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Theorem cjth 11588
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 9742 . . . 4  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotasbc 6320 . . . 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 15 . . 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 11587 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
5 dfsbcq 2993 . . . 4  |-  ( ( * `  A )  =  ( iota_ x  e.  CC ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )  -> 
( [. ( * `  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 ) ) )
64, 5syl 15 . . 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 ) ) )
73, 6mpbird 223 . 2  |-  ( A  e.  CC  ->  [. (
* `  A )  /  x ]. ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )
8 fvex 5539 . . 3  |-  ( * `
 A )  e. 
_V
9 oveq2 5866 . . . . 5  |-  ( x  =  ( * `  A )  ->  ( A  +  x )  =  ( A  +  ( * `  A
) ) )
109eleq1d 2349 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( A  +  x
)  e.  RR  <->  ( A  +  ( * `  A ) )  e.  RR ) )
11 oveq2 5866 . . . . . 6  |-  ( x  =  ( * `  A )  ->  ( A  -  x )  =  ( A  -  ( * `  A
) ) )
1211oveq2d 5874 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
_i  x.  ( A  -  x ) )  =  ( _i  x.  ( A  -  ( * `  A ) ) ) )
1312eleq1d 2349 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( _i  x.  ( A  -  x )
)  e.  RR  <->  ( _i  x.  ( A  -  (
* `  A )
) )  e.  RR ) )
1410, 13anbi12d 691 . . 3  |-  ( x  =  ( * `  A )  ->  (
( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) ) )
158, 14sbcie 3025 . 2  |-  ( [. ( * `  A
)  /  x ]. ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR )  <-> 
( ( A  +  ( * `  A
) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
167, 15sylib 188 1  |-  ( A  e.  CC  ->  (
( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!wreu 2545   [.wsbc 2991   ` cfv 5255  (class class class)co 5858   iota_crio 6297   CCcc 8735   RRcr 8736   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   *ccj 11581
This theorem is referenced by:  recl  11595  crre  11599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-cj 11584
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