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Theorem dvsincos 19344
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 cnex 8834 . . . . . . 7  |-  CC  e.  _V
21prid2 3748 . . . . . 6  |-  CC  e.  { RR ,  CC }
32a1i 10 . . . . 5  |-  (  T. 
->  CC  e.  { RR ,  CC } )
4 ax-icn 8812 . . . . . . . . . 10  |-  _i  e.  CC
54a1i 10 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  _i  e.  CC )
6 simpr 447 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  x  e.  CC )
75, 6mulcld 8871 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
_i  x.  x )  e.  CC )
8 efcl 12380 . . . . . . . 8  |-  ( ( _i  x.  x )  e.  CC  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
97, 8syl 15 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
10 ine0 9231 . . . . . . . 8  |-  _i  =/=  0
1110a1i 10 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  _i  =/=  0 )
129, 5, 11divcld 9552 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  /  _i )  e.  CC )
134negcli 9130 . . . . . . . . . 10  |-  -u _i  e.  CC
14 mulcl 8837 . . . . . . . . . 10  |-  ( (
-u _i  e.  CC  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
1513, 6, 14sylancr 644 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
16 efcl 12380 . . . . . . . . 9  |-  ( (
-u _i  x.  x
)  e.  CC  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1715, 16syl 15 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1817, 5, 11divcld 9552 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1918negcld 9160 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  -u (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
2012, 19addcld 8870 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  e.  CC )
219, 17addcld 8870 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
229, 5mulcld 8871 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  x.  _i )  e.  CC )
23 efcl 12380 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2423adantl 452 . . . . . . . . 9  |-  ( (  T.  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
25 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
2625a1i 10 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  CC )  ->  1  e.  CC )
273dvmptid 19322 . . . . . . . . . . 11  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
284a1i 10 . . . . . . . . . . 11  |-  (  T. 
->  _i  e.  CC )
293, 6, 26, 27, 28dvmptcmul 19329 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  ( _i  x.  1 ) ) )
304mulid1i 8855 . . . . . . . . . . 11  |-  ( _i  x.  1 )  =  _i
3130mpteq2i 4119 . . . . . . . . . 10  |-  ( x  e.  CC  |->  ( _i  x.  1 ) )  =  ( x  e.  CC  |->  _i )
3229, 31syl6eq 2344 . . . . . . . . 9  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  _i ) )
33 eff 12379 . . . . . . . . . . . . 13  |-  exp : CC
--> CC
3433a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  exp : CC --> CC )
3534feqmptd 5591 . . . . . . . . . . 11  |-  (  T. 
->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3635oveq2d 5890 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) ) )
37 dvef 19343 . . . . . . . . . . 11  |-  ( CC 
_D  exp )  =  exp
3837, 35syl5eq 2340 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  exp )  =  ( y  e.  CC  |->  ( exp `  y
) ) )
3936, 38eqtr3d 2330 . . . . . . . . 9  |-  (  T. 
->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
40 fveq2 5541 . . . . . . . . 9  |-  ( y  =  ( _i  x.  x )  ->  ( exp `  y )  =  ( exp `  (
_i  x.  x )
) )
413, 3, 7, 5, 24, 24, 32, 39, 40, 40dvmptco 19337 . . . . . . . 8  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  x.  _i ) ) )
4210a1i 10 . . . . . . . 8  |-  (  T. 
->  _i  =/=  0 )
433, 9, 22, 41, 28, 42dvmptdivc 19330 . . . . . . 7  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i ) ) )
449, 5, 11divcan4d 9558 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i )  =  ( exp `  (
_i  x.  x )
) )
4544mpteq2dva 4122 . . . . . . 7  |-  (  T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  (
_i  x.  x )
) ) )
4643, 45eqtrd 2328 . . . . . 6  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )
47 mulcl 8837 . . . . . . . . . 10  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  -u _i  e.  CC )  ->  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4817, 13, 47sylancl 643 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4948, 5, 11divcld 9552 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  e.  CC )
5013a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  -u _i  e.  CC )
5113a1i 10 . . . . . . . . . . . 12  |-  (  T. 
->  -u _i  e.  CC )
523, 6, 26, 27, 51dvmptcmul 19329 . . . . . . . . . . 11  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  ( -u _i  x.  1 ) ) )
5313mulid1i 8855 . . . . . . . . . . . 12  |-  ( -u _i  x.  1 )  = 
-u _i
5453mpteq2i 4119 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  ( -u _i  x.  1 ) )  =  ( x  e.  CC  |->  -u _i )
5552, 54syl6eq 2344 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  -u _i ) )
56 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  ( -u _i  x.  x )  ->  ( exp `  y )  =  ( exp `  ( -u _i  x.  x ) ) )
573, 3, 15, 50, 24, 24, 55, 39, 56, 56dvmptco 19337 . . . . . . . . 9  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) ) )
583, 17, 48, 57, 28, 42dvmptdivc 19330 . . . . . . . 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 ) ) )
593, 18, 49, 58dvmptneg 19331 . . . . . . 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 ) ) )
6048, 5, 11divneg2d 9566 . . . . . . . . 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 ) )
614, 10negne0i 9137 . . . . . . . . . . 11  |-  -u _i  =/=  0
6261a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  -u _i  =/=  0 )
6317, 50, 62divcan4d 9558 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6460, 63eqtrd 2328 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6564mpteq2dva 4122 . . . . . . 7  |-  (  T. 
