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Theorem prarloclemarch 7194
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7193 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 7168 . . . 4  |-  ( B  e.  Q.  ->  ( *Q `  B )  e. 
Q. )
2 mulclnq 7152 . . . 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 7193 . . 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 503 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  A  e.  Q. )
7 1pi 7091 . . . . . . . . . . 11  |-  1o  e.  N.
8 opelxpi 4541 . . . . . . . . . . 11  |-  ( ( x  e.  N.  /\  1o  e.  N. )  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
97, 8mpan2 421 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
10 enqex 7136 . . . . . . . . . . 11  |-  ~Q  e.  _V
1110ecelqsi 6451 . . . . . . . . . 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 7124 . . . . . . . . 9  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1412, 13eleqtrrdi 2211 . . . . . . . 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 504 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  B  e.  Q. )
17 mulclnq 7152 . . . . . . 7  |-  ( ( [ <. x ,  1o >. ]  ~Q  e.  Q.  /\  B  e.  Q. )  ->  ( [ <. x ,  1o >. ]  ~Q  .Q  B )  e.  Q. )
1815, 16, 17syl2anc 408 . . . . . 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 7177 . . . . . 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 1201 . . . . 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 7159 . . . . . . 7  |-  ( ( ( *Q `  B
)  e.  Q.  /\  A  e.  Q. )  ->  ( ( *Q `  B )  .Q  A
)  =  ( A  .Q  ( *Q `  B ) ) )
2319, 6, 22syl2anc 408 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  A )  =  ( A  .Q  ( *Q
`  B ) ) )
24 mulcomnqg 7159 . . . . . . . 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 408 . . . . . . 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 7160 . . . . . . . . 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 1201 . . . . . . . 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 7169 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  ( B  .Q  ( *Q `  B ) )  =  1Q )
2928oveq2d 5758 . . . . . . . . 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 7165 . . . . . . . . 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 2154 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3425, 33eqtrd 2150 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3523, 34breq12d 3912 . . . . 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 2511 . 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 1316    e. wcel 1465   E.wrex 2394   <.cop 3500   class class class wbr 3899    X. cxp 4507   ` cfv 5093  (class class class)co 5742   1oc1o 6274   [cec 6395   /.cqs 6396   N.cnpi 7048    ~Q ceq 7055   Q.cnq 7056   1Qc1q 7057    .Q cmq 7059   *Qcrq 7060    <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129
This theorem is referenced by:  prarloclemarch2  7195
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