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Theorem dvsincos 22674
Description: Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)
Assertion
Ref Expression
dvsincos  |-  ( ( CC  _D  sin )  =  cos  /\  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )

Proof of Theorem dvsincos
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnelprrecn 9615 . . . . . 6  |-  CC  e.  { RR ,  CC }
21a1i 11 . . . . 5  |-  ( T. 
->  CC  e.  { RR ,  CC } )
3 ax-icn 9581 . . . . . . . . . 10  |-  _i  e.  CC
43a1i 11 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  _i  e.  CC )
5 simpr 459 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  x  e.  CC )
64, 5mulcld 9646 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
_i  x.  x )  e.  CC )
7 efcl 14027 . . . . . . . 8  |-  ( ( _i  x.  x )  e.  CC  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
86, 7syl 17 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
9 ine0 10033 . . . . . . . 8  |-  _i  =/=  0
109a1i 11 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  _i  =/=  0 )
118, 4, 10divcld 10361 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  /  _i )  e.  CC )
12 negicn 9857 . . . . . . . . . 10  |-  -u _i  e.  CC
13 mulcl 9606 . . . . . . . . . 10  |-  ( (
-u _i  e.  CC  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
1412, 5, 13sylancr 661 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
15 efcl 14027 . . . . . . . . 9  |-  ( (
-u _i  x.  x
)  e.  CC  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1614, 15syl 17 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1716, 4, 10divcld 10361 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1817negcld 9954 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1911, 18addcld 9645 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  e.  CC )
208, 16addcld 9645 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
218, 4mulcld 9646 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  x.  _i )  e.  CC )
22 efcl 14027 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2322adantl 464 . . . . . . . . 9  |-  ( ( T.  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
24 1cnd 9642 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  CC )  ->  1  e.  CC )
252dvmptid 22652 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
263a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  _i  e.  CC )
272, 5, 24, 25, 26dvmptcmul 22659 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  ( _i  x.  1 ) ) )
283mulid1i 9628 . . . . . . . . . . 11  |-  ( _i  x.  1 )  =  _i
2928mpteq2i 4478 . . . . . . . . . 10  |-  ( x  e.  CC  |->  ( _i  x.  1 ) )  =  ( x  e.  CC  |->  _i )
3027, 29syl6eq 2459 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  _i ) )
31 eff 14026 . . . . . . . . . . . . 13  |-  exp : CC
--> CC
3231a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  exp : CC --> CC )
3332feqmptd 5902 . . . . . . . . . . 11  |-  ( T. 
->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3433oveq2d 6294 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) ) )
35 dvef 22673 . . . . . . . . . . 11  |-  ( CC 
_D  exp )  =  exp
3635, 33syl5eq 2455 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( y  e.  CC  |->  ( exp `  y
) ) )
3734, 36eqtr3d 2445 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
38 fveq2 5849 . . . . . . . . 9  |-  ( y  =  ( _i  x.  x )  ->  ( exp `  y )  =  ( exp `  (
_i  x.  x )
) )
392, 2, 6, 4, 23, 23, 30, 37, 38, 38dvmptco 22667 . . . . . . . 8  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  x.  _i ) ) )
409a1i 11 . . . . . . . 8  |-  ( T. 
->  _i  =/=  0 )
412, 8, 21, 39, 26, 40dvmptdivc 22660 . . . . . . 7  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i ) ) )
428, 4, 10divcan4d 10367 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i )  =  ( exp `  (
_i  x.  x )
) )
4342mpteq2dva 4481 . . . . . . 7  |-  ( T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  (
_i  x.  x )
) ) )
4441, 43eqtrd 2443 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )
45 mulcl 9606 . . . . . . . . . 10  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  -u _i  e.  CC )  ->  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4616, 12, 45sylancl 660 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4746, 4, 10divcld 10361 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  e.  CC )
4812a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  -u _i  e.  CC )
4912a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  -u _i  e.  CC )
502, 5, 24, 25, 49dvmptcmul 22659 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  ( -u _i  x.  1 ) ) )
5112mulid1i 9628 . . . . . . . . . . . 12  |-  ( -u _i  x.  1 )  = 
-u _i
5251mpteq2i 4478 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  ( -u _i  x.  1 ) )  =  ( x  e.  CC  |->  -u _i )
5350, 52syl6eq 2459 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  -u _i ) )
54 fveq2 5849 . . . . . . . . . 10  |-  ( y  =  ( -u _i  x.  x )  ->  ( exp `  y )  =  ( exp `  ( -u _i  x.  x ) ) )
552, 2, 14, 48, 23, 23, 53, 37, 54, 54dvmptco 22667 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) ) )
562, 16, 46, 55, 26, 40dvmptdivc 22660 . . . . . . . 8  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )  =  ( x  e.  CC  |->  ( ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i ) ) )
572, 17, 47, 56dvmptneg 22661 . . . . . . 7  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )  =  ( x  e.  CC  |->  -u ( ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i ) ) )
5846, 4, 10divneg2d 10375 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i ) )
593, 9negne0i 9930 . . . . . . . . . . 11  |-  -u _i  =/=  0
6059a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  -u _i  =/=  0 )
6116, 48, 60divcan4d 10367 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6258, 61eqtrd 2443 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6362mpteq2dva 4481 . . . . . . 7  |-  ( T. 
->  ( x  e.  CC  |->  -u ( ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )
6457, 63eqtrd 2443 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )
652, 11, 8, 44, 18, 16, 64dvmptadd 22655 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) ) ) )
66 2cnd 10649 . . . . 5  |-  ( T. 
