Step |
Hyp |
Ref |
Expression |
1 |
|
df-sin |
|- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
2 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
3 |
2
|
subcn |
|- - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
4 |
3
|
a1i |
|- ( T. -> - e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
5 |
|
efcn |
|- exp e. ( CC -cn-> CC ) |
6 |
5
|
a1i |
|- ( T. -> exp e. ( CC -cn-> CC ) ) |
7 |
|
ax-icn |
|- _i e. CC |
8 |
|
eqid |
|- ( x e. CC |-> ( _i x. x ) ) = ( x e. CC |-> ( _i x. x ) ) |
9 |
8
|
mulc1cncf |
|- ( _i e. CC -> ( x e. CC |-> ( _i x. x ) ) e. ( CC -cn-> CC ) ) |
10 |
7 9
|
mp1i |
|- ( T. -> ( x e. CC |-> ( _i x. x ) ) e. ( CC -cn-> CC ) ) |
11 |
6 10
|
cncfmpt1f |
|- ( T. -> ( x e. CC |-> ( exp ` ( _i x. x ) ) ) e. ( CC -cn-> CC ) ) |
12 |
|
negicn |
|- -u _i e. CC |
13 |
|
eqid |
|- ( x e. CC |-> ( -u _i x. x ) ) = ( x e. CC |-> ( -u _i x. x ) ) |
14 |
13
|
mulc1cncf |
|- ( -u _i e. CC -> ( x e. CC |-> ( -u _i x. x ) ) e. ( CC -cn-> CC ) ) |
15 |
12 14
|
mp1i |
|- ( T. -> ( x e. CC |-> ( -u _i x. x ) ) e. ( CC -cn-> CC ) ) |
16 |
6 15
|
cncfmpt1f |
|- ( T. -> ( x e. CC |-> ( exp ` ( -u _i x. x ) ) ) e. ( CC -cn-> CC ) ) |
17 |
2 4 11 16
|
cncfmpt2f |
|- ( T. -> ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) e. ( CC -cn-> CC ) ) |
18 |
|
cncff |
|- ( ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) e. ( CC -cn-> CC ) -> ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) : CC --> CC ) |
19 |
17 18
|
syl |
|- ( T. -> ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) : CC --> CC ) |
20 |
|
eqid |
|- ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) = ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) |
21 |
20
|
fmpt |
|- ( A. x e. CC ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) e. CC <-> ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) : CC --> CC ) |
22 |
19 21
|
sylibr |
|- ( T. -> A. x e. CC ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) e. CC ) |
23 |
|
eqidd |
|- ( T. -> ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) = ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) ) |
24 |
|
eqidd |
|- ( T. -> ( y e. CC |-> ( y / ( 2 x. _i ) ) ) = ( y e. CC |-> ( y / ( 2 x. _i ) ) ) ) |
25 |
|
oveq1 |
|- ( y = ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) -> ( y / ( 2 x. _i ) ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
26 |
22 23 24 25
|
fmptcof |
|- ( T. -> ( ( y e. CC |-> ( y / ( 2 x. _i ) ) ) o. ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) ) = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) ) |
27 |
|
2mulicn |
|- ( 2 x. _i ) e. CC |
28 |
|
2muline0 |
|- ( 2 x. _i ) =/= 0 |
29 |
|
eqid |
|- ( y e. CC |-> ( y / ( 2 x. _i ) ) ) = ( y e. CC |-> ( y / ( 2 x. _i ) ) ) |
30 |
29
|
divccncf |
|- ( ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) =/= 0 ) -> ( y e. CC |-> ( y / ( 2 x. _i ) ) ) e. ( CC -cn-> CC ) ) |
31 |
27 28 30
|
mp2an |
|- ( y e. CC |-> ( y / ( 2 x. _i ) ) ) e. ( CC -cn-> CC ) |
32 |
31
|
a1i |
|- ( T. -> ( y e. CC |-> ( y / ( 2 x. _i ) ) ) e. ( CC -cn-> CC ) ) |
33 |
17 32
|
cncfco |
|- ( T. -> ( ( y e. CC |-> ( y / ( 2 x. _i ) ) ) o. ( x e. CC |-> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) ) ) e. ( CC -cn-> CC ) ) |
34 |
26 33
|
eqeltrrd |
|- ( T. -> ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) e. ( CC -cn-> CC ) ) |
35 |
34
|
mptru |
|- ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) e. ( CC -cn-> CC ) |
36 |
1 35
|
eqeltri |
|- sin e. ( CC -cn-> CC ) |