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

Theorem sinval 12650
Description: Value of the sine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
sinval  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )

Proof of Theorem sinval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6028 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5672 . . . 4  |-  ( x  =  A  ->  ( exp `  ( _i  x.  x ) )  =  ( exp `  (
_i  x.  A )
) )
3 oveq2 6028 . . . . 5  |-  ( x  =  A  ->  ( -u _i  x.  x )  =  ( -u _i  x.  A ) )
43fveq2d 5672 . . . 4  |-  ( x  =  A  ->  ( exp `  ( -u _i  x.  x ) )  =  ( exp `  ( -u _i  x.  A ) ) )
52, 4oveq12d 6038 . . 3  |-  ( x  =  A  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  =  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )
65oveq1d 6035 . 2  |-  ( x  =  A  ->  (
( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
7 df-sin 12599 . 2  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
8 ovex 6045 . 2  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  e. 
_V
96, 7, 8fvmpt 5745 1  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   _ici 8925    x. cmul 8928    - cmin 9223   -ucneg 9224    / cdiv 9609   2c2 9981   expce 12591   sincsin 12593
This theorem is referenced by:  tanval2  12661  resinval  12663  sinneg  12674  efival  12680  sinhval  12682  sinadd  12692  dvsincos  19732  sinper  20256  sineq0  20296  efeq1  20298  sinasin  20596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-sin 12599
  Copyright terms: Public domain W3C validator