ( - 𝐴 + ( 2 · π ) ) |
10 |
|
negsubdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 2 · π ) ∈ ℂ ) → - ( 𝐴 − ( 2 · π ) ) = ( - 𝐴 + ( 2 · π ) ) ) |
11 |
|
negsubdi2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 2 · π ) ∈ ℂ ) → - ( 𝐴 − ( 2 · π ) ) = ( ( 2 · π ) − 𝐴 ) ) |
12 |
10 11
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 2 · π ) ∈ ℂ ) → ( - 𝐴 + ( 2 · π ) ) = ( ( 2 · π ) − 𝐴 ) ) |
13 |
7 12
|
mpan2 |
⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 + ( 2 · π ) ) = ( ( 2 · π ) − 𝐴 ) ) |
14 |
9 13
|
syl5eq |
⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 + ( 1 · ( 2 · π ) ) ) = ( ( 2 · π ) − 𝐴 ) ) |
15 |
14
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( - 𝐴 + ( 1 · ( 2 · π ) ) ) ) = ( sin ‘ ( ( 2 · π ) − 𝐴 ) ) ) |
16 |
4 15
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ - 𝐴 ) = ( sin ‘ ( ( 2 · π ) − 𝐴 ) ) ) |
17 |
|
sinneg |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ - 𝐴 ) = - ( sin ‘ 𝐴 ) ) |
18 |
16 17
|
eqtr3d |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( ( 2 · π ) − 𝐴 ) ) = - ( sin ‘ 𝐴 ) ) |