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Theorem cjth 12911
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 10542 . . . 4  |-  ( A  e.  CC  ->  E! x  e.  CC  (
( A  +  x
)  e.  RR  /\  ( _i  x.  ( A  -  x )
)  e.  RR ) )
2 riotasbc 6271 . . . 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 16 . . 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 12910 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  =  ( iota_ x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
5 dfsbcq 3338 . . . 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 16 . . 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 232 . 2  |-  ( A  e.  CC  ->  [. (
* `  A )  /  x ]. ( ( A  +  x )  e.  RR  /\  (
_i  x.  ( A  -  x ) )  e.  RR ) )
8 fvex 5881 . . 3  |-  ( * `
 A )  e. 
_V
9 oveq2 6302 . . . . 5  |-  ( x  =  ( * `  A )  ->  ( A  +  x )  =  ( A  +  ( * `  A
) ) )
109eleq1d 2536 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( A  +  x
)  e.  RR  <->  ( A  +  ( * `  A ) )  e.  RR ) )
11 oveq2 6302 . . . . . 6  |-  ( x  =  ( * `  A )  ->  ( A  -  x )  =  ( A  -  ( * `  A
) ) )
1211oveq2d 6310 . . . . 5  |-  ( x  =  ( * `  A )  ->  (
_i  x.  ( A  -  x ) )  =  ( _i  x.  ( A  -  ( * `  A ) ) ) )
1312eleq1d 2536 . . . 4  |-  ( x  =  ( * `  A )  ->  (
( _i  x.  ( A  -  x )
)  e.  RR  <->  ( _i  x.  ( A  -  (
* `  A )
) )  e.  RR ) )
1410, 13anbi12d 710 . . 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 3371 . 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 196 1  |-  ( A  e.  CC  ->  (
( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E!wreu 2819   [.wsbc 3336   ` cfv 5593   iota_crio 6254  (class class class)co 6294   CCcc 9500   RRcr 9501   _ici 9504    + caddc 9505    x. cmul 9507    - cmin 9815   *ccj 12904
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-cj 12907
This theorem is referenced by:  recl  12918  crre  12922
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