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Theorem dvsincos 19870
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 9076 . . . . . . 7  |-  CC  e.  _V
21prid2 3915 . . . . . 6  |-  CC  e.  { RR ,  CC }
32a1i 11 . . . . 5  |-  (  T. 
->  CC  e.  { RR ,  CC } )
4 ax-icn 9054 . . . . . . . . . 10  |-  _i  e.  CC
54a1i 11 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  _i  e.  CC )
6 simpr 449 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  x  e.  CC )
75, 6mulcld 9113 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
_i  x.  x )  e.  CC )
8 efcl 12690 . . . . . . . 8  |-  ( ( _i  x.  x )  e.  CC  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
97, 8syl 16 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  ( exp `  ( _i  x.  x ) )  e.  CC )
10 ine0 9474 . . . . . . . 8  |-  _i  =/=  0
1110a1i 11 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  _i  =/=  0 )
129, 5, 11divcld 9795 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  /  _i )  e.  CC )
134negcli 9373 . . . . . . . . . 10  |-  -u _i  e.  CC
14 mulcl 9079 . . . . . . . . . 10  |-  ( (
-u _i  e.  CC  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
1513, 6, 14sylancr 646 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  ( -u _i  x.  x )  e.  CC )
16 efcl 12690 . . . . . . . . 9  |-  ( (
-u _i  x.  x
)  e.  CC  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1715, 16syl 16 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  ( exp `  ( -u _i  x.  x ) )  e.  CC )
1817, 5, 11divcld 9795 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1918negcld 9403 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  -u (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
2012, 19addcld 9112 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  e.  CC )
219, 17addcld 9112 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
229, 5mulcld 9113 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  x.  _i )  e.  CC )
23 efcl 12690 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2423adantl 454 . . . . . . . . 9  |-  ( (  T.  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
25 ax-1cn 9053 . . . . . . . . . . . 12  |-  1  e.  CC
2625a1i 11 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  CC )  ->  1  e.  CC )
273dvmptid 19848 . . . . . . . . . . 11  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
284a1i 11 . . . . . . . . . . 11  |-  (  T. 
->  _i  e.  CC )
293, 6, 26, 27, 28dvmptcmul 19855 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  ( _i  x.  1 ) ) )
304mulid1i 9097 . . . . . . . . . . 11  |-  ( _i  x.  1 )  =  _i
3130mpteq2i 4295 . . . . . . . . . 10  |-  ( x  e.  CC  |->  ( _i  x.  1 ) )  =  ( x  e.  CC  |->  _i )
3229, 31syl6eq 2486 . . . . . . . . 9  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  _i ) )
33 eff 12689 . . . . . . . . . . . . 13  |-  exp : CC
--> CC
3433a1i 11 . . . . . . . . . . . 12  |-  (  T. 
->  exp : CC --> CC )
3534feqmptd 5782 . . . . . . . . . . 11  |-  (  T. 
->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3635oveq2d 6100 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) ) )
37 dvef 19869 . . . . . . . . . . 11  |-  ( CC 
_D  exp )  =  exp
3837, 35syl5eq 2482 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  exp )  =  ( y  e.  CC  |->  ( exp `  y
) ) )
3936, 38eqtr3d 2472 . . . . . . . . 9  |-  (  T. 
->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
40 fveq2 5731 . . . . . . . . 9  |-  ( y  =  ( _i  x.  x )  ->  ( exp `  y )  =  ( exp `  (
_i  x.  x )
) )
413, 3, 7, 5, 24, 24, 32, 39, 40, 40dvmptco 19863 . . . . . . . 8  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )  =  ( x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  x.  _i ) ) )
4210a1i 11 . . . . . . . 8  |-  (  T. 
->  _i  =/=  0 )
433, 9, 22, 41, 28, 42dvmptdivc 19856 . . . . . . 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 9801 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i )  =  ( exp `  (
_i  x.  x )
) )
4544mpteq2dva 4298 . . . . . . 7  |-  (  T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  (
_i  x.  x )
) ) )
4643, 45eqtrd 2470 . . . . . 6  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )
47 mulcl 9079 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  e.  CC )
4948, 5, 11divcld 9795 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  e.  CC )
5013a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  -u _i  e.  CC )
5113a1i 11 . . . . . . . . . . . 12  |-  (  T. 
