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Theorem archsr 7554
Description: For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression  [ <. ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  },  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R is the embedding of the positive integer  x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
Assertion
Ref Expression
archsr  |-  ( A  e.  R.  ->  E. x  e.  N.  A  <R  [ <. (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
Distinct variable group:    A, l, u, x

Proof of Theorem archsr
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7499 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3900 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  <->  A  <R  [
<. ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
32rexbidv 2413 . 2  |-  ( [
<. z ,  w >. ]  ~R  =  A  -> 
( E. x  e. 
N.  [ <. z ,  w >. ]  ~R  <R  [
<. ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  <->  E. x  e.  N.  A  <R  [ <. (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
)
4 1pr 7326 . . . . . . 7  |-  1P  e.  P.
5 addclpr 7309 . . . . . . 7  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  +P.  1P )  e.  P. )
64, 5mpan2 419 . . . . . 6  |-  ( z  e.  P.  ->  (
z  +P.  1P )  e.  P. )
7 archpr 7415 . . . . . 6  |-  ( ( z  +P.  1P )  e.  P.  ->  E. x  e.  N.  ( z  +P. 
1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
86, 7syl 14 . . . . 5  |-  ( z  e.  P.  ->  E. x  e.  N.  ( z  +P. 
1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
98adantr 272 . . . 4  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  E. x  e.  N.  ( z  +P.  1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >. )
10 nnprlu 7325 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
1110adantl 273 . . . . . . . . 9  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
12 addclpr 7309 . . . . . . . . 9  |-  ( (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P.  /\  1P  e.  P. )  ->  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
1311, 4, 12sylancl 407 . . . . . . . 8  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e.  P. )
14 simplr 502 . . . . . . . 8  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  w  e.  P. )
15 ltaddpr 7369 . . . . . . . 8  |-  ( ( ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P.  /\  w  e.  P. )  ->  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  <P 
( ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  +P.  w ) )
1613, 14, 15syl2anc 406 . . . . . . 7  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  <P  ( ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  +P.  w ) )
17 addcomprg 7350 . . . . . . . 8  |-  ( ( w  e.  P.  /\  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )  ->  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )  =  ( ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  +P.  w ) )
1814, 13, 17syl2anc 406 . . . . . . 7  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )  =  ( ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  +P.  w ) )
1916, 18breqtrrd 3924 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  <P  ( w  +P.  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )
20 ltaddpr 7369 . . . . . . . 8  |-  ( (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P.  /\  1P  e.  P. )  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  <P  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
)
2111, 4, 20sylancl 407 . . . . . . 7  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  <P  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
)
22 ltsopr 7368 . . . . . . . . 9  |-  <P  Or  P.
23 ltrelpr 7277 . . . . . . . . 9  |-  <P  C_  ( P.  X.  P. )
2422, 23sotri 4902 . . . . . . . 8  |-  ( ( ( z  +P.  1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  /\  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  <P  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
)  ->  ( z  +P.  1P )  <P  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )
2524expcom 115 . . . . . . 7  |-  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  <P  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  ->  ( ( z  +P. 
1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  ->  ( z  +P.  1P )  <P  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )
2621, 25syl 14 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( ( z  +P.  1P )  <P  <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  ->  ( z  +P. 
1P )  <P  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )
2722, 23sotri 4902 . . . . . . 7  |-  ( ( ( z  +P.  1P )  <P  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  /\  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  <P 
( w  +P.  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )  ->  ( z  +P.  1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )
2827expcom 115 . . . . . 6  |-  ( (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  <P 
( w  +P.  ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )  ->  ( ( z  +P.  1P )  <P 
( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  -> 
( z  +P.  1P )  <P  ( w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
2919, 26, 28sylsyld 58 . . . . 5  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( ( z  +P.  1P )  <P  <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  ->  ( z  +P. 
1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
3029reximdva 2509 . . . 4  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  ( E. x  e. 
N.  ( z  +P. 
1P )  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  ->  E. x  e.  N.  ( z  +P. 
1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
319, 30mpd 13 . . 3  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  E. x  e.  N.  ( z  +P.  1P )  <P  ( w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) )
32 simpl 108 . . . . 5  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( z  e. 
P.  /\  w  e.  P. ) )
334a1i 9 . . . . 5  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  1P  e.  P. )
34 ltsrprg 7519 . . . . 5  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  <->  ( z  +P.  1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
3532, 13, 33, 34syl12anc 1197 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  <->  ( z  +P.  1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
3635rexbidva 2409 . . 3  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  ( E. x  e. 
N.  [ <. z ,  w >. ]  ~R  <R  [
<. ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  <->  E. x  e.  N.  ( z  +P. 
1P )  <P  (
w  +P.  ( <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ) ) )
3731, 36mpbird 166 . 2  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  E. x  e.  N.  [
<. z ,  w >. ]  ~R  <R  [ <. ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
381, 3, 37ecoptocl 6482 1  |-  ( A  e.  R.  ->  E. x  e.  N.  A  <R  [ <. (
<. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   {cab 2101   E.wrex 2392   <.cop 3498   class class class wbr 3897  (class class class)co 5740   1oc1o 6272   [cec 6393   N.cnpi 7044    ~Q ceq 7051    <Q cltq 7057   P.cnp 7063   1Pc1p 7064    +P. cpp 7065    <P cltp 7067    ~R cer 7068   R.cnr 7069    <R cltr 7075
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-2o 6280  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-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-iltp 7242  df-enr 7498  df-nr 7499  df-ltr 7502
This theorem is referenced by:  axarch  7663
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