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Theorem archsr 7306
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 7252 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3840 . . 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 2381 . 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 7092 . . . . . . 7  |-  1P  e.  P.
5 addclpr 7075 . . . . . . 7  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  +P.  1P )  e.  P. )
64, 5mpan2 416 . . . . . 6  |-  ( z  e.  P.  ->  (
z  +P.  1P )  e.  P. )
7 archpr 7181 . . . . . 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 270 . . . 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 7091 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
1110adantl 271 . . . . . . . . 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 7075 . . . . . . . . 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 404 . . . . . . . 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 497 . . . . . . . 8  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  w  e.  P. )
15 ltaddpr 7135 . . . . . . . 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 403 . . . . . . 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 7116 . . . . . . . 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 403 . . . . . . 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 3863 . . . . . 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 7135 . . . . . . . 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 404 . . . . . . 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 7134 . . . . . . . . 9  |-  <P  Or  P.
23 ltrelpr 7043 . . . . . . . . 9  |-  <P  C_  ( P.  X.  P. )
2422, 23sotri 4814 . . . . . . . 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 114 . . . . . . 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 4814 . . . . . . 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 114 . . . . . 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 57 . . . . 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 2475 . . . 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 107 . . . . 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 7272 . . . . 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 1172 . . . 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 2377 . . 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 165 . 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 6359 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3444   class class class wbr 3837  (class class class)co 5634   1oc1o 6156   [cec 6270   N.cnpi 6810    ~Q ceq 6817    <Q cltq 6823   P.cnp 6829   1Pc1p 6830    +P. cpp 6831    <P cltp 6833    ~R cer 6834   R.cnr 6835    <R cltr 6841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-i1p 7005  df-iplp 7006  df-iltp 7008  df-enr 7251  df-nr 7252  df-ltr 7255
This theorem is referenced by:  axarch  7405
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