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Theorem archsr 7020
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 6966 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3796 . . 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 2370 . 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 6806 . . . . . . 7  |-  1P  e.  P.
5 addclpr 6789 . . . . . . 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 6895 . . . . . 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 6805 . . . . . . . . . 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 6789 . . . . . . . . 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 6849 . . . . . . . 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 6830 . . . . . . . 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 3819 . . . . . 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 6849 . . . . . . . 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 6848 . . . . . . . . 9  |-  <P  Or  P.
23 ltrelpr 6757 . . . . . . . . 9  |-  <P  C_  ( P.  X.  P. )
2422, 23sotri 4750 . . . . . . . 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 4750 . . . . . . 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 2464 . . . 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 6986 . . . . 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 1168 . . . 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 2366 . . 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 6259 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 1285    e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3409   class class class wbr 3793  (class class class)co 5543   1oc1o 6058   [cec 6170   N.cnpi 6524    ~Q ceq 6531    <Q cltq 6537   P.cnp 6543   1Pc1p 6544    +P. cpp 6545    <P cltp 6547    ~R cer 6548   R.cnr 6549    <R cltr 6555
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-iltp 6722  df-enr 6965  df-nr 6966  df-ltr 6969
This theorem is referenced by:  axarch  7119
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