MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  archnq Structured version   Unicode version

Theorem archnq 9347
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 9292 . . . 4  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
2 xp1st 6803 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
31, 2syl 16 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
4 1pi 9250 . . 3  |-  1o  e.  N.
5 addclpi 9259 . . 3  |-  ( ( ( 1st `  A
)  e.  N.  /\  1o  e.  N. )  -> 
( ( 1st `  A
)  +N  1o )  e.  N. )
63, 4, 5sylancl 660 . 2  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  +N  1o )  e.  N. )
7 xp2nd 6804 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
81, 7syl 16 . . . . 5  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
9 mulclpi 9260 . . . . 5  |-  ( ( ( ( 1st `  A
)  +N  1o )  e.  N.  /\  ( 2nd `  A )  e. 
N. )  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
106, 8, 9syl2anc 659 . . . 4  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
11 eqid 2454 . . . . . . 7  |-  ( ( 1st `  A )  +N  1o )  =  ( ( 1st `  A
)  +N  1o )
12 oveq2 6278 . . . . . . . . 9  |-  ( x  =  1o  ->  (
( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) )
1312eqeq1d 2456 . . . . . . . 8  |-  ( x  =  1o  ->  (
( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) 
<->  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) ) )
1413rspcev 3207 . . . . . . 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 670 . . . . . 6  |-  E. x  e.  N.  ( ( 1st `  A )  +N  x
)  =  ( ( 1st `  A )  +N  1o )
16 ltexpi 9269 . . . . . 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 233 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  e.  N. )  -> 
( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o ) )
183, 6, 17syl2anc 659 . . . 4  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( 1st `  A
)  +N  1o ) )
19 nlt1pi 9273 . . . . 5  |-  -.  ( 2nd `  A )  <N  1o
20 ltmpi 9271 . . . . . . 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 9253 . . . . . . . 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 4451 . . . . . 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 253 . . . . 5  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) ) )
2619, 25mtbii 300 . . . 4  |-  ( A  e.  Q.  ->  -.  ( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) )
27 ltsopi 9255 . . . . 5  |-  <N  Or  N.
28 ltrelpi 9256 . . . . 5  |-  <N  C_  ( N.  X.  N. )
2927, 28sotri3 5385 . . . 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 1226 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) ) )
31 pinq 9294 . . . . . 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 9310 . . . . 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 666 . . . 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 6298 . . . . . . . 8  |-  ( ( 1st `  A )  +N  1o )  e. 
_V
364elexi 3116 . . . . . . . 8  |-  1o  e.  _V
3735, 36op2nd 6782 . . . . . . 7  |-  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  1o
3837oveq2i 6281 . . . . . 6  |-  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  =  ( ( 1st `  A )  .N  1o )
39 mulidpi 9253 . . . . . . 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 2507 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  =  ( 1st `  A ) )
4235, 36op1st 6781 . . . . . . 7  |-  ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  ( ( 1st `  A )  +N  1o )
4342oveq1i 6280 . . . . . 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 4452 . . . 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 253 . . 3  |-  ( A  e.  Q.  ->  ( A  <Q  <. ( ( 1st `  A )  +N  1o ) ,  1o >.  <->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) ) )
4730, 46mpbird 232 . 2  |-  ( A  e.  Q.  ->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
48 opeq1 4203 . . . 4  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  <. x ,  1o >.  =  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
4948breq2d 4451 . . 3  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  ( A  <Q  <.
x ,  1o >.  <->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)
5049rspcev 3207 . 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 659 1  |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439    X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   1oc1o 7115   N.cnpi 9211    +N cpli 9212    .N cmi 9213    <N clti 9214   Q.cnq 9219    <Q cltq 9225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-ltpq 9277  df-nq 9279  df-ltnq 9285
This theorem is referenced by:  prlem934  9400
  Copyright terms: Public domain W3C validator