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Theorem prarloclemarch 6670
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 6669 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 6644 . . . 4  |-  ( B  e.  Q.  ->  ( *Q `  B )  e. 
Q. )
2 mulclnq 6628 . . . 4  |-  ( ( A  e.  Q.  /\  ( *Q `  B )  e.  Q. )  -> 
( A  .Q  ( *Q `  B ) )  e.  Q. )
31, 2sylan2 280 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  ( *Q `  B ) )  e.  Q. )
4 archnqq 6669 . . 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 496 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  A  e.  Q. )
7 1pi 6567 . . . . . . . . . . 11  |-  1o  e.  N.
8 opelxpi 4402 . . . . . . . . . . 11  |-  ( ( x  e.  N.  /\  1o  e.  N. )  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
97, 8mpan2 416 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
10 enqex 6612 . . . . . . . . . . 11  |-  ~Q  e.  _V
1110ecelqsi 6226 . . . . . . . . . 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 6600 . . . . . . . . 9  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1412, 13syl6eleqr 2173 . . . . . . . 8  |-  ( x  e.  N.  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
1514adantl 271 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
16 simplr 497 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  B  e.  Q. )
17 mulclnq 6628 . . . . . . 7  |-  ( ( [ <. x ,  1o >. ]  ~Q  e.  Q.  /\  B  e.  Q. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  B )  e.  Q. )
1815, 16, 17syl2anc 403 . . . . . 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 6653 . . . . . 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 1170 . . . . 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 6635 . . . . . . 7  |-  ( ( ( *Q `  B
)  e.  Q.  /\  A  e.  Q. )  ->  ( ( *Q `  B )  .Q  A
)  =  ( A  .Q  ( *Q `  B ) ) )
2319, 6, 22syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  A )  =  ( A  .Q  ( *Q
`  B ) ) )
24 mulcomnqg 6635 . . . . . . . 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 403 . . . . . . 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 6636 . . . . . . . . 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 1170 . . . . . . . 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 6645 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  ( B  .Q  ( *Q `  B ) )  =  1Q )
2928oveq2d 5559 . . . . . . . . 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 6641 . . . . . . . . 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 2118 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3425, 33eqtrd 2114 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3523, 34breq12d 3806 . . . . 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 186 . . . 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 156 . . 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 2464 . 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350   <.cop 3409   class class class wbr 3793    X. cxp 4369   ` cfv 4932  (class class class)co 5543   1oc1o 6058   [cec 6170   /.cqs 6171   N.cnpi 6524    ~Q ceq 6531   Q.cnq 6532   1Qc1q 6533    .Q cmq 6535   *Qcrq 6536    <Q cltq 6537
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-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-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-mpq 6597  df-enq 6599  df-nqqs 6600  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605
This theorem is referenced by:  prarloclemarch2  6671
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