MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cjval Unicode version

Theorem cjval 11836
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 6029 . . . . 5  |-  ( y  =  A  ->  (
y  +  x )  =  ( A  +  x ) )
21eleq1d 2455 . . . 4  |-  ( y  =  A  ->  (
( y  +  x
)  e.  RR  <->  ( A  +  x )  e.  RR ) )
3 oveq1 6029 . . . . . 6  |-  ( y  =  A  ->  (
y  -  x )  =  ( A  -  x ) )
43oveq2d 6038 . . . . 5  |-  ( y  =  A  ->  (
_i  x.  ( y  -  x ) )  =  ( _i  x.  ( A  -  x )
) )
54eleq1d 2455 . . . 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 6489 . 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 11833 . 2  |-  *  =  ( y  e.  CC  |->  ( iota_ x  e.  CC ( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) ) )
9 riotaex 6491 . 2  |-  ( iota_ x  e.  CC ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )  e. 
_V
107, 8, 9fvmpt 5747 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 5396  (class class class)co 6022   iota_crio 6480   CCcc 8923   RRcr 8924   _ici 8927    + caddc 8928    x. cmul 8930    - cmin 9225   *ccj 11830
This theorem is referenced by:  cjth  11837  remim  11851
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 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-riota 6487  df-cj 11833
  Copyright terms: Public domain W3C validator