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Theorem archsr 7810
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 7755 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 4021 . . 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 2491 . 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 7582 . . . . . . 7  |-  1P  e.  P.
5 addclpr 7565 . . . . . . 7  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  +P.  1P )  e.  P. )
64, 5mpan2 425 . . . . . 6  |-  ( z  e.  P.  ->  (
z  +P.  1P )  e.  P. )
7 archpr 7671 . . . . . 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 276 . . . 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 7581 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
1110adantl 277 . . . . . . . . 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 7565 . . . . . . . . 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 413 . . . . . . . 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 528 . . . . . . . 8  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  x  e.  N. )  ->  w  e.  P. )
15 ltaddpr 7625 . . . . . . . 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 411 . . . . . . 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 7606 . . . . . . . 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 411 . . . . . . 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 4046 . . . . . 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 7625 . . . . . . . 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 413 . . . . . . 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 7624 . . . . . . . . 9  |-  <P  Or  P.
23 ltrelpr 7533 . . . . . . . . 9  |-  <P  C_  ( P.  X.  P. )
2422, 23sotri 5042 . . . . . . . 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 116 . . . . . . 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 5042 . . . . . . 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 116 . . . . . 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 2592 . . . 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 109 . . . . 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 7775 . . . . 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 1247 . . . 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 2487 . . 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 167 . 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 6647 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   E.wrex 2469   <.cop 3610   class class class wbr 4018  (class class class)co 5895   1oc1o 6433   [cec 6556   N.cnpi 7300    ~Q ceq 7307    <Q cltq 7313   P.cnp 7319   1Pc1p 7320    +P. cpp 7321    <P cltp 7323    ~R cer 7324   R.cnr 7325    <R cltr 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-i1p 7495  df-iplp 7496  df-iltp 7498  df-enr 7754  df-nr 7755  df-ltr 7758
This theorem is referenced by:  axarch  7919
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