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Theorem archrecpr 7213
Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
Assertion
Ref Expression
archrecpr  |-  ( A  e.  P.  ->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A )
Distinct variable groups:    A, j    j,
l, u
Allowed substitution hints:    A( u, l)

Proof of Theorem archrecpr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prop 7024 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prml 7026 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
31, 2syl 14 . . 3  |-  ( A  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
4 archrecnq 7212 . . . . 5  |-  ( x  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
54ad2antrl 474 . . . 4  |-  ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
61ad2antrr 472 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
7 simplrr 503 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  x  e.  ( 1st `  A ) )
8 prcdnql 7033 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  -> 
( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
) ) )
96, 7, 8syl2anc 403 . . . . 5  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  <Q  x  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
) ) )
109reximdva 2475 . . . 4  |-  ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  ->  ( E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A ) ) )
115, 10mpd 13 . . 3  |-  ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A ) )
123, 11rexlimddv 2493 . 2  |-  ( A  e.  P.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
) )
13 nnnq 6971 . . . . . 6  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
14 recclnq 6941 . . . . . 6  |-  ( [
<. j ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
1513, 14syl 14 . . . . 5  |-  ( j  e.  N.  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
1615adantl 271 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
17 simpl 107 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  A  e.  P. )
18 nqprl 7100 . . . 4  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  A  e.  P. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
)  <->  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A ) )
1916, 17, 18syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
)  <->  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A ) )
2019rexbidva 2377 . 2  |-  ( A  e.  P.  ->  ( E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A )  <->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A ) )
2112, 20mpbid 145 1  |-  ( A  e.  P.  ->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3447   class class class wbr 3843   ` cfv 5010   1stc1st 5901   2ndc2nd 5902   1oc1o 6166   [cec 6280   N.cnpi 6821    ~Q ceq 6828   Q.cnq 6829   *Qcrq 6833    <Q cltq 6834   P.cnp 6840    <P cltp 6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-rq 6901  df-ltnqqs 6902  df-inp 7015  df-iltp 7019
This theorem is referenced by:  caucvgprprlemlim  7260
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