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Theorem cjval 12894
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 6289 . . . . 5  |-  ( y  =  A  ->  (
y  +  x )  =  ( A  +  x ) )
21eleq1d 2536 . . . 4  |-  ( y  =  A  ->  (
( y  +  x
)  e.  RR  <->  ( A  +  x )  e.  RR ) )
3 oveq1 6289 . . . . . 6  |-  ( y  =  A  ->  (
y  -  x )  =  ( A  -  x ) )
43oveq2d 6298 . . . . 5  |-  ( y  =  A  ->  (
_i  x.  ( y  -  x ) )  =  ( _i  x.  ( A  -  x )
) )
54eleq1d 2536 . . . 4  |-  ( y  =  A  ->  (
( _i  x.  (
y  -  x ) )  e.  RR  <->  ( _i  x.  ( A  -  x
) )  e.  RR ) )
62, 5anbi12d 710 . . 3  |-  ( y  =  A  ->  (
( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR )  <-> 
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) ) )
76riotabidv 6245 . 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 12891 . 2  |-  *  =  ( y  e.  CC  |->  ( iota_ x  e.  CC  ( ( y  +  x )  e.  RR  /\  ( _i  x.  (
y  -  x ) )  e.  RR ) ) )
9 riotaex 6247 . 2  |-  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )  e. 
_V
107, 8, 9fvmpt 5948 1  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586   iota_crio 6242  (class class class)co 6282   CCcc 9486   RRcr 9487   _ici 9490    + caddc 9491    x. cmul 9493    - cmin 9801   *ccj 12888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-riota 6243  df-ov 6285  df-cj 12891
This theorem is referenced by:  cjth  12895  remim  12909
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