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Theorem archnq 8604
Description: For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
archnq  |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >. )
Distinct variable group:    x, A

Proof of Theorem archnq
StepHypRef Expression
1 elpqn 8549 . . . 4  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
2 xp1st 6149 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
31, 2syl 15 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
4 1pi 8507 . . 3  |-  1o  e.  N.
5 addclpi 8516 . . 3  |-  ( ( ( 1st `  A
)  e.  N.  /\  1o  e.  N. )  -> 
( ( 1st `  A
)  +N  1o )  e.  N. )
63, 4, 5sylancl 643 . 2  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  +N  1o )  e.  N. )
7 xp2nd 6150 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
81, 7syl 15 . . . . 5  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
9 mulclpi 8517 . . . . 5  |-  ( ( ( ( 1st `  A
)  +N  1o )  e.  N.  /\  ( 2nd `  A )  e. 
N. )  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
106, 8, 9syl2anc 642 . . . 4  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
11 eqid 2283 . . . . . . 7  |-  ( ( 1st `  A )  +N  1o )  =  ( ( 1st `  A
)  +N  1o )
12 oveq2 5866 . . . . . . . . 9  |-  ( x  =  1o  ->  (
( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) )
1312eqeq1d 2291 . . . . . . . 8  |-  ( x  =  1o  ->  (
( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) 
<->  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) ) )
1413rspcev 2884 . . . . . . 7  |-  ( ( 1o  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) )  ->  E. x  e.  N.  ( ( 1st `  A )  +N  x
)  =  ( ( 1st `  A )  +N  1o ) )
154, 11, 14mp2an 653 . . . . . 6  |-  E. x  e.  N.  ( ( 1st `  A )  +N  x
)  =  ( ( 1st `  A )  +N  1o )
16 ltexpi 8526 . . . . . 6  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  e.  N. )  -> 
( ( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o ) 
<->  E. x  e.  N.  ( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) ) )
1715, 16mpbiri 224 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  e.  N. )  -> 
( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o ) )
183, 6, 17syl2anc 642 . . . 4  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( 1st `  A
)  +N  1o ) )
19 nlt1pi 8530 . . . . 5  |-  -.  ( 2nd `  A )  <N  1o
20 ltmpi 8528 . . . . . . 7  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( ( 1st `  A
)  +N  1o )  .N  1o ) ) )
216, 20syl 15 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( ( 1st `  A
)  +N  1o )  .N  1o ) ) )
22 mulidpi 8510 . . . . . . . 8  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
236, 22syl 15 . . . . . . 7  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
2423breq2d 4035 . . . . . 6  |-  ( A  e.  Q.  ->  (
( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( ( 1st `  A
)  +N  1o )  .N  1o )  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) ) )
2521, 24bitrd 244 . . . . 5  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) ) )
2619, 25mtbii 293 . . . 4  |-  ( A  e.  Q.  ->  -.  ( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) )
27 ltsopi 8512 . . . . 5  |-  <N  Or  N.
28 ltrelpi 8513 . . . . 5  |-  <N  C_  ( N.  X.  N. )
2927, 28sotri3 5073 . . . 4  |-  ( ( ( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  e.  N.  /\  ( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o )  /\  -.  ( ( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) )  ->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) )
3010, 18, 26, 29syl3anc 1182 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) ) )
31 pinq 8551 . . . . . 6  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
326, 31syl 15 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
33 ordpinq 8567 . . . . 5  |-  ( ( A  e.  Q.  /\  <.
( ( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )  ->  ( A  <Q  <.
( ( 1st `  A
)  +N  1o ) ,  1o >.  <->  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  <N  ( ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) ) ) )
3432, 33mpdan 649 . . . 4  |-  ( A  e.  Q.  ->  ( A  <Q  <. ( ( 1st `  A )  +N  1o ) ,  1o >.  <->  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  <N  ( ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) ) ) )
35 ovex 5883 . . . . . . . 8  |-  ( ( 1st `  A )  +N  1o )  e. 
_V
364elexi 2797 . . . . . . . 8  |-  1o  e.  _V
3735, 36op2nd 6129 . . . . . . 7  |-  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  1o
3837oveq2i 5869 . . . . . 6  |-  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  =  ( ( 1st `  A )  .N  1o )
39 mulidpi 8510 . . . . . . 7  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
403, 39syl 15 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
4138, 40syl5eq 2327 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  =  ( 1st `  A ) )
4235, 36op1st 6128 . . . . . . 7  |-  ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  ( ( 1st `  A )  +N  1o )
4342oveq1i 5868 . . . . . 6  |-  ( ( 1st `  <. (
( 1st `  A
)  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )
4443a1i 10 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) )
4541, 44breq12d 4036 . . . 4  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  <N  ( ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) )  <->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) ) )
4634, 45bitrd 244 . . 3  |-  ( A  e.  Q.  ->  ( A  <Q  <. ( ( 1st `  A )  +N  1o ) ,  1o >.  <->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) ) )
4730, 46mpbird 223 . 2  |-  ( A  e.  Q.  ->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
48 opeq1 3796 . . . 4  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  <. x ,  1o >.  =  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
4948breq2d 4035 . . 3  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  ( A  <Q  <.
x ,  1o >.  <->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)
5049rspcev 2884 . 2  |-  ( ( ( ( 1st `  A
)  +N  1o )  e.  N.  /\  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )  ->  E. x  e.  N.  A  <Q  <. x ,  1o >. )
516, 47, 50syl2anc 642 1  |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   1oc1o 6472   N.cnpi 8466    +N cpli 8467    .N cmi 8468    <N clti 8469   Q.cnq 8474    <Q cltq 8480
This theorem is referenced by:  prlem934  8657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-ltpq 8534  df-nq 8536  df-ltnq 8542
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