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Theorem creur 9620
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
StepHypRef Expression
1 cnre 8716 . 2  |-  ( A  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w ) ) )
2 cru 9618 . . . . . . . . . . 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 441 . . . . . . . . . 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 2255 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
5 ancom 439 . . . . . . . . . 10  |-  ( ( y  =  w  /\  x  =  z )  <->  ( x  =  z  /\  y  =  w )
)
63, 4, 53bitr4g 281 . . . . . . . . 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 632 . . . . . . . 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 2524 . . . . . . 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 230 . . . . . . . . 9  |-  ( y  =  w  ->  (
x  =  z  <->  x  =  z ) )
109ceqsrexv 2838 . . . . . . . 8  |-  ( w  e.  RR  ->  ( E. y  e.  RR  ( y  =  w  /\  x  =  z )  <->  x  =  z
) )
1110ad2antlr 710 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  x  e.  RR )  ->  ( E. y  e.  RR  ( y  =  w  /\  x  =  z )  <->  x  =  z ) )
128, 11bitrd 246 . . . . . 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 2588 . . . . 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 2895 . . . . 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 458 . . . 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 2259 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2528 . . . . 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 2683 . . . 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 215 . . 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 2634 . 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 17 1  |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   E.wrex 2510   E!wreu 2511  (class class class)co 5710   CCcc 8615   RRcr 8616   _ici 8619    + caddc 8620    x. cmul 8622
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304
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