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Theorem cjth 10746
Description: The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
cjth  |-  ( A  e.  CC  ->  (
( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )

Proof of Theorem cjth
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cju 8832 . . . 4  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotasbc 5795 . . . 4  |-  ( 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 ) )  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )
31, 2syl 14 . . 3  |-  ( A  e.  CC  ->  [. ( iota_ x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
4 cjval 10745 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
54sbceq1d 2942 . . 3  |-  ( A  e.  CC  ->  ( [. ( * `  A
)  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <->  [. ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) ) )
63, 5mpbird 166 . 2  |-  ( A  e.  CC  ->  [. (
* `  A )  /  x ]. ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )
7 riotacl 5794 . . . . 5  |-  ( 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 )
81, 7syl 14 . . . 4  |-  ( A  e.  CC  ->  ( iota_ x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )  e.  CC )
94, 8eqeltrd 2234 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
10 oveq2 5832 . . . . . 6  |-  ( x  =  ( * `  A )  ->  ( A  +  x )  =  ( A  +  ( * `  A
) ) )
1110eleq1d 2226 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( A  +  x
)  e.  RR  <->  ( A  +  ( * `  A ) )  e.  RR ) )
12 oveq2 5832 . . . . . . 7  |-  ( x  =  ( * `  A )  ->  ( A  -  x )  =  ( A  -  ( * `  A
) ) )
1312oveq2d 5840 . . . . . 6  |-  ( x  =  ( * `  A )  ->  (
_i  x.  ( A  -  x ) )  =  ( _i  x.  ( A  -  ( * `  A ) ) ) )
1413eleq1d 2226 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( _i  x.  ( A  -  x )
)  e.  RR  <->  ( _i  x.  ( A  -  (
* `  A )
) )  e.  RR ) )
1511, 14anbi12d 465 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
1615sbcieg 2969 . . 3  |-  ( ( * `  A )  e.  CC  ->  ( [. ( * `  A
)  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
179, 16syl 14 . 2  |-  ( A  e.  CC  ->  ( [. ( * `  A
)  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
186, 17mpbid 146 1  |-  ( A  e.  CC  ->  (
( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E!wreu 2437   [.wsbc 2937   ` cfv 5170   iota_crio 5779  (class class class)co 5824   CCcc 7730   RRcr 7731   _ici 7734    + caddc 7735    x. cmul 7737    - cmin 8046   *ccj 10739
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-ltxr 7917  df-sub 8048  df-neg 8049  df-reap 8450  df-cj 10742
This theorem is referenced by:  recl  10753  crre  10757
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