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

Theorem sinval 13402
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 6098 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5692 . . . 4  |-  ( x  =  A  ->  ( exp `  ( _i  x.  x ) )  =  ( exp `  (
_i  x.  A )
) )
3 oveq2 6098 . . . . 5  |-  ( x  =  A  ->  ( -u _i  x.  x )  =  ( -u _i  x.  A ) )
43fveq2d 5692 . . . 4  |-  ( x  =  A  ->  ( exp `  ( -u _i  x.  x ) )  =  ( exp `  ( -u _i  x.  A ) ) )
52, 4oveq12d 6108 . . 3  |-  ( x  =  A  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  =  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )
65oveq1d 6105 . 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 13351 . 2  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
8 ovex 6115 . 2  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  e. 
_V
96, 7, 8fvmpt 5771 1  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   CCcc 9276   _ici 9280    x. cmul 9283    - cmin 9591   -ucneg 9592    / cdiv 9989   2c2 10367   expce 13343   sincsin 13345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-sin 13351
This theorem is referenced by:  tanval2  13413  resinval  13415  sinneg  13426  efival  13432  sinhval  13434  sinadd  13444  dvsincos  21353  sinper  21886  sineq0  21926  efeq1  21928  sinasin  22227  sineq0ALT  31490
  Copyright terms: Public domain W3C validator