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

Definition df-sin 12351
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 12345 . 2  class  sin
2 vx . . 3  set  x
3 cc 8735 . . 3  class  CC
4 ci 8739 . . . . . . 7  class  _i
52cv 1622 . . . . . . 7  class  x
6 cmul 8742 . . . . . . 7  class  x.
74, 5, 6co 5858 . . . . . 6  class  ( _i  x.  x )
8 ce 12343 . . . . . 6  class  exp
97, 8cfv 5255 . . . . 5  class  ( exp `  ( _i  x.  x
) )
104cneg 9038 . . . . . . 7  class  -u _i
1110, 5, 6co 5858 . . . . . 6  class  ( -u _i  x.  x )
1211, 8cfv 5255 . . . . 5  class  ( exp `  ( -u _i  x.  x ) )
13 cmin 9037 . . . . 5  class  -
149, 12, 13co 5858 . . . 4  class  ( ( exp `  ( _i  x.  x ) )  -  ( exp `  ( -u _i  x.  x ) ) )
15 c2 9795 . . . . 5  class  2
1615, 4, 6co 5858 . . . 4  class  ( 2  x.  _i )
17 cdiv 9423 . . . 4  class  /
1814, 16, 17co 5858 . . 3  class  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) )
192, 3, 18cmpt 4077 . 2  class  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  -  ( exp `  ( -u _i  x.  x ) ) )  /  ( 2  x.  _i ) ) )
201, 19wceq 1623 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  12402  sinf  12404  dvsincos  19328  sincn  19820
  Copyright terms: Public domain W3C validator