Step |
Hyp |
Ref |
Expression |
1 |
|
asincl |
|- ( A e. CC -> ( arcsin ` A ) e. CC ) |
2 |
|
sinval |
|- ( ( arcsin ` A ) e. CC -> ( sin ` ( arcsin ` A ) ) = ( ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) / ( 2 x. _i ) ) ) |
3 |
1 2
|
syl |
|- ( A e. CC -> ( sin ` ( arcsin ` A ) ) = ( ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) / ( 2 x. _i ) ) ) |
4 |
|
ax-icn |
|- _i e. CC |
5 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
6 |
4 5
|
mpan |
|- ( A e. CC -> ( _i x. A ) e. CC ) |
7 |
6
|
negcld |
|- ( A e. CC -> -u ( _i x. A ) e. CC ) |
8 |
|
ax-1cn |
|- 1 e. CC |
9 |
|
sqcl |
|- ( A e. CC -> ( A ^ 2 ) e. CC ) |
10 |
|
subcl |
|- ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC ) |
11 |
8 9 10
|
sylancr |
|- ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC ) |
12 |
11
|
sqrtcld |
|- ( A e. CC -> ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC ) |
13 |
6 7 12
|
pnpcan2d |
|- ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) - ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) = ( ( _i x. A ) - -u ( _i x. A ) ) ) |
14 |
|
efiasin |
|- ( A e. CC -> ( exp ` ( _i x. ( arcsin ` A ) ) ) = ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) |
15 |
|
mulneg12 |
|- ( ( _i e. CC /\ ( arcsin ` A ) e. CC ) -> ( -u _i x. ( arcsin ` A ) ) = ( _i x. -u ( arcsin ` A ) ) ) |
16 |
4 1 15
|
sylancr |
|- ( A e. CC -> ( -u _i x. ( arcsin ` A ) ) = ( _i x. -u ( arcsin ` A ) ) ) |
17 |
|
asinneg |
|- ( A e. CC -> ( arcsin ` -u A ) = -u ( arcsin ` A ) ) |
18 |
17
|
oveq2d |
|- ( A e. CC -> ( _i x. ( arcsin ` -u A ) ) = ( _i x. -u ( arcsin ` A ) ) ) |
19 |
16 18
|
eqtr4d |
|- ( A e. CC -> ( -u _i x. ( arcsin ` A ) ) = ( _i x. ( arcsin ` -u A ) ) ) |
20 |
19
|
fveq2d |
|- ( A e. CC -> ( exp ` ( -u _i x. ( arcsin ` A ) ) ) = ( exp ` ( _i x. ( arcsin ` -u A ) ) ) ) |
21 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
22 |
|
efiasin |
|- ( -u A e. CC -> ( exp ` ( _i x. ( arcsin ` -u A ) ) ) = ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) |
23 |
21 22
|
syl |
|- ( A e. CC -> ( exp ` ( _i x. ( arcsin ` -u A ) ) ) = ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) |
24 |
|
mulneg2 |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) ) |
25 |
4 24
|
mpan |
|- ( A e. CC -> ( _i x. -u A ) = -u ( _i x. A ) ) |
26 |
|
sqneg |
|- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) ) |
27 |
26
|
oveq2d |
|- ( A e. CC -> ( 1 - ( -u A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) ) |
28 |
27
|
fveq2d |
|- ( A e. CC -> ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) = ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) |
29 |
25 28
|
oveq12d |
|- ( A e. CC -> ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) |
30 |
20 23 29
|
3eqtrd |
|- ( A e. CC -> ( exp ` ( -u _i x. ( arcsin ` A ) ) ) = ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) |
31 |
14 30
|
oveq12d |
|- ( A e. CC -> ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) = ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) - ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) ) ) |
32 |
6
|
2timesd |
|- ( A e. CC -> ( 2 x. ( _i x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) ) |
33 |
|
2cn |
|- 2 e. CC |
34 |
|
mulass |
|- ( ( 2 e. CC /\ _i e. CC /\ A e. CC ) -> ( ( 2 x. _i ) x. A ) = ( 2 x. ( _i x. A ) ) ) |
35 |
33 4 34
|
mp3an12 |
|- ( A e. CC -> ( ( 2 x. _i ) x. A ) = ( 2 x. ( _i x. A ) ) ) |
36 |
6 6
|
subnegd |
|- ( A e. CC -> ( ( _i x. A ) - -u ( _i x. A ) ) = ( ( _i x. A ) + ( _i x. A ) ) ) |
37 |
32 35 36
|
3eqtr4d |
|- ( A e. CC -> ( ( 2 x. _i ) x. A ) = ( ( _i x. A ) - -u ( _i x. A ) ) ) |
38 |
13 31 37
|
3eqtr4d |
|- ( A e. CC -> ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) = ( ( 2 x. _i ) x. A ) ) |
39 |
|
mulcl |
|- ( ( _i e. CC /\ ( arcsin ` A ) e. CC ) -> ( _i x. ( arcsin ` A ) ) e. CC ) |
40 |
4 1 39
|
sylancr |
|- ( A e. CC -> ( _i x. ( arcsin ` A ) ) e. CC ) |
41 |
|
efcl |
|- ( ( _i x. ( arcsin ` A ) ) e. CC -> ( exp ` ( _i x. ( arcsin ` A ) ) ) e. CC ) |
42 |
40 41
|
syl |
|- ( A e. CC -> ( exp ` ( _i x. ( arcsin ` A ) ) ) e. CC ) |
43 |
|
negicn |
|- -u _i e. CC |
44 |
|
mulcl |
|- ( ( -u _i e. CC /\ ( arcsin ` A ) e. CC ) -> ( -u _i x. ( arcsin ` A ) ) e. CC ) |
45 |
43 1 44
|
sylancr |
|- ( A e. CC -> ( -u _i x. ( arcsin ` A ) ) e. CC ) |
46 |
|
efcl |
|- ( ( -u _i x. ( arcsin ` A ) ) e. CC -> ( exp ` ( -u _i x. ( arcsin ` A ) ) ) e. CC ) |
47 |
45 46
|
syl |
|- ( A e. CC -> ( exp ` ( -u _i x. ( arcsin ` A ) ) ) e. CC ) |
48 |
42 47
|
subcld |
|- ( A e. CC -> ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) e. CC ) |
49 |
|
id |
|- ( A e. CC -> A e. CC ) |
50 |
|
2mulicn |
|- ( 2 x. _i ) e. CC |
51 |
50
|
a1i |
|- ( A e. CC -> ( 2 x. _i ) e. CC ) |
52 |
|
2muline0 |
|- ( 2 x. _i ) =/= 0 |
53 |
52
|
a1i |
|- ( A e. CC -> ( 2 x. _i ) =/= 0 ) |
54 |
48 49 51 53
|
divmul2d |
|- ( A e. CC -> ( ( ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) / ( 2 x. _i ) ) = A <-> ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) = ( ( 2 x. _i ) x. A ) ) ) |
55 |
38 54
|
mpbird |
|- ( A e. CC -> ( ( ( exp ` ( _i x. ( arcsin ` A ) ) ) - ( exp ` ( -u _i x. ( arcsin ` A ) ) ) ) / ( 2 x. _i ) ) = A ) |
56 |
3 55
|
eqtrd |
|- ( A e. CC -> ( sin ` ( arcsin ` A ) ) = A ) |