ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elreal2 Unicode version

Theorem elreal2 7661
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
Assertion
Ref Expression
elreal2  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )

Proof of Theorem elreal2
StepHypRef Expression
1 df-r 7653 . . 3  |-  RR  =  ( R.  X.  { 0R } )
21eleq2i 2207 . 2  |-  ( A  e.  RR  <->  A  e.  ( R.  X.  { 0R } ) )
3 xp1st 6070 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 1st `  A
)  e.  R. )
4 1st2nd2 6080 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
5 xp2nd 6071 . . . . . . 7  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  e.  { 0R } )
6 elsni 3549 . . . . . . 7  |-  ( ( 2nd `  A )  e.  { 0R }  ->  ( 2nd `  A
)  =  0R )
75, 6syl 14 . . . . . 6  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  =  0R )
87opeq2d 3719 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  A ) ,  0R >. )
94, 8eqtrd 2173 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  0R >. )
103, 9jca 304 . . 3  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
11 eleq1 2203 . . . . 5  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  <. ( 1st `  A ) ,  0R >.  e.  ( R.  X.  { 0R } ) ) )
12 0r 7581 . . . . . . . 8  |-  0R  e.  R.
1312elexi 2701 . . . . . . 7  |-  0R  e.  _V
1413snid 3562 . . . . . 6  |-  0R  e.  { 0R }
15 opelxp 4576 . . . . . 6  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  0R  e.  { 0R } ) )
1614, 15mpbiran2 926 . . . . 5  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. )
1711, 16syl6bb 195 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. ) )
1817biimparc 297 . . 3  |-  ( ( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. )  ->  A  e.  ( R.  X.  { 0R } ) )
1910, 18impbii 125 . 2  |-  ( A  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
202, 19bitri 183 1  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   {csn 3531   <.cop 3534    X. cxp 4544   ` cfv 5130   1stc1st 6043   2ndc2nd 6044   R.cnr 7128   0Rc0r 7129   RRcr 7642
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-inp 7297  df-i1p 7298  df-enr 7557  df-nr 7558  df-0r 7562  df-r 7653
This theorem is referenced by:  ltresr2  7671  axrnegex  7710  axpre-suploclemres  7732
  Copyright terms: Public domain W3C validator