->  2  e.  CC )
67 2ne0 10669 . . . . . 6  |-  2  =/=  0
6867a1i 11 . . . . 5  |-  ( T. 
->  2  =/=  0
)
692, 19, 20, 65, 66, 68dvmptdivc 22660 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( ( ( exp `  ( _i  x.  x
) )  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2 ) ) )  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) )
70 df-sin 14014 . . . . . 6  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
718, 16subcld 9967 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
72 2cnd 10649 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  e.  CC )
7367a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  =/=  0 )
7471, 4, 72, 10, 73divdiv1d 10392 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 )  =  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( _i  x.  2 ) ) )
75 2cn 10647 . . . . . . . . . . 11  |-  2  e.  CC
763, 75mulcomi 9632 . . . . . . . . . 10  |-  ( _i  x.  2 )  =  ( 2  x.  _i )
7776oveq2i 6289 . . . . . . . . 9  |-  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( _i  x.  2 ) )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) )
7874, 77syl6eq 2459 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 )  =  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
798, 16, 4, 10divsubdird 10400 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  =  ( ( ( exp `  ( _i  x.  x
) )  /  _i )  -  ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )
8011, 17negsubd 9973 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  =  ( ( ( exp `  (
_i  x.  x )
)  /  _i )  -  ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )
8179, 80eqtr4d 2446 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  =  ( ( ( exp `  ( _i  x.  x
) )  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )
8281oveq1d 6293 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 )  =  ( ( ( ( exp `  ( _i  x.  x
) )  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2 ) )
8378, 82eqtr3d 2445 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) )  =  ( ( ( ( exp `  ( _i  x.  x ) )  /  _i )  + 
-u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) )
8483mpteq2dva 4481 . . . . . 6  |-  ( T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8570, 84syl5eq 2455 . . . . 5  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8685oveq2d 6294 . . . 4  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( ( exp `  ( _i  x.  x ) )  /  _i )  + 
-u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) ) )
87 df-cos 14015 . . . . 5  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
8887a1i 11 . . . 4  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) )
8969, 86, 883eqtr4d 2453 . . 3  |-  ( T. 
->  ( CC  _D  sin )  =  cos )
9021, 46addcld 9645 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  e.  CC )
912, 8, 21, 39, 16, 46, 55dvmptadd 22655 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) ) ) )  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) ) ) )
922, 20, 90, 91, 66, 68dvmptdivc 22660 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) )  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  / 
2 ) ) )
9388oveq2d 6294 . . . 4  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) ) )
9471, 4, 10divcld 10361 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  e.  CC )
9594, 72, 73divnegd 10374 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 )  =  (
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
96 sinval 14066 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9796adantl 464 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9897, 78eqtr4d 2446 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
9998negeqd 9850 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  = 
-u ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 ) )
1003negnegi 9925 . . . . . . . . . 10  |-  -u -u _i  =  _i
101100oveq2i 6289 . . . . . . . . 9  |-  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u -u _i )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  _i )
102 mulneg2 10035 . . . . . . . . . 10  |-  ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC  /\  -u _i  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u -u _i )  = 
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i ) )
10371, 12, 102sylancl 660 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u -u _i )  = 
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i ) )
104101, 103syl5eqr 2457 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  _i )  = 
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i ) )
105 mulcl 9606 . . . . . . . . . . 11  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  _i  e.  CC )  -> 
( ( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10616, 3, 105sylancl 660 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10721, 106negsubd 9973 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )  =  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  -  ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) ) )
108 mulneg2 10035 . . . . . . . . . . 11  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  _i  e.  CC )  -> 
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  =  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )
10916, 3, 108sylancl 660 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  =  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )
110109oveq2d 6294 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  =  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) ) )
1118, 16, 4subdird 10054 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  _i )  =  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  -  ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) ) )
112107, 110, 1113eqtr4d 2453 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  _i ) )
11371, 4, 10divrecd 10364 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  ( 1  /  _i ) ) )
114 irec 12312 . . . . . . . . . . 11  |-  ( 1  /  _i )  = 
-u _i
115114oveq2i 6289 . . . . . . . . . 10  |-  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  ( 1  /  _i ) )  =  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i )
116113, 115syl6eq 2459 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i ) )
117116negeqd 9850 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  = 
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  x.  -u _i ) )
118104, 112, 1173eqtr4d 2453 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  = 
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i ) )
119118oveq1d 6293 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( ( exp `  ( _i  x.  x
) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  / 
2 )  =  (
-u ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
12095, 99, 1193eqtr4d 2453 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  /  2
) )
121120mpteq2dva 4481 . . . 4  |-  ( T. 
->  ( x  e.  CC  |->  -u ( sin `  x
) )  =  ( x  e.  CC  |->  ( ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  / 
2 ) ) )
12292, 93, 1213eqtr4d 2453 . . 3  |-  ( T. 
->  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) )
12389, 122jca 530 . 2  |-  ( T. 
->  ( ( CC  _D  sin )  =  cos  /\  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) ) )
124123trud 1414 1  |-  ( ( CC  _D  sin )  =  cos  /\  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842    =/= wne 2598   {cpr 3974    |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523   _ici 9524    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842    / cdiv 10247   2c2 10626   expce 14006   sincsin 14008   cosccos 14009    _D cdv 22559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563
This theorem is referenced by:  dvsin  22675  dvcos  22676
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