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Theorem dvsincos 22114
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 9581 . . . . . 6  |-  CC  e.  { RR ,  CC }
21a1i 11 . . . . 5  |-  ( T. 
->  CC  e.  { RR ,  CC } )
3 ax-icn 9547 . . . . . . . . . 10  |-  _i  e.  CC
43a1i 11 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  _i  e.  CC )
5 simpr 461 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  x  e.  CC )
64, 5mulcld 9612 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
_i  x.  x )  e.  CC )
7 efcl 13673 . . . . . . . 8  |-  ( ( _i  x.  x )  e.  CC  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
86, 7syl 16 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
9 ine0 9988 . . . . . . . 8  |-  _i  =/=  0
109a1i 11 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  _i  =/=  0 )
118, 4, 10divcld 10316 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  /  _i )  e.  CC )
12 negicn 9817 . . . . . . . . . 10  |-  -u _i  e.  CC
13 mulcl 9572 . . . . . . . . . 10  |-  ( (
-u _i  e.  CC  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
1412, 5, 13sylancr 663 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
15 efcl 13673 . . . . . . . . 9  |-  ( (
-u _i  x.  x
)  e.  CC  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1614, 15syl 16 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1716, 4, 10divcld 10316 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1817negcld 9913 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1911, 18addcld 9611 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  e.  CC )
208, 16addcld 9611 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
218, 4mulcld 9612 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  x.  _i )  e.  CC )
22 efcl 13673 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2322adantl 466 . . . . . . . . 9  |-  ( ( T.  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
24 1cnd 9608 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  CC )  ->  1  e.  CC )
252dvmptid 22092 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
263a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  _i  e.  CC )
272, 5, 24, 25, 26dvmptcmul 22099 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  ( _i  x.  1 ) ) )
283mulid1i 9594 . . . . . . . . . . 11  |-  ( _i  x.  1 )  =  _i
2928mpteq2i 4530 . . . . . . . . . 10  |-  ( x  e.  CC  |->  ( _i  x.  1 ) )  =  ( x  e.  CC  |->  _i )
3027, 29syl6eq 2524 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  _i ) )
31 eff 13672 . . . . . . . . . . . . 13  |-  exp : CC
--> CC
3231a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  exp : CC --> CC )
3332feqmptd 5918 . . . . . . . . . . 11  |-  ( T. 
->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3433oveq2d 6298 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) ) )
35 dvef 22113 . . . . . . . . . . 11  |-  ( CC 
_D  exp )  =  exp
3635, 33syl5eq 2520 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( y  e.  CC  |->  ( exp `  y
) ) )
3734, 36eqtr3d 2510 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
38 fveq2 5864 . . . . . . . . 9  |-  ( y  =  ( _i  x.  x )  ->  ( exp `  y )  =  ( exp `  (
_i  x.  x )
) )
392, 2, 6, 4, 23, 23, 30, 37, 38, 38dvmptco 22107 . . . . . . . 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 22100 . . . . . . 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 10322 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i )  =  ( exp `  (
_i  x.  x )
) )
4342mpteq2dva 4533 . . . . . . 7  |-  ( T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  (
_i  x.  x )
) ) )
4441, 43eqtrd 2508 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )
45 mulcl 9572 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4746, 4, 10divcld 10316 . . . . . . . 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 22099 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  ( -u _i  x.  1 ) ) )
5112mulid1i 9594 . . . . . . . . . . . 12  |-  ( -u _i  x.  1 )  = 
-u _i
5251mpteq2i 4530 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  ( -u _i  x.  1 ) )  =  ( x  e.  CC  |->  -u _i )
5350, 52syl6eq 2524 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  -u _i ) )
54 fveq2 5864 . . . . . . . . . 10  |-  ( y  =  ( -u _i  x.  x )  ->  ( exp `  y )  =  ( exp `  ( -u _i  x.  x ) ) )
552, 2, 14, 48, 23, 23, 53, 37, 54, 54dvmptco 22107 . . . . . . . . 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 22100 . . . . . . . 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 22101 . . . . . . 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 10330 . . . . . . . . 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 9890 . . . . . . . . . . 11  |-  -u _i  =/=  0
6059a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  -u _i  =/=  0 )
6116, 48, 60divcan4d 10322 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6258, 61eqtrd 2508 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6362mpteq2dva 4533 . . . . . . 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 2508 . . . . . 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 22095 . . . . 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 10604 . . . . 5  |-  ( T. 
->  2  e.  CC )
67 2ne0 10624 . . . . . 6  |-  2  =/=  0
6867a1i 11 . . . . 5  |-  ( T. 
->  2  =/=  0
)
692, 19, 20, 65, 66, 68dvmptdivc 22100 . . . 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 13660 . . . . . 6  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
718, 16subcld 9926 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
72 2cnd 10604 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  e.  CC )
7367a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  =/=  0 )
7471, 4, 72, 10, 73divdiv1d 10347 . . . . . . . . 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 10602 . . . . . . . . . . 11  |-  2  e.  CC
763, 75mulcomi 9598 . . . . . . . . . 10  |-  ( _i  x.  2 )  =  ( 2  x.  _i )
7776oveq2i 6293 . . . . . . . . 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 2524 . . . . . . . 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 10355 . . . . . . . . . 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 9932 . . . . . . . . . 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 2511 . . . . . . . . 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 6297 . . . . . . . 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 2510 . . . . . . 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 4533 . . . . . 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 2520 . . . . 5  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8685oveq2d 6298 . . . 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 13661 . . . . 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 2518 . . 3  |-  ( T. 
->  ( CC  _D  sin )  =  cos )
9021, 46addcld 9611 . . . . 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 22095 . . . . 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 22100 . . . 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 6298 . . . 4  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) ) )
9471, 4, 10divcld 10316 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  e.  CC )
9594, 72, 73divnegd 10329 . . . . . 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 13711 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9796adantl 466 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9897, 78eqtr4d 2511 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
9998negeqd 9810 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  = 
-u ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 ) )
1003negnegi 9885 . . . . . . . . . 10  |-  -u -u _i  =  _i
101100oveq2i 6293 . . . . . . . . 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 9990 . . . . . . . . . 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 662 . . . . . . . . 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 2522 . . . . . . . 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 9572 . . . . . . . . . . 11  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  _i  e.  CC )  -> 
( ( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10616, 3, 105sylancl 662 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10721, 106negsubd 9932 . . . . . . . . 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 9990 . . . . . . . . . . 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 662 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  =  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )
110109oveq2d 6298 . . . . . . . . 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 10009 . . . . . . . . 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 2518 . . . . . . . 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 10319 . . . . . . . . . 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 12229 . . . . . . . . . . 11  |-  ( 1  /  _i )  = 
-u _i
115114oveq2i 6293 . . . . . . . . . 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 2524 . . . . . . . . 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 9810 . . . . . . . 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 2518 . . . . . . 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 6297 . . . . . 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 2518 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  /  2
) )
121120mpteq2dva 4533 . . . 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 2518 . . 3  |-  ( T. 
->  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) )
12389, 122jca 532 . 2  |-  ( T. 
->  ( ( CC  _D  sin )  =  cos  /\  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) ) )
124123trud 1388 1  |-  ( ( CC  _D  sin )  =  cos  /\  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   {cpr 4029    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   _ici 9490    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   2c2 10581   expce 13652   sincsin 13654   cosccos 13655    _D cdv 21999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-fac 12316  df-bc 12343  df-hash 12368  df-shft 12857  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-sum 13465  df-ef 13658  df-sin 13660  df-cos 13661  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003
This theorem is referenced by:  dvsin  22115  dvcos  22116
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