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Theorem archnq 8783
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 8728 . . . 4  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
2 xp1st 6308 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
31, 2syl 16 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
4 1pi 8686 . . 3  |-  1o  e.  N.
5 addclpi 8695 . . 3  |-  ( ( ( 1st `  A
)  e.  N.  /\  1o  e.  N. )  -> 
( ( 1st `  A
)  +N  1o )  e.  N. )
63, 4, 5sylancl 644 . 2  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  +N  1o )  e.  N. )
7 xp2nd 6309 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
81, 7syl 16 . . . . 5  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
9 mulclpi 8696 . . . . 5  |-  ( ( ( ( 1st `  A
)  +N  1o )  e.  N.  /\  ( 2nd `  A )  e. 
N. )  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
106, 8, 9syl2anc 643 . . . 4  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
11 eqid 2380 . . . . . . 7  |-  ( ( 1st `  A )  +N  1o )  =  ( ( 1st `  A
)  +N  1o )
12 oveq2 6021 . . . . . . . . 9  |-  ( x  =  1o  ->  (
( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) )
1312eqeq1d 2388 . . . . . . . 8  |-  ( x  =  1o  ->  (
( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) 
<->  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) ) )
1413rspcev 2988 . . . . . . 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 654 . . . . . 6  |-  E. x  e.  N.  ( ( 1st `  A )  +N  x
)  =  ( ( 1st `  A )  +N  1o )
16 ltexpi 8705 . . . . . 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 225 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  e.  N. )  -> 
( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o ) )
183, 6, 17syl2anc 643 . . . 4  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( 1st `  A
)  +N  1o ) )
19 nlt1pi 8709 . . . . 5  |-  -.  ( 2nd `  A )  <N  1o
20 ltmpi 8707 . . . . . . 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 16 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( ( 1st `  A
)  +N  1o )  .N  1o ) ) )
22 mulidpi 8689 . . . . . . . 8  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
236, 22syl 16 . . . . . . 7  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
2423breq2d 4158 . . . . . 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 245 . . . . 5  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) ) )
2619, 25mtbii 294 . . . 4  |-  ( A  e.  Q.  ->  -.  ( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) )
27 ltsopi 8691 . . . . 5  |-  <N  Or  N.
28 ltrelpi 8692 . . . . 5  |-  <N  C_  ( N.  X.  N. )
2927, 28sotri3 5197 . . . 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 1184 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) ) )
31 pinq 8730 . . . . . 6  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
326, 31syl 16 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
33 ordpinq 8746 . . . . 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 650 . . . 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 6038 . . . . . . . 8  |-  ( ( 1st `  A )  +N  1o )  e. 
_V
364elexi 2901 . . . . . . . 8  |-  1o  e.  _V
3735, 36op2nd 6288 . . . . . . 7  |-  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  1o
3837oveq2i 6024 . . . . . 6  |-  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  =  ( ( 1st `  A )  .N  1o )
39 mulidpi 8689 . . . . . . 7  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
403, 39syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
4138, 40syl5eq 2424 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  =  ( 1st `  A ) )
4235, 36op1st 6287 . . . . . . 7  |-  ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  ( ( 1st `  A )  +N  1o )
4342oveq1i 6023 . . . . . 6  |-  ( ( 1st `  <. (
( 1st `  A
)  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )
4443a1i 11 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )  .N  ( 2nd `  A
) )  =  ( ( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) )
4541, 44breq12d 4159 . . . 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 245 . . 3  |-  ( A  e.  Q.  ->  ( A  <Q  <. ( ( 1st `  A )  +N  1o ) ,  1o >.  <->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) ) )
4730, 46mpbird 224 . 2  |-  ( A  e.  Q.  ->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
48 opeq1 3919 . . . 4  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  <. x ,  1o >.  =  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
4948breq2d 4158 . . 3  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  ( A  <Q  <.
x ,  1o >.  <->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)
5049rspcev 2988 . 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 643 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2643   <.cop 3753   class class class wbr 4146    X. cxp 4809   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   1oc1o 6646   N.cnpi 8645    +N cpli 8646    .N cmi 8647    <N clti 8648   Q.cnq 8653    <Q cltq 8659
This theorem is referenced by:  prlem934  8836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-ltpq 8713  df-nq 8715  df-ltnq 8721
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