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Theorem cjth 10850
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 8916 . . . 4  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotasbc 5845 . . . 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 10849 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
54sbceq1d 2967 . . 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 167 . 2  |-  ( A  e.  CC  ->  [. (
* `  A )  /  x ]. ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )
7 riotacl 5844 . . . . 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 2254 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
10 oveq2 5882 . . . . . 6  |-  ( x  =  ( * `  A )  ->  ( A  +  x )  =  ( A  +  ( * `  A
) ) )
1110eleq1d 2246 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( A  +  x
)  e.  RR  <->  ( A  +  ( * `  A ) )  e.  RR ) )
12 oveq2 5882 . . . . . . 7  |-  ( x  =  ( * `  A )  ->  ( A  -  x )  =  ( A  -  ( * `  A
) ) )
1312oveq2d 5890 . . . . . 6  |-  ( x  =  ( * `  A )  ->  (
_i  x.  ( A  -  x ) )  =  ( _i  x.  ( A  -  ( * `  A ) ) ) )
1413eleq1d 2246 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( _i  x.  ( A  -  x )
)  e.  RR  <->  ( _i  x.  ( A  -  (
* `  A )
) )  e.  RR ) )
1511, 14anbi12d 473 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
1615sbcieg 2995 . . 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 147 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   E!wreu 2457   [.wsbc 2962   ` cfv 5216   iota_crio 5829  (class class class)co 5874   CCcc 7808   RRcr 7809   _ici 7812    + caddc 7813    x. cmul 7815    - cmin 8126   *ccj 10843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-sub 8128  df-neg 8129  df-reap 8530  df-cj 10846
This theorem is referenced by:  recl  10857  crre  10861
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