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Theorem elreal2 9505
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 9498 . . 3  |-  RR  =  ( R.  X.  { 0R } )
21eleq2i 2545 . 2  |-  ( A  e.  RR  <->  A  e.  ( R.  X.  { 0R } ) )
3 xp1st 6811 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 1st `  A
)  e.  R. )
4 1st2nd2 6818 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. )
5 xp2nd 6812 . . . . . . 7  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  e.  { 0R } )
6 elsni 4052 . . . . . . 7  |-  ( ( 2nd `  A )  e.  { 0R }  ->  ( 2nd `  A
)  =  0R )
75, 6syl 16 . . . . . 6  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  =  0R )
87opeq2d 4220 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  A ) ,  0R >. )
94, 8eqtrd 2508 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  0R >. )
103, 9jca 532 . . 3  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
11 eleq1 2539 . . . . 5  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  <. ( 1st `  A ) ,  0R >.  e.  ( R.  X.  { 0R } ) ) )
12 0r 9453 . . . . . . . 8  |-  0R  e.  R.
1312elexi 3123 . . . . . . 7  |-  0R  e.  _V
1413snid 4055 . . . . . 6  |-  0R  e.  { 0R }
15 opelxp 5028 . . . . . 6  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  0R  e.  { 0R } ) )
1614, 15mpbiran2 917 . . . . 5  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. )
1711, 16syl6bb 261 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. ) )
1817biimparc 487 . . 3  |-  ( ( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. )  ->  A  e.  ( R.  X.  { 0R } ) )
1910, 18impbii 188 . 2  |-  ( A  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
202, 19bitri 249 1  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027   <.cop 4033    X. cxp 4997   ` cfv 5586   1stc1st 6779   2ndc2nd 6780   R.cnr 9239   0Rc0r 9240   RRcr 9487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-1p 9356  df-enr 9429  df-nr 9430  df-0r 9434  df-r 9498
This theorem is referenced by:  ltresr2  9514  axrnegex  9535  axpre-sup  9542
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