ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  archrecpr Unicode version

Theorem archrecpr 6820
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 6631 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prml 6633 . . . 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 6819 . . . . 5  |-  ( x  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
54ad2antrl 467 . . . 4  |-  ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
61ad2antrr 465 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
7 simplrr 496 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  x  e.  ( 1st `  A ) )
8 prcdnql 6640 . . . . . 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 397 . . . . 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 2438 . . . 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 2454 . 2  |-  ( A  e.  P.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
) )
13 nnnq 6578 . . . . . 6  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
14 recclnq 6548 . . . . . 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 266 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
17 simpl 106 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  A  e.  P. )
18 nqprl 6707 . . . 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 397 . . 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 2340 . 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 139 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 101    <-> wb 102    e. wcel 1409   {cab 2042   E.wrex 2324   <.cop 3406   class class class wbr 3792   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    ~Q ceq 6435   Q.cnq 6436   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  caucvgprprlemlim  6867
  Copyright terms: Public domain W3C validator