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Theorem irrapx1 26882
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 12805 . . . 4  |-  QQ  ~~  NN
2 nnenom 11311 . . . 4  |-  NN  ~~  om
31, 2entri 7153 . . 3  |-  QQ  ~~  om
43, 2pm3.2i 442 . 2  |-  ( QQ 
~~  om  /\  NN  ~~  om )
5 ssrab2 3420 . . . . . 6  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  QQ
6 qssre 10576 . . . . . 6  |-  QQ  C_  RR
75, 6sstri 3349 . . . . 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 3461 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR+ )
109rpred 10640 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR )
11 eldifn 3462 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  QQ )
12 elrabi 3082 . . . . 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 26881 . . . . . 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 2781 . . . 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 26873 . . . 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 26870 . 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 1725   A.wral 2697   E.wrex 2698   {crab 2701    \ cdif 3309    C_ wss 3312   class class class wbr 4204   omcom 4837   ` cfv 5446  (class class class)co 6073    ~~ cen 7098   Fincfn 7101   RRcr 8981   0cc0 8982    < clt 9112    - cmin 9283   -ucneg 9284   NNcn 9992   2c2 10041   QQcq 10566   RR+crp 10604   ^cexp 11374   abscabs 12031  denomcdenom 13118
This theorem is referenced by:  pellexlem4  26886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ico 10914  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-numer 13119  df-denom 13120
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