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Theorem cjval 11899
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 6080 . . . . 5  |-  ( y  =  A  ->  (
y  +  x )  =  ( A  +  x ) )
21eleq1d 2501 . . . 4  |-  ( y  =  A  ->  (
( y  +  x
)  e.  RR  <->  ( A  +  x )  e.  RR ) )
3 oveq1 6080 . . . . . 6  |-  ( y  =  A  ->  (
y  -  x )  =  ( A  -  x ) )
43oveq2d 6089 . . . . 5  |-  ( y  =  A  ->  (
_i  x.  ( y  -  x ) )  =  ( _i  x.  ( A  -  x )
) )
54eleq1d 2501 . . . 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 6543 . 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 11896 . 2  |-  *  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) ) )
9 riotaex 6545 . 2  |-  ( iota_ x  e.  CC ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )  e. 
_V
107, 8, 9fvmpt 5798 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   iota_crio 6534   CCcc 8980   RRcr 8981   _ici 8984    + caddc 8985    x. cmul 8987    - cmin 9283   *ccj 11893
This theorem is referenced by:  cjth  11900  remim  11914
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-cj 11896
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