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Theorem creui 8104
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 7177 . 2  |-  ( A  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w ) ) )
2 simpr 108 . . . . 5  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  w  e.  RR )
3 eqcom 2084 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
4 cru 7769 . . . . . . . . . . 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 264 . . . . . . . . . 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 190 . . . . . . . . 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 536 . . . . . . . 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 2366 . . . . . . 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 170 . . . . . . . . 9  |-  ( x  =  z  ->  (
y  =  w  <->  y  =  w ) )
109ceqsrexv 2726 . . . . . . . 8  |-  ( z  e.  RR  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w
)  <->  y  =  w ) )
1110ad2antrr 472 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w )  <->  y  =  w ) )
128, 11bitrd 186 . . . . . 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 2435 . . . . 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 2784 . . . . 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 403 . . . 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 2088 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2370 . . . . 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 2538 . . . 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 155 . . 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 2483 . 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   E!wreu 2351  (class class class)co 5543   CCcc 7041   RRcr 7042   _ici 7045    + caddc 7046    x. cmul 7048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349  df-reap 7742
This theorem is referenced by: (None)
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