Step |
Hyp |
Ref |
Expression |
1 |
|
asinval |
⊢ ( 𝐴 ∈ ℂ → ( arcsin ‘ 𝐴 ) = ( - i · ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ) |
2 |
1
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( arcsin ‘ 𝐴 ) ) = ( ℜ ‘ ( - i · ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ) ) |
3 |
|
ax-icn |
⊢ i ∈ ℂ |
4 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ ) |
5 |
3 4
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( i · 𝐴 ) ∈ ℂ ) |
6 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
7 |
|
sqcl |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
8 |
|
subcl |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝐴 ↑ 2 ) ∈ ℂ ) → ( 1 − ( 𝐴 ↑ 2 ) ) ∈ ℂ ) |
9 |
6 7 8
|
sylancr |
⊢ ( 𝐴 ∈ ℂ → ( 1 − ( 𝐴 ↑ 2 ) ) ∈ ℂ ) |
10 |
9
|
sqrtcld |
⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ∈ ℂ ) |
11 |
5 10
|
addcld |
⊢ ( 𝐴 ∈ ℂ → ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ∈ ℂ ) |
12 |
|
asinlem |
⊢ ( 𝐴 ∈ ℂ → ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ≠ 0 ) |
13 |
11 12
|
logcld |
⊢ ( 𝐴 ∈ ℂ → ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ∈ ℂ ) |
14 |
|
imre |
⊢ ( ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ∈ ℂ → ( ℑ ‘ ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) = ( ℜ ‘ ( - i · ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ) ) |
15 |
13 14
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) = ( ℜ ‘ ( - i · ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ) ) |
16 |
2 15
|
eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( arcsin ‘ 𝐴 ) ) = ( ℑ ‘ ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ) |
17 |
|
asinlem3 |
⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ℜ ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) |
18 |
|
argrege0 |
⊢ ( ( ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ∈ ℂ ∧ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ≠ 0 ∧ 0 ≤ ( ℜ ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) → ( ℑ ‘ ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ∈ ( - ( π / 2 ) [,] ( π / 2 ) ) ) |
19 |
11 12 17 18
|
syl3anc |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( log ‘ ( ( i · 𝐴 ) + ( √ ‘ ( 1 − ( 𝐴 ↑ 2 ) ) ) ) ) ) ∈ ( - ( π / 2 ) [,] ( π / 2 ) ) ) |
20 |
16 19
|
eqeltrd |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( arcsin ‘ 𝐴 ) ) ∈ ( - ( π / 2 ) [,] ( π / 2 ) ) ) |