->  -u _i  e.  CC )
523, 6, 26, 27, 51dvmptcmul 19855 . . . . . . . . . . 11  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  ( -u _i  x.  1 ) ) )
5313mulid1i 9097 . . . . . . . . . . . 12  |-  ( -u _i  x.  1 )  = 
-u _i
5453mpteq2i 4295 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  ( -u _i  x.  1 ) )  =  ( x  e.  CC  |->  -u _i )
5552, 54syl6eq 2486 . . . . . . . . . 10  |-  (  T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  -u _i ) )
56 fveq2 5731 . . . . . . . . . 10  |-  ( y  =  ( -u _i  x.  x )  ->  ( exp `  y )  =  ( exp `  ( -u _i  x.  x ) ) )
573, 3, 15, 50, 24, 24, 55, 39, 56, 56dvmptco 19863 . . . . . . . . 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 19856 . . . . . . . 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 19857 . . . . . . 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 9809 . . . . . . . . 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 9380 . . . . . . . . . . 11  |-  -u _i  =/=  0
6261a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  -u _i  =/=  0 )
6317, 50, 62divcan4d 9801 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6460, 63eqtrd 2470 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6564mpteq2dva 4298 . . . . . . 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 2470 . . . . . 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 19851 . . . . 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 10075 . . . . . 6  |-  2  e.  CC
6968a1i 11 . . . . 5  |-  (  T. 
->  2  e.  CC )
70 2ne0 10088 . . . . . 6  |-  2  =/=  0
7170a1i 11 . . . . 5  |-  (  T. 
->  2  =/=  0
)
723, 20, 21, 67, 69, 71dvmptdivc 19856 . . . 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 12677 . . . . . 6  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
749, 17subcld 9416 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
7568a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  2  e.  CC )
7670a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  2  =/=  0 )
7774, 5, 75, 11, 76divdiv1d 9826 . . . . . . . . 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 9101 . . . . . . . . . 10  |-  ( _i  x.  2 )  =  ( 2  x.  _i )
7978oveq2i 6095 . . . . . . . . 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 2486 . . . . . . . 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 9834 . . . . . . . . . 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 9422 . . . . . . . . . 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 2473 . . . . . . . . 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 6099 . . . . . . . 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 2472 . . . . . . 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 4298 . . . . . 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 2482 . . . . 5  |-  (  T. 
->  sin  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8887oveq2d 6100 . . . 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 12678 . . . . 5  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
9089a1i 11 . . . 4  |-  (  T. 
->  cos  =  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) )
9172, 88, 903eqtr4d 2480 . . 3  |-  (  T. 
->  ( CC  _D  sin )  =  cos )
9222, 48addcld 9112 . . . . 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 19851 . . . . 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 19856 . . . 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 6100 . . . 4  |-  (  T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) ) )
9674, 5, 11divcld 9795 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  e.  CC )
9796, 75, 76divnegd 9808 . . . . . 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 12728 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
9998adantl 454 . . . . . . . 8  |-  ( (  T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
10099, 80eqtr4d 2473 . . . . . . 7  |-  ( (  T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
101100negeqd 9305 . . . . . 6  |-  ( (  T.  /\  x  e.  CC )  ->  -u ( sin `  x )  = 
-u ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 ) )
1024negnegi 9375 . . . . . . . . . 10  |-  -u -u _i  =  _i
103102oveq2i 6095 . . . . . . . . 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 9476 . . . . . . . . . 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 645 . . . . . . . . 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 2484 . . . . . . . 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 9079 . . . . . . . . . . 11  |-  ( ( ( exp `  ( -u _i  x.  x ) )  e.  CC  /\  _i  e.  CC )  -> 
( ( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10817, 4, 107sylancl 645 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  _i )  e.  CC )
10922, 108negsubd 9422 . . . . . . . . 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 9476 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  x.  -u _i )  =  -u ( ( exp `  ( -u _i  x.  x ) )  x.  _i ) )
112111oveq2d 6100 . . . . . . . . 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 9495 . . . . . . . . 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 2480 . . . . . . . 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 9798 . . . . . . . . . 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 11485 . . . . . . . . . . 11  |-  ( 1  /  _i )  = 
-u _i
117116oveq2i 6095 . . . . . . . . . 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 2486 . . . . . . . . 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 9305 . . . . . . . 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 2480 . . . . . . 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 6099 . . . . . 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 2480 . . . . 5  |-  ( (  T.  /\  x  e.  CC )  ->  -u ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  /  2
) )
123122mpteq2dva 4298 . . . 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 2480 . . 3  |-  (  T. 
->  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) )
12591, 124jca 520 . 2  |-  (  T. 
->  ( ( CC  _D  sin )  =  cos  /\  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x
) ) ) )
126125trud 1333 1  |-  ( ( CC  _D  sin )  =  cos  /\  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    T. wtru 1326    = wceq 1653    e. wcel 1726    =/= wne 2601   {cpr 3817    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   _ici 8997    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297    / cdiv 9682   2c2 10054   expce 12669   sincsin 12671   cosccos 12672    _D cdv 19755
This theorem is referenced by:  dvsin  19871  dvcos  19872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759
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