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Theorem irrapx1 26325
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 12492 . . . 4  |-  QQ  ~~  NN
2 nnenom 11042 . . . 4  |-  NN  ~~  om
31, 2entri 6915 . . 3  |-  QQ  ~~  om
43, 2pm3.2i 441 . 2  |-  ( QQ 
~~  om  /\  NN  ~~  om )
5 ssrab2 3258 . . . . . 6  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  QQ
6 qssre 10326 . . . . . 6  |-  QQ  C_  RR
75, 6sstri 3188 . . . . 5  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  RR
87a1i 10 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  C_  RR )
9 eldifi 3298 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR+ )
109rpred 10390 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR )
11 eldifn 3299 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  QQ )
125sseli 3176 . . . . 5  |-  ( A  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  ->  A  e.  QQ )
1311, 12nsyl 113 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  { y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) } )
14 irrapxlem6 26324 . . . . . 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 457 . . . . 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 2626 . . . 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 26316 . . . 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 1185 . . 3  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  e.  Fin )
1918, 5jctil 523 . 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 26313 . 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 644 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 358    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   class class class wbr 4023   omcom 4656   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   RRcr 8736   0cc0 8737    < clt 8867    - cmin 9037   -ucneg 9038   NNcn 9746   2c2 9795   QQcq 10316   RR+crp 10354   ^cexp 11104   abscabs 11719  denomcdenom 12805
This theorem is referenced by:  pellexlem4  26329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ico 10662  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807
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