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Theorem elreal2 8941
Description: Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elreal2  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )

Proof of Theorem elreal2
StepHypRef Expression
1 df-r 8934 . . 3  |-  RR  =  ( R.  X.  { 0R } )
21eleq2i 2452 . 2  |-  ( A  e.  RR  <->  A  e.  ( R.  X.  { 0R } ) )
3 xp1st 6316 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 1st `  A
)  e.  R. )
4 1st2nd2 6326 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
5 xp2nd 6317 . . . . . . 7  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  e.  { 0R } )
6 elsni 3782 . . . . . . 7  |-  ( ( 2nd `  A )  e.  { 0R }  ->  ( 2nd `  A
)  =  0R )
75, 6syl 16 . . . . . 6  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  =  0R )
87opeq2d 3934 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  A ) ,  0R >. )
94, 8eqtrd 2420 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  0R >. )
103, 9jca 519 . . 3  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
11 eleq1 2448 . . . . 5  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  <. ( 1st `  A ) ,  0R >.  e.  ( R.  X.  { 0R } ) ) )
12 0r 8889 . . . . . . . 8  |-  0R  e.  R.
1312elexi 2909 . . . . . . 7  |-  0R  e.  _V
1413snid 3785 . . . . . 6  |-  0R  e.  { 0R }
15 opelxp 4849 . . . . . 6  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  0R  e.  { 0R } ) )
1614, 15mpbiran2 886 . . . . 5  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. )
1711, 16syl6bb 253 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. ) )
1817biimparc 474 . . 3  |-  ( ( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. )  ->  A  e.  ( R.  X.  { 0R } ) )
1910, 18impbii 181 . 2  |-  ( A  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
202, 19bitri 241 1  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3758   <.cop 3761    X. cxp 4817   ` cfv 5395   1stc1st 6287   2ndc2nd 6288   R.cnr 8676   0Rc0r 8677   RRcr 8923
This theorem is referenced by:  ltresr2  8950  axrnegex  8971  axpre-sup  8978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ec 6844  df-qs 6848  df-ni 8683  df-pli 8684  df-mi 8685  df-lti 8686  df-plpq 8719  df-mpq 8720  df-ltpq 8721  df-enq 8722  df-nq 8723  df-erq 8724  df-plq 8725  df-mq 8726  df-1nq 8727  df-rq 8728  df-ltnq 8729  df-np 8792  df-1p 8793  df-enr 8868  df-nr 8869  df-0r 8873  df-r 8934
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