->  ( x  e.  CC  |->  -u ( ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )
6659, 65eqtrd 2328 . . . . . 6  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( -u _i  x.  x ) ) ) )
673, 12, 9, 46, 19, 17, 66dvmptadd 19325 . . . . 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 ) ) ) ) )
68 2cn 9832 . . . . . 6  |-  2  e.  CC
6968a1i 10 . . . . 5  |-  (  T. 
->  2  e.  CC )
70 2ne0 9845 . . . . . 6  |-  2  =/=  0
7170a1i 10 . . . . 5  |-  (  T. 
->  2  =/=  0
)
723, 20, 21, 67, 69, 71dvmptdivc 19330 . . . 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 ) ) )
73 df-sin 12367 . . . . . 6  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
749, 17subcld 9173 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
7568a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  2  e.  CC )
7670a1i 10 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  2  =/=  0 )
7774, 5, 75, 11, 76divdiv1d 9583 . . . . . . . . 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 ) ) )
784, 68mulcomi 8859 . . . . . . . . . 10  |-  ( _i  x.  2 )  =  ( 2  x.  _i )
7978oveq2i 5885 . . . . . . . . 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 ) )
8077, 79syl6eq 2344 . . . . . . . 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 ) ) )
819, 17, 5, 11divsubdird 9591 . . . . . . . . . 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 ) ) )
8212, 18negsubd 9179 . . . . . . . . . 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 ) ) )
8381, 82eqtr4d 2331 . . . . . . . . 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 ) ) )
8483oveq1d 5889 . . . . . . . 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 ) )
8580, 84eqtr3d 2330 . . . . . . 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
) )
8685mpteq2dva 4122 . . . . . 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
) ) )
8773, 86syl5eq 2340 . . . . 5  |-  (  T. 
->  sin  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8887oveq2d 5890 . . . 4  |-  (  T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( ( exp `  ( _i  x.  x ) )  /  _i )  + 
-u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) ) )
89 df-cos 12368 . . . . 5  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
9089a1i 10 . . . 4  |-  (  T. 
->  cos  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) )
9172, 88, 903eqtr4d 2338 . . 3  |-  (  T. 
->  ( CC  _D  sin )  =  cos )
9222, 48addcld 8870 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  e.  CC )
933, 9, 22, 41, 17, 48, 57dvmptadd 19325 . . . . 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 ) ) ) )
943, 21, 92, 93, 69, 71dvmptdivc 19330 . . . 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 ) ) )
9590oveq2d 5890 . . . 4  |-  (  T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) ) )
9674, 5, 11divcld 9552 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  e.  CC )
9796, 75, 76divnegd 9565 . . . . . 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 ) )
98 sinval 12418 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9998adantl 452 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
10099, 80eqtr4d 2331 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
101100negeqd 9062 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  -u ( sin `  x )  = 
-u ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 ) )
1024negnegi 9132 . . . . . . . . . 10  |-  -u -u _i  =  _i
103102oveq2i 5885 . . . . . . . . 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 )
104 mulneg2 9233 . . . . . . . . . 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 ) )
10574, 13, 104sylancl 643 . . . . . . . . 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 ) )
106103, 105syl5eqr 2342 . . . . . . . 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 ) )
107 mulcl 8837 . . . . . . . . . . 11  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  _i  e.  CC )  -> 
( ( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10817, 4, 107sylancl 643 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10922, 108negsubd 9179 . . . . . . . . 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 ) ) )
110 mulneg2 9233 . . . . . . . . . . 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 ) )
11117, 4, 110sylancl 643 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  =  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )
112111oveq2d 5890 . . . . . . . . 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 ) ) )
1139, 17, 5subdird 9252 . . . . . . . . 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 ) ) )
114109, 112, 1133eqtr4d 2338 . . . . . . . 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 ) )
11574, 5, 11divrecd 9555 . . . . . . . . . 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 ) ) )
116 irec 11218 . . . . . . . . . . 11  |-  ( 1  /  _i )  = 
-u _i
117116oveq2i 5885 . . . . . . . . . 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 )
118115, 117syl6eq 2344 . . . . . . . . 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 ) )
119118negeqd 9062 . . . . . . . 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 ) )
120106, 114, 1193eqtr4d 2338 . . . . . . 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 ) )
121120oveq1d 5889 . . . . . 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 ) )
12297, 101, 1213eqtr4d 2338 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  -u ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  /  2
) )
123122mpteq2dva 4122 . . . 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 ) ) )
12494, 95, 1233eqtr4d 2338 . . 3  |-  (  T. 
->  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) )
12591, 124jca 518 . 2  |-  (  T. 
->  ( ( CC  _D  sin )  =  cos  /\  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) ) )
126125trud 1314 1  |-  ( ( CC  _D  sin )  =  cos  /\  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   {cpr 3654    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   expce 12359   sincsin 12361   cosccos 12362    _D cdv 19229
This theorem is referenced by:  dvsin  19345  dvcos  19346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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