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Theorem archpr 7415
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 7325. (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 7247 . . 3  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 7250 . . 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 7189 . . . 4  |-  ( z  e.  Q.  ->  E. x  e.  N.  z  <Q  [ <. x ,  1o >. ]  ~Q  )
54ad2antrl 479 . . 3  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  E. x  e.  N.  z  <Q  [ <. x ,  1o >. ]  ~Q  )
6 simprl 503 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  z  e.  Q. )
76ad2antrr 477 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  z  e.  Q. )
8 simprr 504 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A ) ) )  ->  z  e.  ( 2nd `  A ) )
98ad2antrr 477 . . . . . . 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 109 . . . . . . . 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 2661 . . . . . . . . 9  |-  z  e. 
_V
12 breq1 3900 . . . . . . . . 9  |-  ( l  =  z  ->  (
l  <Q  [ <. x ,  1o >. ]  ~Q  <->  z  <Q  [
<. x ,  1o >. ]  ~Q  ) )
13 ltnqex 7321 . . . . . . . . . 10  |-  { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  }  e.  _V
14 gtnqex 7322 . . . . . . . . . 10  |-  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u }  e.  _V
1513, 14op1st 6010 . . . . . . . . 9  |-  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )  =  { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  }
1611, 12, 15elab2 2803 . . . . . . . 8  |-  ( z  e.  ( 1st `  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )  <->  z  <Q  [ <. x ,  1o >. ]  ~Q  )
1710, 16sylibr 133 . . . . . . 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 2178 . . . . . . . . 9  |-  ( w  =  z  ->  (
w  e.  ( 2nd `  A )  <->  z  e.  ( 2nd `  A ) ) )
19 eleq1 2178 . . . . . . . . 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 462 . . . . . . . 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 2761 . . . . . . 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 1197 . . . . . 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 505 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  ( z  e.  Q.  /\  z  e.  ( 2nd `  A
) ) )  /\  x  e.  N. )  /\  z  <Q  [ <. x ,  1o >. ]  ~Q  )  ->  A  e.  P. )
24 nnprlu 7325 . . . . . . . 8  |-  ( x  e.  N.  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
2524ad2antlr 478 . . . . . . 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 7278 . . . . . . 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 406 . . . . . 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 166 . . . . 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 114 . . . 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 2509 . . 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 2529 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 103    <-> wb 104    e. wcel 1463   {cab 2101   E.wrex 2392   <.cop 3498   class class class wbr 3897   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   1oc1o 6272   [cec 6393   N.cnpi 7044    ~Q ceq 7051   Q.cnq 7052    <Q cltq 7057   P.cnp 7063    <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-iltp 7242
This theorem is referenced by:  archsr  7554
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