Step |
Hyp |
Ref |
Expression |
1 |
|
sinasin |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ ( arcsin ‘ 𝐴 ) ) = 𝐴 ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( sin ‘ ( arcsin ‘ 𝐴 ) ) = 𝐴 ) |
3 |
|
fveqeq2 |
⊢ ( ( arcsin ‘ 𝐴 ) = 𝐵 → ( ( sin ‘ ( arcsin ‘ 𝐴 ) ) = 𝐴 ↔ ( sin ‘ 𝐵 ) = 𝐴 ) ) |
4 |
2 3
|
syl5ibcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( ( arcsin ‘ 𝐴 ) = 𝐵 → ( sin ‘ 𝐵 ) = 𝐴 ) ) |
5 |
|
asinsin |
⊢ ( ( 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( arcsin ‘ ( sin ‘ 𝐵 ) ) = 𝐵 ) |
6 |
5
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( arcsin ‘ ( sin ‘ 𝐵 ) ) = 𝐵 ) |
7 |
|
fveqeq2 |
⊢ ( ( sin ‘ 𝐵 ) = 𝐴 → ( ( arcsin ‘ ( sin ‘ 𝐵 ) ) = 𝐵 ↔ ( arcsin ‘ 𝐴 ) = 𝐵 ) ) |
8 |
6 7
|
syl5ibcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( ( sin ‘ 𝐵 ) = 𝐴 → ( arcsin ‘ 𝐴 ) = 𝐵 ) ) |
9 |
4 8
|
impbid |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℜ ‘ 𝐵 ) ∈ ( - ( π / 2 ) (,) ( π / 2 ) ) ) → ( ( arcsin ‘ 𝐴 ) = 𝐵 ↔ ( sin ‘ 𝐵 ) = 𝐴 ) ) |