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Theorem cosval 12679
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 6048 . . . . 5  |-  ( x  =  A  ->  (
_i  x.  x )  =  ( _i  x.  A ) )
21fveq2d 5691 . . . 4  |-  ( x  =  A  ->  ( exp `  ( _i  x.  x ) )  =  ( exp `  (
_i  x.  A )
) )
3 oveq2 6048 . . . . 5  |-  ( x  =  A  ->  ( -u _i  x.  x )  =  ( -u _i  x.  A ) )
43fveq2d 5691 . . . 4  |-  ( x  =  A  ->  ( exp `  ( -u _i  x.  x ) )  =  ( exp `  ( -u _i  x.  A ) ) )
52, 4oveq12d 6058 . . 3  |-  ( x  =  A  ->  (
( exp `  (
_i  x.  x )
)  +  ( exp `  ( -u _i  x.  x ) ) )  =  ( ( exp `  ( _i  x.  A
) )  +  ( exp `  ( -u _i  x.  A ) ) ) )
65oveq1d 6055 . 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 12628 . 2  |-  cos  =  ( x  e.  CC  |->  ( ( ( exp `  ( _i  x.  x
) )  +  ( exp `  ( -u _i  x.  x ) ) )  /  2 ) )
8 ovex 6065 . 2  |-  ( ( ( exp `  (
_i  x.  A )
)  +  ( exp `  ( -u _i  x.  A ) ) )  /  2 )  e. 
_V
96, 7, 8fvmpt 5765 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 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   _ici 8948    + caddc 8949    x. cmul 8951   -ucneg 9248    / cdiv 9633   2c2 10005   expce 12619   cosccos 12622
This theorem is referenced by:  tanval2  12689  tanval3  12690  recosval  12692  cosneg  12703  efival  12708  coshval  12711  cosadd  12721  cosper  20343  pige3  20378  cosargd  20456  asinsin  20685  cosasin  20697  cosatan  20714
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-cos 12628
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