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Theorem cjval 11827
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 oveq1 6020 . . . . 5  |-  ( y  =  A  ->  (
y  +  x )  =  ( A  +  x ) )
21eleq1d 2446 . . . 4  |-  ( y  =  A  ->  (
( y  +  x
)  e.  RR  <->  ( A  +  x )  e.  RR ) )
3 oveq1 6020 . . . . . 6  |-  ( y  =  A  ->  (
y  -  x )  =  ( A  -  x ) )
43oveq2d 6029 . . . . 5  |-  ( y  =  A  ->  (
_i  x.  ( y  -  x ) )  =  ( _i  x.  ( A  -  x )
) )
54eleq1d 2446 . . . 4  |-  ( y  =  A  ->  (
( _i  x.  (
y  -  x ) )  e.  RR  <->  ( _i  x.  ( A  -  x
) )  e.  RR ) )
62, 5anbi12d 692 . . 3  |-  ( y  =  A  ->  (
( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR )  <-> 
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) ) )
76riotabidv 6480 . 2  |-  ( 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 ) ) )
8 df-cj 11824 . 2  |-  *  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) ) )
9 riotaex 6482 . 2  |-  ( iota_ x  e.  CC ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )  e. 
_V
107, 8, 9fvmpt 5738 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 359    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   iota_crio 6471   CCcc 8914   RRcr 8915   _ici 8918    + caddc 8919    x. cmul 8921    - cmin 9216   *ccj 11821
This theorem is referenced by:  cjth  11828  remim  11842
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-reu 2649  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-riota 6478  df-cj 11824
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