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Theorem cjth 9860
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 8094 . . . 4  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotasbc 5508 . . . 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 9859 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
54sbceq1d 2821 . . 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 165 . 2  |-  ( A  e.  CC  ->  [. (
* `  A )  /  x ]. ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )
7 riotacl 5507 . . . . 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 2156 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
10 oveq2 5545 . . . . . 6  |-  ( x  =  ( * `  A )  ->  ( A  +  x )  =  ( A  +  ( * `  A
) ) )
1110eleq1d 2148 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( A  +  x
)  e.  RR  <->  ( A  +  ( * `  A ) )  e.  RR ) )
12 oveq2 5545 . . . . . . 7  |-  ( x  =  ( * `  A )  ->  ( A  -  x )  =  ( A  -  ( * `  A
) ) )
1312oveq2d 5553 . . . . . 6  |-  ( x  =  ( * `  A )  ->  (
_i  x.  ( A  -  x ) )  =  ( _i  x.  ( A  -  ( * `  A ) ) ) )
1413eleq1d 2148 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
( _i  x.  ( A  -  x )
)  e.  RR  <->  ( _i  x.  ( A  -  (
* `  A )
) )  e.  RR ) )
1511, 14anbi12d 457 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
1615sbcieg 2847 . . 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 145 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   E!wreu 2351   [.wsbc 2816   ` cfv 4926   iota_crio 5492  (class class class)co 5537   CCcc 7030   RRcr 7031   _ici 7034    + caddc 7035    x. cmul 7037    - cmin 7335   *ccj 9853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-ltxr 7209  df-sub 7337  df-neg 7338  df-reap 7731  df-cj 9856
This theorem is referenced by:  recl  9867  crre  9871
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