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Theorem archpr 6799
Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer  x is embedded into the reals as described at nnprlu 6709. (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
archpr  |-  ( A  e.  P.  ->  E. x  e.  N.  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
Distinct variable group:    A, l, u, x

Proof of Theorem archpr
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6631 . . 3  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 6634 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. z  e.  Q.  z  e.  ( 2nd `  A ) )
31, 2syl 14 . 2  |-  ( A  e.  P.  ->  E. z  e.  Q.  z  e.  ( 2nd `  A ) )
4 archnqq 6573 . . . 4  |-  ( z  e.  Q.  ->  E. x  e.  N.  z  <Q  [ <. x ,  1o >. ]  ~Q  )
54ad2antrl 467 . . 3  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  E. x  e.  N.  z  <Q  [ <. x ,  1o >. ]  ~Q  )
6 simprl 491 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  z  e.  Q. )
76ad2antrr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  z  e.  Q. )
8 simprr 492 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  z  e.  ( 2nd `  A ) )
98ad2antrr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  z  e.  ( 2nd `  A ) )
10 simpr 107 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  z  <Q  [ <. x ,  1o >. ]  ~Q  )
11 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
12 breq1 3795 . . . . . . . . 9  |-  ( l  =  z  ->  (
l  <Q  [ <. x ,  1o >. ]  ~Q  <->  z  <Q  [
<. x ,  1o >. ]  ~Q  ) )
13 ltnqex 6705 . . . . . . . . . 10  |-  { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  }  e.  _V
14 gtnqex 6706 . . . . . . . . . 10  |-  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u }  e.  _V
1513, 14op1st 5801 . . . . . . . . 9  |-  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )  =  { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  }
1611, 12, 15elab2 2713 . . . . . . . 8  |-  ( z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )  <->  z  <Q  [ <. x ,  1o >. ]  ~Q  )
1710, 16sylibr 141 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) )
18 eleq1 2116 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  ( 2nd `  A )  <->  z  e.  ( 2nd `  A ) ) )
19 eleq1 2116 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )  <->  z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) )
2018, 19anbi12d 450 . . . . . . . 8  |-  ( w  =  z  ->  (
( w  e.  ( 2nd `  A )  /\  w  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) )  <->  ( z  e.  ( 2nd `  A
)  /\  z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) ) )
2120rspcev 2673 . . . . . . 7  |-  ( ( z  e.  Q.  /\  ( z  e.  ( 2nd `  A )  /\  z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) )  ->  E. w  e.  Q.  ( w  e.  ( 2nd `  A )  /\  w  e.  ( 1st ` 
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) )
227, 9, 17, 21syl12anc 1144 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  E. w  e.  Q.  ( w  e.  ( 2nd `  A )  /\  w  e.  ( 1st ` 
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) )
23 simplll 493 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  A  e.  P. )
24 nnprlu 6709 . . . . . . . 8  |-  ( x  e.  N.  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
2524ad2antlr 466 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
26 ltdfpr 6662 . . . . . . 7  |-  ( ( A  e.  P.  /\  <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )  -> 
( A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. 
<->  E. w  e.  Q.  ( w  e.  ( 2nd `  A )  /\  w  e.  ( 1st ` 
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) ) )
2723, 25, 26syl2anc 397 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  ( A  <P  <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  <->  E. w  e.  Q.  ( w  e.  ( 2nd `  A )  /\  w  e.  ( 1st ` 
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) ) ) )
2822, 27mpbird 160 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
2928ex 112 . . . 4  |-  ( ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  /\  x  e.  N. )  ->  ( z  <Q  [ <. x ,  1o >. ]  ~Q  ->  A  <P 
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) )
3029reximdva 2438 . . 3  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  ( E. x  e.  N.  z  <Q  [ <. x ,  1o >. ]  ~Q  ->  E. x  e.  N.  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. ) )
315, 30mpd 13 . 2  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  E. x  e.  N.  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
323, 31rexlimddv 2454 1  |-  ( A  e.  P.  ->  E. x  e.  N.  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
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    <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:  archsr  6924
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