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Theorem archrecpr 7465
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 7276 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prml 7278 . . . 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 7464 . . . . 5  |-  ( x  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
54ad2antrl 481 . . . 4  |-  ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  x )
61ad2antrr 479 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
7 simplrr 525 . . . . . 6  |-  ( ( ( A  e.  P.  /\  ( x  e.  Q.  /\  x  e.  ( 1st `  A ) ) )  /\  j  e.  N. )  ->  x  e.  ( 1st `  A ) )
8 prcdnql 7285 . . . . . 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 408 . . . . 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 2532 . . . 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 2552 . 2  |-  ( A  e.  P.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  ( 1st `  A
) )
13 nnnq 7223 . . . . . 6  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
14 recclnq 7193 . . . . . 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 275 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
17 simpl 108 . . . 4  |-  ( ( A  e.  P.  /\  j  e.  N. )  ->  A  e.  P. )
18 nqprl 7352 . . . 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 408 . . 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 2432 . 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 146 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 103    <-> wb 104    e. wcel 1480   {cab 2123   E.wrex 2415   <.cop 3525   class class class wbr 3924   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073    ~Q ceq 7080   Q.cnq 7081   *Qcrq 7085    <Q cltq 7086   P.cnp 7092    <P cltp 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271
This theorem is referenced by:  caucvgprprlemlim  7512
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