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Theorem creur 10536
Description: The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
creur  |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Distinct variable group:    x, y, A

Proof of Theorem creur
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 9595 . 2  |-  ( A  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w ) ) )
2 cru 10534 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) )  =  ( z  +  ( _i  x.  w ) )  <-> 
( x  =  z  /\  y  =  w ) ) )
32ancoms 453 . . . . . . . . . 10  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) )  =  ( z  +  ( _i  x.  w ) )  <-> 
( x  =  z  /\  y  =  w ) ) )
4 eqcom 2452 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
5 ancom 450 . . . . . . . . . 10  |-  ( ( y  =  w  /\  x  =  z )  <->  ( x  =  z  /\  y  =  w )
)
63, 4, 53bitr4g 288 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <-> 
( y  =  w  /\  x  =  z ) ) )
76anassrs 648 . . . . . . . 8  |-  ( ( ( ( z  e.  RR  /\  w  e.  RR )  /\  x  e.  RR )  /\  y  e.  RR )  ->  (
( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  ( y  =  w  /\  x  =  z ) ) )
87rexbidva 2951 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  x  e.  RR )  ->  ( E. y  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <->  E. y  e.  RR  ( y  =  w  /\  x  =  z ) ) )
9 biidd 237 . . . . . . . . 9  |-  ( y  =  w  ->  (
x  =  z  <->  x  =  z ) )
109ceqsrexv 3219 . . . . . . . 8  |-  ( w  e.  RR  ->  ( E. y  e.  RR  ( y  =  w  /\  x  =  z )  <->  x  =  z
) )
1110ad2antlr 726 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  x  e.  RR )  ->  ( E. y  e.  RR  ( y  =  w  /\  x  =  z )  <->  x  =  z ) )
128, 11bitrd 253 . . . . . 6  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  x  e.  RR )  ->  ( E. y  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <-> 
x  =  z ) )
1312ralrimiva 2857 . . . . 5  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  A. x  e.  RR  ( E. y  e.  RR  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  x  =  z ) )
14 reu6i 3276 . . . . 5  |-  ( ( z  e.  RR  /\  A. x  e.  RR  ( E. y  e.  RR  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  x  =  z ) )  ->  E! x  e.  RR  E. y  e.  RR  (
z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) ) )
1513, 14syldan 470 . . . 4  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  E! x  e.  RR  E. y  e.  RR  (
z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) ) )
16 eqeq1 2447 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2954 . . . . 5  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )  <->  E. y  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) ) ) )
1817reubidv 3028 . . . 4  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  <->  E! x  e.  RR  E. y  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) ) ) )
1915, 18syl5ibrcom 222 . . 3  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  ( A  =  ( z  +  ( _i  x.  w ) )  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) ) )
2019rexlimivv 2940 . 2  |-  ( E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w
) )  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
211, 20syl 16 1  |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   E!wreu 2795  (class class class)co 6281   CCcc 9493   RRcr 9494   _ici 9497    + caddc 9498    x. cmul 9500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213
This theorem is referenced by: (None)
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