Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  irrapx1 Unicode version

Theorem irrapx1 26575
Description: Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
irrapx1  |-  ( A  e.  ( RR+  \  QQ )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  ~~  NN )
Distinct variable group:    y, A

Proof of Theorem irrapx1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qnnen 12733 . . . 4  |-  QQ  ~~  NN
2 nnenom 11239 . . . 4  |-  NN  ~~  om
31, 2entri 7090 . . 3  |-  QQ  ~~  om
43, 2pm3.2i 442 . 2  |-  ( QQ 
~~  om  /\  NN  ~~  om )
5 ssrab2 3364 . . . . . 6  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  QQ
6 qssre 10509 . . . . . 6  |-  QQ  C_  RR
75, 6sstri 3293 . . . . 5  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  RR
87a1i 11 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  C_  RR )
9 eldifi 3405 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR+ )
109rpred 10573 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR )
11 eldifn 3406 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  QQ )
12 elrabi 3026 . . . . 5  |-  ( A  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  ->  A  e.  QQ )
1311, 12nsyl 115 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  { y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) } )
14 irrapxlem6 26574 . . . . . 6  |-  ( ( A  e.  RR+  /\  a  e.  RR+ )  ->  E. b  e.  { y  e.  QQ  |  ( 0  < 
y  /\  ( abs `  ( y  -  A
) )  <  (
(denom `  y ) ^ -u 2 ) ) }  ( abs `  (
b  -  A ) )  <  a )
159, 14sylan 458 . . . . 5  |-  ( ( A  e.  ( RR+  \  QQ )  /\  a  e.  RR+ )  ->  E. b  e.  { y  e.  QQ  |  ( 0  < 
y  /\  ( abs `  ( y  -  A
) )  <  (
(denom `  y ) ^ -u 2 ) ) }  ( abs `  (
b  -  A ) )  <  a )
1615ralrimiva 2725 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  A. a  e.  RR+  E. b  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ( abs `  ( b  -  A ) )  < 
a )
17 rencldnfi 26566 . . . 4  |-  ( ( ( { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  C_  RR  /\  A  e.  RR  /\ 
-.  A  e.  {
y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) } )  /\  A. a  e.  RR+  E. b  e.  {
y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ( abs `  ( b  -  A ) )  < 
a )  ->  -.  { y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  e.  Fin )
188, 10, 13, 16, 17syl31anc 1187 . . 3  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  e.  Fin )
1918, 5jctil 524 . 2  |-  ( A  e.  ( RR+  \  QQ )  ->  ( { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  QQ  /\  -.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  e.  Fin ) )
20 ctbnfien 26563 . 2  |-  ( ( ( QQ  ~~  om  /\  NN  ~~  om )  /\  ( { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  C_  QQ  /\  -.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  e.  Fin ) )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ~~  NN )
214, 19, 20sylancr 645 1  |-  ( A  e.  ( RR+  \  QQ )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  ~~  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2642   E.wrex 2643   {crab 2646    \ cdif 3253    C_ wss 3256   class class class wbr 4146   omcom 4778   ` cfv 5387  (class class class)co 6013    ~~ cen 7035   Fincfn 7038   RRcr 8915   0cc0 8916    < clt 9046    - cmin 9216   -ucneg 9217   NNcn 9925   2c2 9974   QQcq 10499   RR+crp 10537   ^cexp 11302   abscabs 11959  denomcdenom 13046
This theorem is referenced by:  pellexlem4  26579
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-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  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-se 4476  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-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-acn 7755  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-ico 10847  df-fz 10969  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927  df-numer 13047  df-denom 13048
  Copyright terms: Public domain W3C validator