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Theorem cjval 10617
Description: The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
cjval  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
Distinct variable group:    x, A

Proof of Theorem cjval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cju 8719 . . 3  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotacl 5744 . . 3  |-  ( E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  ->  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )  e.  CC )
31, 2syl 14 . 2  |-  ( A  e.  CC  ->  ( iota_ x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )  e.  CC )
4 oveq1 5781 . . . . . 6  |-  ( y  =  A  ->  (
y  +  x )  =  ( A  +  x ) )
54eleq1d 2208 . . . . 5  |-  ( y  =  A  ->  (
( y  +  x
)  e.  RR  <->  ( A  +  x )  e.  RR ) )
6 oveq1 5781 . . . . . . 7  |-  ( y  =  A  ->  (
y  -  x )  =  ( A  -  x ) )
76oveq2d 5790 . . . . . 6  |-  ( y  =  A  ->  (
_i  x.  ( y  -  x ) )  =  ( _i  x.  ( A  -  x )
) )
87eleq1d 2208 . . . . 5  |-  ( y  =  A  ->  (
( _i  x.  (
y  -  x ) )  e.  RR  <->  ( _i  x.  ( A  -  x
) )  e.  RR ) )
95, 8anbi12d 464 . . . 4  |-  ( y  =  A  ->  (
( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR )  <-> 
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) ) )
109riotabidv 5732 . . 3  |-  ( y  =  A  ->  ( iota_ x  e.  CC  (
( y  +  x
)  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) ) )
11 df-cj 10614 . . 3  |-  *  =  ( y  e.  CC  |->  ( iota_ x  e.  CC  ( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) ) )
1210, 11fvmptg 5497 . 2  |-  ( ( A  e.  CC  /\  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )  e.  CC )  ->  ( * `  A )  =  (
iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) ) )
133, 12mpdan 417 1  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   E!wreu 2418   ` cfv 5123   iota_crio 5729  (class class class)co 5774   CCcc 7618   RRcr 7619   _ici 7622    + caddc 7623    x. cmul 7625    - cmin 7933   *ccj 10611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-sub 7935  df-neg 7936  df-reap 8337  df-cj 10614
This theorem is referenced by:  cjth  10618  remim  10632
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