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Theorem reval 11911
Description: The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
reval  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( ( A  +  ( * `  A ) )  / 
2 ) )

Proof of Theorem reval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( x  =  A  ->  (
* `  x )  =  ( * `  A ) )
2 oveq12 6090 . . . 4  |-  ( ( x  =  A  /\  ( * `  x
)  =  ( * `
 A ) )  ->  ( x  +  ( * `  x
) )  =  ( A  +  ( * `
 A ) ) )
31, 2mpdan 650 . . 3  |-  ( x  =  A  ->  (
x  +  ( * `
 x ) )  =  ( A  +  ( * `  A
) ) )
43oveq1d 6096 . 2  |-  ( x  =  A  ->  (
( x  +  ( * `  x ) )  /  2 )  =  ( ( A  +  ( * `  A ) )  / 
2 ) )
5 df-re 11905 . 2  |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `  x
) )  /  2
) )
6 ovex 6106 . 2  |-  ( ( A  +  ( * `
 A ) )  /  2 )  e. 
_V
74, 5, 6fvmpt 5806 1  |-  ( A  e.  CC  ->  (
Re `  A )  =  ( ( A  +  ( * `  A ) )  / 
2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993    / cdiv 9677   2c2 10049   *ccj 11901   Recre 11902
This theorem is referenced by:  recl  11915  ref  11917  crre  11919  addcj  11953  sqreulem  12163  recosval  12737  dvmptre  19855  cosargd  20503  lnopunilem1  23513  cnre2csqima  24309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-re 11905
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