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Theorem archnq 8620
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 8565 . . . 4  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
2 xp1st 6165 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
31, 2syl 15 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
4 1pi 8523 . . 3  |-  1o  e.  N.
5 addclpi 8532 . . 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 6166 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
81, 7syl 15 . . . . 5  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
9 mulclpi 8533 . . . . 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 2296 . . . . . . 7  |-  ( ( 1st `  A )  +N  1o )  =  ( ( 1st `  A
)  +N  1o )
12 oveq2 5882 . . . . . . . . 9  |-  ( x  =  1o  ->  (
( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) )
1312eqeq1d 2304 . . . . . . . 8  |-  ( x  =  1o  ->  (
( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) 
<->  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) ) )
1413rspcev 2897 . . . . . . 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 8542 . . . . . 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 8546 . . . . 5  |-  -.  ( 2nd `  A )  <N  1o
20 ltmpi 8544 . . . . . . 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 8526 . . . . . . . 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 4051 . . . . . 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 8528 . . . . 5  |-  <N  Or  N.
28 ltrelpi 8529 . . . . 5  |-  <N  C_  ( N.  X.  N. )
2927, 28sotri3 5089 . . . 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 8567 . . . . . 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 8583 . . . . 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 5899 . . . . . . . 8  |-  ( ( 1st `  A )  +N  1o )  e. 
_V
364elexi 2810 . . . . . . . 8  |-  1o  e.  _V
3735, 36op2nd 6145 . . . . . . 7  |-  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  1o
3837oveq2i 5885 . . . . . 6  |-  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  =  ( ( 1st `  A )  .N  1o )
39 mulidpi 8526 . . . . . . 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 2340 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  =  ( 1st `  A ) )
4235, 36op1st 6144 . . . . . . 7  |-  ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  ( ( 1st `  A )  +N  1o )
4342oveq1i 5884 . . . . . 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 4052 . . . 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 3812 . . . 4  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  <. x ,  1o >.  =  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
4948breq2d 4051 . . 3  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  ( A  <Q  <.
x ,  1o >.  <->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)
5049rspcev 2897 . 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 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   1oc1o 6488   N.cnpi 8482    +N cpli 8483    .N cmi 8484    <N clti 8485   Q.cnq 8490    <Q cltq 8496
This theorem is referenced by:  prlem934  8673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-ltpq 8550  df-nq 8552  df-ltnq 8558
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