ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  creui Unicode version

Theorem creui 8876
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 7916 . 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 2172 . . . . . . . . . 10  |-  ( ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y
) )  <->  ( x  +  ( _i  x.  y ) )  =  ( z  +  ( _i  x.  w ) ) )
4 cru 8521 . . . . . . . . . . 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 566 . . . . . . . 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 2467 . . . . . . 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 2860 . . . . . . . 8  |-  ( z  e.  RR  ->  ( E. x  e.  RR  ( x  =  z  /\  y  =  w
)  <->  y  =  w ) )
1110ad2antrr 485 . . . . . . 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 2543 . . . . 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 2921 . . . . 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 2177 . . . . . 6  |-  ( A  =  ( z  +  ( _i  x.  w
) )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  <->  ( z  +  ( _i  x.  w ) )  =  ( x  +  ( _i  x.  y ) ) ) )
1716rexbidv 2471 . . . . 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 2653 . . . 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 2593 . 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 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   E!wreu 2450  (class class class)co 5853   CCcc 7772   RRcr 7773   _ici 7776    + caddc 7777    x. cmul 7779
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093  df-reap 8494
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator