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Theorem prarloclemarch 7380
Description: A version of the Archimedean property. This variation is "stronger" than archnqq 7379 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 7354 . . . 4  |-  ( B  e.  Q.  ->  ( *Q `  B )  e. 
Q. )
2 mulclnq 7338 . . . 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 7379 . . 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 524 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  A  e.  Q. )
7 1pi 7277 . . . . . . . . . . 11  |-  1o  e.  N.
8 opelxpi 4643 . . . . . . . . . . 11  |-  ( ( x  e.  N.  /\  1o  e.  N. )  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
97, 8mpan2 423 . . . . . . . . . 10  |-  ( x  e.  N.  ->  <. x ,  1o >.  e.  ( N.  X.  N. ) )
10 enqex 7322 . . . . . . . . . . 11  |-  ~Q  e.  _V
1110ecelqsi 6567 . . . . . . . . . 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 7310 . . . . . . . . 9  |-  Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
1412, 13eleqtrrdi 2264 . . . . . . . 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 525 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  B  e.  Q. )
17 mulclnq 7338 . . . . . . 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 7363 . . . . . 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 1233 . . . . 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 7345 . . . . . . 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 7345 . . . . . . . 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 7346 . . . . . . . . 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 1233 . . . . . . . 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 7355 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  ( B  .Q  ( *Q `  B ) )  =  1Q )
2928oveq2d 5869 . . . . . . . . 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 7351 . . . . . . . . 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 2207 . . . . . . 7  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( [
<. x ,  1o >. ]  ~Q  .Q  B )  .Q  ( *Q `  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3425, 33eqtrd 2203 . . . . . 6  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  x  e.  N. )  ->  ( ( *Q
`  B )  .Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )  =  [ <. x ,  1o >. ]  ~Q  )
3523, 34breq12d 4002 . . . . 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 2572 . 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 1348    e. wcel 2141   E.wrex 2449   <.cop 3586   class class class wbr 3989    X. cxp 4609   ` cfv 5198  (class class class)co 5853   1oc1o 6388   [cec 6511   /.cqs 6512   N.cnpi 7234    ~Q ceq 7241   Q.cnq 7242   1Qc1q 7243    .Q cmq 7245   *Qcrq 7246    <Q cltq 7247
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315
This theorem is referenced by:  prarloclemarch2  7381
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