Step |
Hyp |
Ref |
Expression |
1 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
2 |
1
|
oveq1i |
|- ( ( _i x. _i ) x. A ) = ( -u 1 x. A ) |
3 |
|
ax-icn |
|- _i e. CC |
4 |
|
mulass |
|- ( ( _i e. CC /\ _i e. CC /\ A e. CC ) -> ( ( _i x. _i ) x. A ) = ( _i x. ( _i x. A ) ) ) |
5 |
3 3 4
|
mp3an12 |
|- ( A e. CC -> ( ( _i x. _i ) x. A ) = ( _i x. ( _i x. A ) ) ) |
6 |
|
mulm1 |
|- ( A e. CC -> ( -u 1 x. A ) = -u A ) |
7 |
2 5 6
|
3eqtr3a |
|- ( A e. CC -> ( _i x. ( _i x. A ) ) = -u A ) |
8 |
7
|
fveq2d |
|- ( A e. CC -> ( exp ` ( _i x. ( _i x. A ) ) ) = ( exp ` -u A ) ) |
9 |
3 3
|
mulneg1i |
|- ( -u _i x. _i ) = -u ( _i x. _i ) |
10 |
1
|
negeqi |
|- -u ( _i x. _i ) = -u -u 1 |
11 |
|
negneg1e1 |
|- -u -u 1 = 1 |
12 |
10 11
|
eqtri |
|- -u ( _i x. _i ) = 1 |
13 |
9 12
|
eqtri |
|- ( -u _i x. _i ) = 1 |
14 |
13
|
oveq1i |
|- ( ( -u _i x. _i ) x. A ) = ( 1 x. A ) |
15 |
|
negicn |
|- -u _i e. CC |
16 |
|
mulass |
|- ( ( -u _i e. CC /\ _i e. CC /\ A e. CC ) -> ( ( -u _i x. _i ) x. A ) = ( -u _i x. ( _i x. A ) ) ) |
17 |
15 3 16
|
mp3an12 |
|- ( A e. CC -> ( ( -u _i x. _i ) x. A ) = ( -u _i x. ( _i x. A ) ) ) |
18 |
|
mulid2 |
|- ( A e. CC -> ( 1 x. A ) = A ) |
19 |
14 17 18
|
3eqtr3a |
|- ( A e. CC -> ( -u _i x. ( _i x. A ) ) = A ) |
20 |
19
|
fveq2d |
|- ( A e. CC -> ( exp ` ( -u _i x. ( _i x. A ) ) ) = ( exp ` A ) ) |
21 |
8 20
|
oveq12d |
|- ( A e. CC -> ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) = ( ( exp ` -u A ) - ( exp ` A ) ) ) |
22 |
21
|
oveq1d |
|- ( A e. CC -> ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) ) |
23 |
|
mulcl |
|- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC ) |
24 |
3 23
|
mpan |
|- ( A e. CC -> ( _i x. A ) e. CC ) |
25 |
|
sinval |
|- ( ( _i x. A ) e. CC -> ( sin ` ( _i x. A ) ) = ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) ) |
26 |
24 25
|
syl |
|- ( A e. CC -> ( sin ` ( _i x. A ) ) = ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) ) |
27 |
|
irec |
|- ( 1 / _i ) = -u _i |
28 |
27
|
negeqi |
|- -u ( 1 / _i ) = -u -u _i |
29 |
3
|
negnegi |
|- -u -u _i = _i |
30 |
28 29
|
eqtri |
|- -u ( 1 / _i ) = _i |
31 |
30
|
oveq1i |
|- ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) |
32 |
|
ine0 |
|- _i =/= 0 |
33 |
3 32
|
reccli |
|- ( 1 / _i ) e. CC |
34 |
|
efcl |
|- ( A e. CC -> ( exp ` A ) e. CC ) |
35 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
36 |
|
efcl |
|- ( -u A e. CC -> ( exp ` -u A ) e. CC ) |
37 |
35 36
|
syl |
|- ( A e. CC -> ( exp ` -u A ) e. CC ) |
38 |
34 37
|
subcld |
|- ( A e. CC -> ( ( exp ` A ) - ( exp ` -u A ) ) e. CC ) |
39 |
38
|
halfcld |
|- ( A e. CC -> ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. CC ) |
40 |
|
mulneg12 |
|- ( ( ( 1 / _i ) e. CC /\ ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. CC ) -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) ) |
41 |
33 39 40
|
sylancr |
|- ( A e. CC -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) ) |
42 |
|
2cnd |
|- ( A e. CC -> 2 e. CC ) |
43 |
|
2ne0 |
|- 2 =/= 0 |
44 |
43
|
a1i |
|- ( A e. CC -> 2 =/= 0 ) |
45 |
38 42 44
|
divnegd |
|- ( A e. CC -> -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( -u ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) |
46 |
34 37
|
negsubdi2d |
|- ( A e. CC -> -u ( ( exp ` A ) - ( exp ` -u A ) ) = ( ( exp ` -u A ) - ( exp ` A ) ) ) |
47 |
46
|
oveq1d |
|- ( A e. CC -> ( -u ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) |
48 |
45 47
|
eqtrd |
|- ( A e. CC -> -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) |
49 |
48
|
oveq2d |
|- ( A e. CC -> ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) ) |
50 |
37 34
|
subcld |
|- ( A e. CC -> ( ( exp ` -u A ) - ( exp ` A ) ) e. CC ) |
51 |
50
|
halfcld |
|- ( A e. CC -> ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) e. CC ) |
52 |
3
|
a1i |
|- ( A e. CC -> _i e. CC ) |
53 |
32
|
a1i |
|- ( A e. CC -> _i =/= 0 ) |
54 |
51 52 53
|
divrec2d |
|- ( A e. CC -> ( ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) / _i ) = ( ( 1 / _i ) x. ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) ) |
55 |
50 42 52 44 53
|
divdiv1d |
|- ( A e. CC -> ( ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) / _i ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) ) |
56 |
49 54 55
|
3eqtr2d |
|- ( A e. CC -> ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) ) |
57 |
41 56
|
eqtrd |
|- ( A e. CC -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) ) |
58 |
31 57
|
eqtr3id |
|- ( A e. CC -> ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) ) |
59 |
22 26 58
|
3eqtr4d |
|- ( A e. CC -> ( sin ` ( _i x. A ) ) = ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) ) |
60 |
59
|
oveq1d |
|- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) / _i ) ) |
61 |
39 52 53
|
divcan3d |
|- ( A e. CC -> ( ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) |
62 |
60 61
|
eqtrd |
|- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) |