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Theorem prarloclemarch 7417
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7416 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 7391 . . . 4  |-  ( B  e.  Q.  ->  ( *Q `  B )  e. 
Q. )
2 mulclnq 7375 . . . 4  |-  ( ( A  e.  Q.  /\  ( *Q `  B )  e.  Q. )  -> 
( A  .Q  ( *Q `  B ) )  e.  Q. )
31, 2sylan2 286 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  ( *Q `  B ) )  e.  Q. )
4 archnqq 7416 . . 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 527 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  A  e.  Q. )
7 1pi 7314 . . . . . . . . . . 11  |-  1o  e.  N.
8 opelxpi 4659 . . . . . . . . . . 11  |-  ( ( x  e.  N.  /\  1o  e.  N. )  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
97, 8mpan2 425 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
10 enqex 7359 . . . . . . . . . . 11  |-  ~Q  e.  _V
1110ecelqsi 6589 . . . . . . . . . 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 7347 . . . . . . . . 9  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1412, 13eleqtrrdi 2271 . . . . . . . 8  |-  ( x  e.  N.  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
1514adantl 277 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  [ <. x ,  1o >. ]  ~Q  e.  Q. )
16 simplr 528 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  B  e.  Q. )
17 mulclnq 7375 . . . . . . 7  |-  ( ( [ <. x ,  1o >. ]  ~Q  e.  Q.  /\  B  e.  Q. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  B )  e.  Q. )
1815, 16, 17syl2anc 411 . . . . . 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 7400 . . . . . 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 1238 . . . . 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 7382 . . . . . . 7  |-  ( ( ( *Q `  B
)  e.  Q.  /\  A  e.  Q. )  ->  ( ( *Q `  B )  .Q  A
)  =  ( A  .Q  ( *Q `  B ) ) )
2319, 6, 22syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  A )  =  ( A  .Q  ( *Q
`  B ) ) )
24 mulcomnqg 7382 . . . . . . . 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 411 . . . . . . 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 7383 . . . . . . . . 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 1238 . . . . . . . 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 7392 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  ( B  .Q  ( *Q `  B ) )  =  1Q )
2928oveq2d 5891 . . . . . . . . 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 7388 . . . . . . . . 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 2214 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3425, 33eqtrd 2210 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3523, 34breq12d 4017 . . . . 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 188 . . . 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 158 . . 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 2579 . 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   <.cop 3596   class class class wbr 4004    X. cxp 4625   ` cfv 5217  (class class class)co 5875   1oc1o 6410   [cec 6533   /.cqs 6534   N.cnpi 7271    ~Q ceq 7278   Q.cnq 7279   1Qc1q 7280    .Q cmq 7282   *Qcrq 7283    <Q cltq 7284
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352
This theorem is referenced by:  prarloclemarch2  7418
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