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Theorem creui 9737
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
Dummy variables  z  w are mutually distinct and distinct from all other variables.

Proof of Theorem creui
StepHypRef Expression
1 cnre 8830 . 2  |-  ( A  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  A  =  ( z  +  ( _i  x.  w ) ) )
2 simpr 449 . . . . 5  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  w  e.  RR )
3 eqcom 2287 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
4 cru 9734 . . . . . . . . . . 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 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 ) ) )
63, 5syl5bb 250 . . . . . . . . 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 784 . . . . . . . 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 2562 . . . . . . 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 230 . . . . . . . . 9  |-  ( x  =  z  ->  (
y  =  w  <->  y  =  w ) )
109ceqsrexv 2903 . . . . . . . 8  |-  ( z  e.  RR  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w
)  <->  y  =  w ) )
1110ad2antrr 708 . . . . . . 7  |-  ( ( ( z  e.  RR  /\  w  e.  RR )  /\  y  e.  RR )  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w )  <->  y  =  w ) )
128, 11bitrd 246 . . . . . 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 2628 . . . . 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 2958 . . . . 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 644 . . . 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 2291 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2566 . . . . 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 2726 . . . 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 215 . . 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 2674 . 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 17 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2545   E.wrex 2546   E!wreu 2547  (class class class)co 5820   CCcc 8731   RRcr 8732   _ici 8735    + caddc 8736    x. cmul 8738
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420
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