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Theorem sinval 13730
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 6302 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5875 . . . 4  |-  ( x  =  A  ->  ( exp `  ( _i  x.  x ) )  =  ( exp `  (
_i  x.  A )
) )
3 oveq2 6302 . . . . 5  |-  ( x  =  A  ->  ( -u _i  x.  x )  =  ( -u _i  x.  A ) )
43fveq2d 5875 . . . 4  |-  ( x  =  A  ->  ( exp `  ( -u _i  x.  x ) )  =  ( exp `  ( -u _i  x.  A ) ) )
52, 4oveq12d 6312 . . 3  |-  ( x  =  A  ->  (
( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  =  ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) ) )
65oveq1d 6309 . 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 13679 . 2  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
8 ovex 6319 . 2  |-  ( ( ( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) )  e. 
_V
96, 7, 8fvmpt 5956 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 1379    e. wcel 1767   ` cfv 5593  (class class class)co 6294   CCcc 9500   _ici 9504    x. cmul 9507    - cmin 9815   -ucneg 9816    / cdiv 10216   2c2 10595   expce 13671   sincsin 13673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-sin 13679
This theorem is referenced by:  tanval2  13741  resinval  13743  sinneg  13754  efival  13760  sinhval  13762  sinadd  13772  dvsincos  22227  sinper  22717  sineq0  22757  efeq1  22759  sinasin  23063  sineq0ALT  33110
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