MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvsincos Structured version   Unicode version

Theorem dvsincos 21453
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 9375 . . . . . 6  |-  CC  e.  { RR ,  CC }
21a1i 11 . . . . 5  |-  ( T. 
->  CC  e.  { RR ,  CC } )
3 ax-icn 9341 . . . . . . . . . 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 9406 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
_i  x.  x )  e.  CC )
7 efcl 13368 . . . . . . . 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 9780 . . . . . . . 8  |-  _i  =/=  0
109a1i 11 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  _i  =/=  0 )
118, 4, 10divcld 10107 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  /  _i )  e.  CC )
12 negicn 9611 . . . . . . . . . 10  |-  -u _i  e.  CC
13 mulcl 9366 . . . . . . . . . 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 13368 . . . . . . . . 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 10107 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1817negcld 9706 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( exp `  ( -u _i  x.  x ) )  /  _i )  e.  CC )
1911, 18addcld 9405 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  e.  CC )
208, 16addcld 9405 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
218, 4mulcld 9406 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  x.  _i )  e.  CC )
22 efcl 13368 . . . . . . . . . 10  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2322adantl 466 . . . . . . . . 9  |-  ( ( T.  /\  y  e.  CC )  ->  ( exp `  y )  e.  CC )
24 1cnd 9402 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  CC )  ->  1  e.  CC )
252dvmptid 21431 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
263a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  _i  e.  CC )
272, 5, 24, 25, 26dvmptcmul 21438 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  ( _i  x.  1 ) ) )
283mulid1i 9388 . . . . . . . . . . 11  |-  ( _i  x.  1 )  =  _i
2928mpteq2i 4375 . . . . . . . . . 10  |-  ( x  e.  CC  |->  ( _i  x.  1 ) )  =  ( x  e.  CC  |->  _i )
3027, 29syl6eq 2491 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( _i  x.  x ) ) )  =  ( x  e.  CC  |->  _i ) )
31 eff 13367 . . . . . . . . . . . . 13  |-  exp : CC
--> CC
3231a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  exp : CC --> CC )
3332feqmptd 5744 . . . . . . . . . . 11  |-  ( T. 
->  exp  =  ( y  e.  CC  |->  ( exp `  y ) ) )
3433oveq2d 6107 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( CC  _D  ( y  e.  CC  |->  ( exp `  y ) ) ) )
35 dvef 21452 . . . . . . . . . . 11  |-  ( CC 
_D  exp )  =  exp
3635, 33syl5eq 2487 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  exp )  =  ( y  e.  CC  |->  ( exp `  y
) ) )
3734, 36eqtr3d 2477 . . . . . . . . 9  |-  ( T. 
->  ( CC  _D  (
y  e.  CC  |->  ( exp `  y ) ) )  =  ( y  e.  CC  |->  ( exp `  y ) ) )
38 fveq2 5691 . . . . . . . . 9  |-  ( y  =  ( _i  x.  x )  ->  ( exp `  y )  =  ( exp `  (
_i  x.  x )
) )
392, 2, 6, 4, 23, 23, 30, 37, 38, 38dvmptco 21446 . . . . . . . 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 21439 . . . . . . 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 10113 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  x.  _i )  /  _i )  =  ( exp `  (
_i  x.  x )
) )
4342mpteq2dva 4378 . . . . . . 7  |-  ( T. 
