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Theorem archnq 9356
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 9301 . . . 4  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
2 xp1st 6781 . . . 4  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
31, 2syl 17 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
4 1pi 9259 . . 3  |-  1o  e.  N.
5 addclpi 9268 . . 3  |-  ( ( ( 1st `  A
)  e.  N.  /\  1o  e.  N. )  -> 
( ( 1st `  A
)  +N  1o )  e.  N. )
63, 4, 5sylancl 666 . 2  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  +N  1o )  e.  N. )
7 xp2nd 6782 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
81, 7syl 17 . . . . 5  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
9 mulclpi 9269 . . . . 5  |-  ( ( ( ( 1st `  A
)  +N  1o )  e.  N.  /\  ( 2nd `  A )  e. 
N. )  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
106, 8, 9syl2anc 665 . . . 4  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  e.  N. )
11 eqid 2428 . . . . . . 7  |-  ( ( 1st `  A )  +N  1o )  =  ( ( 1st `  A
)  +N  1o )
12 oveq2 6257 . . . . . . . . 9  |-  ( x  =  1o  ->  (
( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) )
1312eqeq1d 2430 . . . . . . . 8  |-  ( x  =  1o  ->  (
( ( 1st `  A
)  +N  x )  =  ( ( 1st `  A )  +N  1o ) 
<->  ( ( 1st `  A
)  +N  1o )  =  ( ( 1st `  A )  +N  1o ) ) )
1413rspcev 3125 . . . . . . 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 676 . . . . . 6  |-  E. x  e.  N.  ( ( 1st `  A )  +N  x
)  =  ( ( 1st `  A )  +N  1o )
16 ltexpi 9278 . . . . . 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 236 . . . . 5  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( ( 1st `  A
)  +N  1o )  e.  N. )  -> 
( 1st `  A
)  <N  ( ( 1st `  A )  +N  1o ) )
183, 6, 17syl2anc 665 . . . 4  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( 1st `  A
)  +N  1o ) )
19 nlt1pi 9282 . . . . 5  |-  -.  ( 2nd `  A )  <N  1o
20 ltmpi 9280 . . . . . . 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 17 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( ( 1st `  A
)  +N  1o )  .N  1o ) ) )
22 mulidpi 9262 . . . . . . . 8  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
236, 22syl 17 . . . . . . 7  |-  ( A  e.  Q.  ->  (
( ( 1st `  A
)  +N  1o )  .N  1o )  =  ( ( 1st `  A
)  +N  1o ) )
2423breq2d 4378 . . . . . 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 256 . . . . 5  |-  ( A  e.  Q.  ->  (
( 2nd `  A
)  <N  1o  <->  ( (
( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) ) )
2619, 25mtbii 303 . . . 4  |-  ( A  e.  Q.  ->  -.  ( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) )  <N  (
( 1st `  A
)  +N  1o ) )
27 ltsopi 9264 . . . . 5  |-  <N  Or  N.
28 ltrelpi 9265 . . . . 5  |-  <N  C_  ( N.  X.  N. )
2927, 28sotri3 5192 . . . 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 1264 . . 3  |-  ( A  e.  Q.  ->  ( 1st `  A )  <N 
( ( ( 1st `  A )  +N  1o )  .N  ( 2nd `  A
) ) )
31 pinq 9303 . . . . . 6  |-  ( ( ( 1st `  A
)  +N  1o )  e.  N.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
326, 31syl 17 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  +N  1o ) ,  1o >.  e.  Q. )
33 ordpinq 9319 . . . . 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 672 . . . 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 6277 . . . . . . . 8  |-  ( ( 1st `  A )  +N  1o )  e. 
_V
364elexi 3032 . . . . . . . 8  |-  1o  e.  _V
3735, 36op2nd 6760 . . . . . . 7  |-  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  1o
3837oveq2i 6260 . . . . . 6  |-  ( ( 1st `  A )  .N  ( 2nd `  <. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)  =  ( ( 1st `  A )  .N  1o )
39 mulidpi 9262 . . . . . . 7  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
403, 39syl 17 . . . . . 6  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
4138, 40syl5eq 2474 . . . . 5  |-  ( A  e.  Q.  ->  (
( 1st `  A
)  .N  ( 2nd `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
)  =  ( 1st `  A ) )
4235, 36op1st 6759 . . . . . . 7  |-  ( 1st `  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )  =  ( ( 1st `  A )  +N  1o )
4342oveq1i 6259 . . . . . 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 4379 . . . 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 256 . . 3  |-  ( A  e.  Q.  ->  ( A  <Q  <. ( ( 1st `  A )  +N  1o ) ,  1o >.  <->  ( 1st `  A )  <N  (
( ( 1st `  A
)  +N  1o )  .N  ( 2nd `  A
) ) ) )
4730, 46mpbird 235 . 2  |-  ( A  e.  Q.  ->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
48 opeq1 4130 . . . 4  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  <. x ,  1o >.  =  <. ( ( 1st `  A )  +N  1o ) ,  1o >. )
4948breq2d 4378 . . 3  |-  ( x  =  ( ( 1st `  A )  +N  1o )  ->  ( A  <Q  <.
x ,  1o >.  <->  A  <Q 
<. ( ( 1st `  A
)  +N  1o ) ,  1o >. )
)
5049rspcev 3125 . 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 665 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2715   <.cop 3947   class class class wbr 4366    X. cxp 4794   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750   1oc1o 7130   N.cnpi 9220    +N cpli 9221    .N cmi 9222    <N clti 9223   Q.cnq 9228    <Q cltq 9234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-omul 7142  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-ltpq 9286  df-nq 9288  df-ltnq 9294
This theorem is referenced by:  prlem934  9409
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