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Theorem cosval 12644
Description: Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
cosval  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )

Proof of Theorem cosval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 6021 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5665 . . . 4  |-  ( x  =  A  ->  ( exp `  ( _i  x.  x ) )  =  ( exp `  (
_i  x.  A )
) )
3 oveq2 6021 . . . . 5  |-  ( x  =  A  ->  ( -u _i  x.  x )  =  ( -u _i  x.  A ) )
43fveq2d 5665 . . . 4  |-  ( x  =  A  ->  ( exp `  ( -u _i  x.  x ) )  =  ( exp `  ( -u _i  x.  A ) ) )
52, 4oveq12d 6031 . . 3  |-  ( x  =  A  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  =  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) ) )
65oveq1d 6028 . 2  |-  ( x  =  A  ->  (
( ( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
7 df-cos 12593 . 2  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
8 ovex 6038 . 2  |-  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  e. 
_V
96, 7, 8fvmpt 5738 1  |-  ( A  e.  CC  ->  ( cos `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914   _ici 8918    + caddc 8919    x. cmul 8921   -ucneg 9217    / cdiv 9602   2c2 9974   expce 12584   cosccos 12587
This theorem is referenced by:  tanval2  12654  tanval3  12655  recosval  12657  cosneg  12668  efival  12673  coshval  12676  cosadd  12686  cosper  20250  pige3  20285  cosargd  20363  asinsin  20592  cosasin  20604  cosatan  20621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-cos 12593
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