->  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  x.  _i )  /  _i ) )  =  ( x  e.  CC  |->  ( exp `  (
_i  x.  x )
) ) )
4441, 43eqtrd 2475 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( ( exp `  (
_i  x.  x )
)  /  _i ) ) )  =  ( x  e.  CC  |->  ( exp `  ( _i  x.  x ) ) ) )
45 mulcl 9366 . . . . . . . . . 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 10107 . . . . . . . 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 21438 . . . . . . . . . . 11  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  ( -u _i  x.  1 ) ) )
5112mulid1i 9388 . . . . . . . . . . . 12  |-  ( -u _i  x.  1 )  = 
-u _i
5251mpteq2i 4375 . . . . . . . . . . 11  |-  ( x  e.  CC  |->  ( -u _i  x.  1 ) )  =  ( x  e.  CC  |->  -u _i )
5350, 52syl6eq 2491 . . . . . . . . . 10  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  (
-u _i  x.  x
) ) )  =  ( x  e.  CC  |->  -u _i ) )
54 fveq2 5691 . . . . . . . . . 10  |-  ( y  =  ( -u _i  x.  x )  ->  ( exp `  y )  =  ( exp `  ( -u _i  x.  x ) ) )
552, 2, 14, 48, 23, 23, 53, 37, 54, 54dvmptco 21446 . . . . . . . . 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 21439 . . . . . . . 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 21440 . . . . . . 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 10121 . . . . . . . . 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 9683 . . . . . . . . . . 11  |-  -u _i  =/=  0
6059a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  -u _i  =/=  0 )
6116, 48, 60divcan4d 10113 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  -u _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6258, 61eqtrd 2475 . . . . . . . 8  |-  ( ( T.  /\  x  e.  CC )  ->  -u (
( ( exp `  ( -u _i  x.  x ) )  x.  -u _i )  /  _i )  =  ( exp `  ( -u _i  x.  x ) ) )
6362mpteq2dva 4378 . . . . . . 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 2475 . . . . . 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 21434 . . . . 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 10394 . . . . 5  |-  ( T. 
->  2  e.  CC )
67 2ne0 10414 . . . . . 6  |-  2  =/=  0
6867a1i 11 . . . . 5  |-  ( T. 
->  2  =/=  0
)
692, 19, 20, 65, 66, 68dvmptdivc 21439 . . . 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 13355 . . . . . 6  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
718, 16subcld 9719 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  e.  CC )
72 2cnd 10394 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  e.  CC )
7367a1i 11 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  CC )  ->  2  =/=  0 )
7471, 4, 72, 10, 73divdiv1d 10138 . . . . . . . . 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 10392 . . . . . . . . . . 11  |-  2  e.  CC
763, 75mulcomi 9392 . . . . . . . . . 10  |-  ( _i  x.  2 )  =  ( 2  x.  _i )
7776oveq2i 6102 . . . . . . . . 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 2491 . . . . . . . 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 10146 . . . . . . . . . 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 9725 . . . . . . . . . 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 2478 . . . . . . . . 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 6106 . . . . . . . 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 2477 . . . . . . 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 4378 . . . . . 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 2487 . . . . 5  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( ( ( ( exp `  (
_i  x.  x )
)  /  _i )  +  -u ( ( exp `  ( -u _i  x.  x ) )  /  _i ) )  /  2
) ) )
8685oveq2d 6107 . . . 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 13356 . . . . 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 2485 . . 3  |-  ( T. 
->  ( CC  _D  sin )  =  cos )
9021, 46addcld 9405 . . . . 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 21434 . . . . 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 21439 . . . 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 6107 . . . 4  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) ) ) )
9471, 4, 10divcld 10107 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  e.  CC )
9594, 72, 73divnegd 10120 . . . . . 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 13406 . . . . . . . . 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 2478 . . . . . . 7  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  /  2 ) )
9998negeqd 9604 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  = 
-u ( ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  _i )  / 
2 ) )
1003negnegi 9678 . . . . . . . . . 10  |-  -u -u _i  =  _i
101100oveq2i 6102 . . . . . . . . 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 9782 . . . . . . . . . 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 2489 . . . . . . . 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 9366 . . . . . . . . . . 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 9725 . . . . . . . . 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 9782 . . . . . . . . . . 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 6107 . . . . . . . . 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 9801 . . . . . . . . 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 2485 . . . . . . . 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 10110 . . . . . . . . . 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 11965 . . . . . . . . . . 11  |-  ( 1  /  _i )  = 
-u _i
115114oveq2i 6102 . . . . . . . . . 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 2491 . . . . . . . . 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 9604 . . . . . . . 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 2485 . . . . . . 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 6106 . . . . . 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 2485 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  =  ( ( ( ( exp `  ( _i  x.  x ) )  x.  _i )  +  ( ( exp `  ( -u _i  x.  x ) )  x.  -u _i ) )  /  2
) )
121120mpteq2dva 4378 . . . 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 2485 . . 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 1378 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 1369   T. wtru 1370    e. wcel 1756    =/= wne 2606   {cpr 3879    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   expce 13347   sincsin 13349   cosccos 13350    _D cdv 21338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342
This theorem is referenced by:  dvsin  21454  dvcos  21455
  Copyright terms: Public domain W3C validator