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Definition df-cos 12678
Description: Define the cosine function. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-cos  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )

Detailed syntax breakdown of Definition df-cos
StepHypRef Expression
1 ccos 12672 . 2  class  cos
2 vx . . 3  set  x
3 cc 8993 . . 3  class  CC
4 ci 8997 . . . . . . 7  class  _i
52cv 1652 . . . . . . 7  class  x
6 cmul 9000 . . . . . . 7  class  x.
74, 5, 6co 6084 . . . . . 6  class  ( _i  x.  x )
8 ce 12669 . . . . . 6  class  exp
97, 8cfv 5457 . . . . 5  class  ( exp `  ( _i  x.  x
) )
104cneg 9297 . . . . . . 7  class  -u _i
1110, 5, 6co 6084 . . . . . 6  class  ( -u _i  x.  x )
1211, 8cfv 5457 . . . . 5  class  ( exp `  ( -u _i  x.  x ) )
13 caddc 8998 . . . . 5  class  +
149, 12, 13co 6084 . . . 4  class  ( ( exp `  ( _i  x.  x ) )  +  ( exp `  ( -u _i  x.  x ) ) )
15 c2 10054 . . . 4  class  2
16 cdiv 9682 . . . 4  class  /
1714, 15, 16co 6084 . . 3  class  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 )
182, 3, 17cmpt 4269 . 2  class  ( x  e.  CC  |->  ( ( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
191, 18wceq 1653 1  wff  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
Colors of variables: wff set class
This definition is referenced by:  cosval  12729  cosf  12731  dvsincos  19870  coscn  20366
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