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Theorem elreal2 8770
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 8763 . . 3  |-  RR  =  ( R.  X.  { 0R } )
21eleq2i 2360 . 2  |-  ( A  e.  RR  <->  A  e.  ( R.  X.  { 0R } ) )
3 xp1st 6165 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 1st `  A
)  e.  R. )
4 1st2nd2 6175 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
5 xp2nd 6166 . . . . . . 7  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  e.  { 0R } )
6 elsni 3677 . . . . . . 7  |-  ( ( 2nd `  A )  e.  { 0R }  ->  ( 2nd `  A
)  =  0R )
75, 6syl 15 . . . . . 6  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  =  0R )
87opeq2d 3819 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  A ) ,  0R >. )
94, 8eqtrd 2328 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  0R >. )
103, 9jca 518 . . 3  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
11 eleq1 2356 . . . . 5  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  <. ( 1st `  A ) ,  0R >.  e.  ( R.  X.  { 0R } ) ) )
12 0r 8718 . . . . . . . 8  |-  0R  e.  R.
1312elexi 2810 . . . . . . 7  |-  0R  e.  _V
1413snid 3680 . . . . . 6  |-  0R  e.  { 0R }
15 opelxp 4735 . . . . . 6  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  0R  e.  { 0R } ) )
1614, 15mpbiran2 885 . . . . 5  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. )
1711, 16syl6bb 252 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. ) )
1817biimparc 473 . . 3  |-  ( ( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. )  ->  A  e.  ( R.  X.  { 0R } ) )
1910, 18impbii 180 . 2  |-  ( A  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
202, 19bitri 240 1  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   R.cnr 8505   0Rc0r 8506   RRcr 8752
This theorem is referenced by:  ltresr2  8779  axrnegex  8800  axpre-sup  8807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-enr 8697  df-nr 8698  df-0r 8702  df-r 8763
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