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Theorem prarloclemarch 7340
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7339 in the sense that we provide an integer which is larger than a given rational  A even after being multiplied by a second rational  B. (Contributed by Jim Kingdon, 30-Nov-2019.)
Assertion
Ref Expression
prarloclemarch  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  E. x  e.  N.  A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem prarloclemarch
StepHypRef Expression
1 recclnq 7314 . . . 4  |-  ( B  e.  Q.  ->  ( *Q `  B )  e. 
Q. )
2 mulclnq 7298 . . . 4  |-  ( ( A  e.  Q.  /\  ( *Q `  B )  e.  Q. )  -> 
( A  .Q  ( *Q `  B ) )  e.  Q. )
31, 2sylan2 284 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  ( *Q `  B ) )  e.  Q. )
4 archnqq 7339 . . 3  |-  ( ( A  .Q  ( *Q
`  B ) )  e.  Q.  ->  E. x  e.  N.  ( A  .Q  ( *Q `  B ) )  <Q  [ <. x ,  1o >. ]  ~Q  )
53, 4syl 14 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  E. x  e.  N.  ( A  .Q  ( *Q `  B ) ) 
<Q  [ <. x ,  1o >. ]  ~Q  )
6 simpll 519 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  A  e.  Q. )
7 1pi 7237 . . . . . . . . . . 11  |-  1o  e.  N.
8 opelxpi 4620 . . . . . . . . . . 11  |-  ( ( x  e.  N.  /\  1o  e.  N. )  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
97, 8mpan2 422 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
10 enqex 7282 . . . . . . . . . . 11  |-  ~Q  e.  _V
1110ecelqsi 6536 . . . . . . . . . 10  |-  ( <.
x ,  1o >.  e.  ( N.  X.  N. )  ->  [ <. x ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
129, 11syl 14 . . . . . . . . 9  |-  ( x  e.  N.  ->  [ <. x ,  1o >. ]  ~Q  e.  ( ( N.  X.  N. ) /.  ~Q  )
)
13 df-nqqs 7270 . . . . . . . . 9  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1412, 13eleqtrrdi 2251 . . . . . . . 8  |-  ( x  e.  N.  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
1514adantl 275 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
16 simplr 520 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  B  e.  Q. )
17 mulclnq 7298 . . . . . . 7  |-  ( ( [ <. x ,  1o >. ]  ~Q  e.  Q.  /\  B  e.  Q. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  B )  e.  Q. )
1815, 16, 17syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  B )  e.  Q. )
1916, 1syl 14 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( *Q `  B )  e.  Q. )
20 ltmnqg 7323 . . . . . 6  |-  ( ( A  e.  Q.  /\  ( [ <. x ,  1o >. ]  ~Q  .Q  B
)  e.  Q.  /\  ( *Q `  B )  e.  Q. )  -> 
( A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B
)  <->  ( ( *Q
`  B )  .Q  A )  <Q  (
( *Q `  B
)  .Q  ( [
<. x ,  1o >. ]  ~Q  .Q  B ) ) ) )
216, 18, 19, 20syl3anc 1220 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B
)  <->  ( ( *Q
`  B )  .Q  A )  <Q  (
( *Q `  B
)  .Q  ( [
<. x ,  1o >. ]  ~Q  .Q  B ) ) ) )
22 mulcomnqg 7305 . . . . . . 7  |-  ( ( ( *Q `  B
)  e.  Q.  /\  A  e.  Q. )  ->  ( ( *Q `  B )  .Q  A
)  =  ( A  .Q  ( *Q `  B ) ) )
2319, 6, 22syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  A )  =  ( A  .Q  ( *Q
`  B ) ) )
24 mulcomnqg 7305 . . . . . . . 8  |-  ( ( ( *Q `  B
)  e.  Q.  /\  ( [ <. x ,  1o >. ]  ~Q  .Q  B
)  e.  Q. )  ->  ( ( *Q `  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B
) )  =  ( ( [ <. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) ) )
2519, 18, 24syl2anc 409 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  ( ( [ <. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) ) )
26 mulassnqg 7306 . . . . . . . . 9  |-  ( ( [ <. x ,  1o >. ]  ~Q  e.  Q.  /\  B  e.  Q.  /\  ( *Q `  B )  e.  Q. )  -> 
( ( [ <. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  ( [ <. x ,  1o >. ]  ~Q  .Q  ( B  .Q  ( *Q `  B ) ) ) )
2715, 16, 19, 26syl3anc 1220 . . . . . . . 8  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  ( [ <. x ,  1o >. ]  ~Q  .Q  ( B  .Q  ( *Q `  B ) ) ) )
28 recidnq 7315 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  ( B  .Q  ( *Q `  B ) )  =  1Q )
2928oveq2d 5842 . . . . . . . . 9  |-  ( B  e.  Q.  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  ( B  .Q  ( *Q `  B ) ) )  =  ( [ <. x ,  1o >. ]  ~Q  .Q  1Q ) )
3016, 29syl 14 . . . . . . . 8  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  ( B  .Q  ( *Q `  B ) ) )  =  ( [
<. x ,  1o >. ]  ~Q  .Q  1Q ) )
31 mulidnq 7311 . . . . . . . . 9  |-  ( [
<. x ,  1o >. ]  ~Q  e.  Q.  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  1Q )  =  [ <. x ,  1o >. ]  ~Q  )
3215, 31syl 14 . . . . . . . 8  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  1Q )  =  [ <. x ,  1o >. ]  ~Q  )
3327, 30, 323eqtrd 2194 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3425, 33eqtrd 2190 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3523, 34breq12d 3980 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( ( *Q `  B )  .Q  A )  <Q 
( ( *Q `  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B
) )  <->  ( A  .Q  ( *Q `  B
) )  <Q  [ <. x ,  1o >. ]  ~Q  ) )
3621, 35bitrd 187 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B
)  <->  ( A  .Q  ( *Q `  B ) )  <Q  [ <. x ,  1o >. ]  ~Q  )
)
3736biimprd 157 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( A  .Q  ( *Q `  B ) )  <Q  [ <. x ,  1o >. ]  ~Q  ->  A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) ) )
3837reximdva 2559 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( E. x  e. 
N.  ( A  .Q  ( *Q `  B ) )  <Q  [ <. x ,  1o >. ]  ~Q  ->  E. x  e.  N.  A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) ) )
395, 38mpd 13 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  E. x  e.  N.  A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   <.cop 3564   class class class wbr 3967    X. cxp 4586   ` cfv 5172  (class class class)co 5826   1oc1o 6358   [cec 6480   /.cqs 6481   N.cnpi 7194    ~Q ceq 7201   Q.cnq 7202   1Qc1q 7203    .Q cmq 7205   *Qcrq 7206    <Q cltq 7207
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275
This theorem is referenced by:  prarloclemarch2  7341
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