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Theorem creui 8837
Description: The imaginary 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
creui  |-  ( A  e.  CC  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Distinct variable group:    x, y, A

Proof of Theorem creui
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7877 . 2  |-  ( A  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w ) ) )
2 simpr 109 . . . . 5  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  w  e.  RR )
3 eqcom 2159 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
4 cru 8482 . . . . . . . . . . 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 ) ) )
54ancoms 266 . . . . . . . . . 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 ) ) )
63, 5syl5bb 191 . . . . . . . . 9  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <-> 
( x  =  z  /\  y  =  w ) ) )
76anass1rs 561 . . . . . . . 8  |-  ( ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  /\  x  e.  RR )  ->  (
( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  ( x  =  z  /\  y  =  w ) ) )
87rexbidva 2454 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  ->  ( E. x  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <->  E. x  e.  RR  ( x  =  z  /\  y  =  w
) ) )
9 biidd 171 . . . . . . . . 9  |-  ( x  =  z  ->  (
y  =  w  <->  y  =  w ) )
109ceqsrexv 2842 . . . . . . . 8  |-  ( z  e.  RR  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w
)  <->  y  =  w ) )
1110ad2antrr 480 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w )  <->  y  =  w ) )
128, 11bitrd 187 . . . . . 6  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  ->  ( E. x  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) )  <-> 
y  =  w ) )
1312ralrimiva 2530 . . . . 5  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  A. y  e.  RR  ( E. x  e.  RR  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  y  =  w ) )
14 reu6i 2903 . . . . 5  |-  ( ( w  e.  RR  /\  A. y  e.  RR  ( E. x  e.  RR  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) )  <->  y  =  w ) )  ->  E! y  e.  RR  E. x  e.  RR  (
z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) ) )
152, 13, 14syl2anc 409 . . . 4  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  E! y  e.  RR  E. x  e.  RR  (
z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) ) )
16 eqeq1 2164 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2458 . . . . 5  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) )  <->  E. x  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) ) ) )
1817reubidv 2640 . . . 4  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  <->  E! y  e.  RR  E. x  e.  RR  ( z  +  ( _i  x.  w
) )  =  ( x  +  ( _i  x.  y ) ) ) )
1915, 18syl5ibrcom 156 . . 3  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  ( A  =  ( z  +  ( _i  x.  w ) )  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) ) )
2019rexlimivv 2580 . 2  |-  ( E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w
) )  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
211, 20syl 14 1  |-  ( A  e.  CC  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   E!wreu 2437  (class class class)co 5827   CCcc 7733   RRcr 7734   _ici 7737    + caddc 7738    x. cmul 7740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-iota 5138  df-fun 5175  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-pnf 7917  df-mnf 7918  df-ltxr 7920  df-sub 8053  df-neg 8054  df-reap 8455
This theorem is referenced by: (None)
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