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Theorem elreal2 7828
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 7820 . . 3  |-  RR  =  ( R.  X.  { 0R } )
21eleq2i 2244 . 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 3610 . . . . . . 7  |-  ( ( 2nd `  A )  e.  { 0R }  ->  ( 2nd `  A
)  =  0R )
75, 6syl 14 . . . . . 6  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( 2nd `  A
)  =  0R )
87opeq2d 3785 . . . . 5  |-  ( A  e.  ( R.  X.  { 0R } )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. ( 1st `  A ) ,  0R >. )
94, 8eqtrd 2210 . . . 4  |-  ( A  e.  ( R.  X.  { 0R } )  ->  A  =  <. ( 1st `  A ) ,  0R >. )
103, 9jca 306 . . 3  |-  ( A  e.  ( R.  X.  { 0R } )  -> 
( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
11 eleq1 2240 . . . . 5  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  <. ( 1st `  A ) ,  0R >.  e.  ( R.  X.  { 0R } ) ) )
12 0r 7748 . . . . . . . 8  |-  0R  e.  R.
1312elexi 2749 . . . . . . 7  |-  0R  e.  _V
1413snid 3623 . . . . . 6  |-  0R  e.  { 0R }
15 opelxp 4656 . . . . . 6  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  0R  e.  { 0R } ) )
1614, 15mpbiran2 941 . . . . 5  |-  ( <.
( 1st `  A
) ,  0R >.  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. )
1711, 16bitrdi 196 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  0R >.  -> 
( A  e.  ( R.  X.  { 0R } )  <->  ( 1st `  A )  e.  R. ) )
1817biimparc 299 . . 3  |-  ( ( ( 1st `  A
)  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. )  ->  A  e.  ( R.  X.  { 0R } ) )
1910, 18impbii 126 . 2  |-  ( A  e.  ( R.  X.  { 0R } )  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
202, 19bitri 184 1  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {csn 3592   <.cop 3595    X. cxp 4624   ` cfv 5216   1stc1st 6138   2ndc2nd 6139   R.cnr 7295   0Rc0r 7296   RRcr 7809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-i1p 7465  df-enr 7724  df-nr 7725  df-0r 7729  df-r 7820
This theorem is referenced by:  ltresr2  7838  axrnegex  7877  axpre-suploclemres  7899
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