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Theorem irrapx1 26924
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 12494 . . . 4  |-  QQ  ~~  NN
2 nnenom 11044 . . . 4  |-  NN  ~~  om
31, 2entri 6917 . . 3  |-  QQ  ~~  om
43, 2pm3.2i 441 . 2  |-  ( QQ 
~~  om  /\  NN  ~~  om )
5 ssrab2 3260 . . . . . 6  |-  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  C_  QQ
6 qssre 10328 . . . . . 6  |-  QQ  C_  RR
75, 6sstri 3190 . . . . 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 3300 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR+ )
109rpred 10392 . . . 4  |-  ( A  e.  ( RR+  \  QQ )  ->  A  e.  RR )
11 eldifn 3301 . . . . 5  |-  ( A  e.  ( RR+  \  QQ )  ->  -.  A  e.  QQ )
125sseli 3178 . . . . 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 26923 . . . . . 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 2628 . . . 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 26915 . . . 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 26912 . 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 1686   A.wral 2545   E.wrex 2546   {crab 2549    \ cdif 3151    C_ wss 3154   class class class wbr 4025   omcom 4658   ` cfv 5257  (class class class)co 5860    ~~ cen 6862   Fincfn 6865   RRcr 8738   0cc0 8739    < clt 8869    - cmin 9039   -ucneg 9040   NNcn 9748   2c2 9797   QQcq 10318   RR+crp 10356   ^cexp 11106   abscabs 11721  denomcdenom 12807
This theorem is referenced by:  pellexlem4  26928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-ico 10664  df-fz 10785  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-dvds 12534  df-gcd 12688  df-numer 12808  df-denom 12809
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