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Definition df-sin 12442
Description: Define the sine function. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-sin  |-  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )

Detailed syntax breakdown of Definition df-sin
StepHypRef Expression
1 csin 12436 . 2  class  sin
2 vx . . 3  set  x
3 cc 8822 . . 3  class  CC
4 ci 8826 . . . . . . 7  class  _i
52cv 1641 . . . . . . 7  class  x
6 cmul 8829 . . . . . . 7  class  x.
74, 5, 6co 5942 . . . . . 6  class  ( _i  x.  x )
8 ce 12434 . . . . . 6  class  exp
97, 8cfv 5334 . . . . 5  class  ( exp `  ( _i  x.  x
) )
104cneg 9125 . . . . . . 7  class  -u _i
1110, 5, 6co 5942 . . . . . 6  class  ( -u _i  x.  x )
1211, 8cfv 5334 . . . . 5  class  ( exp `  ( -u _i  x.  x ) )
13 cmin 9124 . . . . 5  class  -
149, 12, 13co 5942 . . . 4  class  ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )
15 c2 9882 . . . . 5  class  2
1615, 4, 6co 5942 . . . 4  class  ( 2  x.  _i )
17 cdiv 9510 . . . 4  class  /
1814, 16, 17co 5942 . . 3  class  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) )
192, 3, 18cmpt 4156 . 2  class  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
201, 19wceq 1642 1  wff  sin  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
Colors of variables: wff set class
This definition is referenced by:  sinval  12493  sinf  12495  dvsincos  19426  sincn  19921
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