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Theorem aaliou 19720
Description: Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 
F in integer coefficients, is not approximable beyond order  N  = deg ( F ) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypotheses
Ref Expression
aalioulem2.a  |-  N  =  (deg `  F )
aalioulem2.b  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
aalioulem2.c  |-  ( ph  ->  N  e.  NN )
aalioulem2.d  |-  ( ph  ->  A  e.  RR )
aalioulem3.e  |-  ( ph  ->  ( F `  A
)  =  0 )
Assertion
Ref Expression
aaliou  |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
Distinct variable groups:    ph, x, p, q    x, A, p, q    x, F, p, q    x, N
Allowed substitution hints:    N( q, p)

Proof of Theorem aaliou
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 aalioulem2.a . . 3  |-  N  =  (deg `  F )
2 aalioulem2.b . . 3  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
3 aalioulem2.c . . 3  |-  ( ph  ->  N  e.  NN )
4 aalioulem2.d . . 3  |-  ( ph  ->  A  e.  RR )
5 aalioulem3.e . . 3  |-  ( ph  ->  ( F `  A
)  =  0 )
61, 2, 3, 4, 5aalioulem6 19719 . 2  |-  ( ph  ->  E. a  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( a  /  (
q ^ N ) )  <_  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
7 rphalfcl 10380 . . . . 5  |-  ( a  e.  RR+  ->  ( a  /  2 )  e.  RR+ )
87adantl 452 . . . 4  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( a  /  2 )  e.  RR+ )
97ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR+ )
10 nnrp 10365 . . . . . . . . . . . . . 14  |-  ( q  e.  NN  ->  q  e.  RR+ )
1110ad2antll 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  q  e.  RR+ )
123nnzd 10118 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
1312ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  N  e.  ZZ )
1411, 13rpexpcld 11270 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( q ^ N )  e.  RR+ )
159, 14rpdivcld 10409 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR+ )
1615rpred 10392 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR )
17 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR+ )
1817, 14rpdivcld 10409 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR+ )
1918rpred 10392 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR )
204adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  RR+ )  ->  A  e.  RR )
21 znq 10322 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  QQ )
22 qre 10323 . . . . . . . . . . . . . 14  |-  ( ( p  /  q )  e.  QQ  ->  (
p  /  q )  e.  RR )
2321, 22syl 15 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  RR )
24 resubcl 9113 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( p  /  q
)  e.  RR )  ->  ( A  -  ( p  /  q
) )  e.  RR )
2520, 23, 24syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  RR )
2625recnd 8863 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  CC )
2726abscld 11920 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( abs `  ( A  -  (
p  /  q ) ) )  e.  RR )
2816, 19, 273jca 1132 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
( a  /  2
)  /  ( q ^ N ) )  e.  RR  /\  (
a  /  ( q ^ N ) )  e.  RR  /\  ( abs `  ( A  -  ( p  /  q
) ) )  e.  RR ) )
299rpred 10392 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR )
30 rpre 10362 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  a  e.  RR )
3130ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR )
32 rphalflt 10382 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  ( a  /  2 )  < 
a )
3332ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  < 
a )
3429, 31, 14, 33ltdiv1dd 10445 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( a  /  (
q ^ N ) ) )
3534anim1i 551 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  a  e.  RR+ )  /\  ( p  e.  ZZ  /\  q  e.  NN ) )  /\  ( a  /  ( q ^ N ) )  <_ 
( abs `  ( A  -  ( p  /  q ) ) ) )  ->  (
( ( a  / 
2 )  /  (
q ^ N ) )  <  ( a  /  ( q ^ N ) )  /\  ( a  /  (
q ^ N ) )  <_  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
3635ex 423 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q ) ) )  ->  ( (
( a  /  2
)  /  ( q ^ N ) )  <  ( a  / 
( q ^ N
) )  /\  (
a  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
37 ltletr 8915 . . . . . . . . 9  |-  ( ( ( ( a  / 
2 )  /  (
q ^ N ) )  e.  RR  /\  ( a  /  (
q ^ N ) )  e.  RR  /\  ( abs `  ( A  -  ( p  / 
q ) ) )  e.  RR )  -> 
( ( ( ( a  /  2 )  /  ( q ^ N ) )  < 
( a  /  (
q ^ N ) )  /\  ( a  /  ( q ^ N ) )  <_ 
( abs `  ( A  -  ( p  /  q ) ) ) )  ->  (
( a  /  2
)  /  ( q ^ N ) )  <  ( abs `  ( A  -  ( p  /  q ) ) ) ) )
3828, 36, 37sylsyld 52 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q ) ) )  ->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
3938orim2d 813 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( ( A  =  ( p  /  q )  \/  ( a  /  (
q ^ N ) )  <_  ( abs `  ( A  -  (
p  /  q ) ) ) )  -> 
( A  =  ( p  /  q )  \/  ( ( a  /  2 )  / 
( q ^ N
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) ) )
4039anassrs 629 . . . . . 6  |-  ( ( ( ( ph  /\  a  e.  RR+ )  /\  p  e.  ZZ )  /\  q  e.  NN )  ->  ( ( A  =  ( p  / 
q )  \/  (
a  /  ( q ^ N ) )  <_  ( abs `  ( A  -  ( p  /  q ) ) ) )  ->  ( A  =  ( p  /  q )  \/  ( ( a  / 
2 )  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) ) )
4140ralimdva 2623 . . . . 5  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  p  e.  ZZ )  ->  ( A. q  e.  NN  ( A  =  (
p  /  q )  \/  ( a  / 
( q ^ N
) )  <_  ( abs `  ( A  -  ( p  /  q
) ) ) )  ->  A. q  e.  NN  ( A  =  (
p  /  q )  \/  ( ( a  /  2 )  / 
( q ^ N
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) ) )
4241ralimdva 2623 . . . 4  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( a  /  ( q ^ N ) )  <_ 
( abs `  ( A  -  ( p  /  q ) ) ) )  ->  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( ( a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
43 oveq1 5867 . . . . . . . 8  |-  ( x  =  ( a  / 
2 )  ->  (
x  /  ( q ^ N ) )  =  ( ( a  /  2 )  / 
( q ^ N
) ) )
4443breq1d 4035 . . . . . . 7  |-  ( x  =  ( a  / 
2 )  ->  (
( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) )  <->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
4544orbi2d 682 . . . . . 6  |-  ( x  =  ( a  / 
2 )  ->  (
( A  =  ( p  /  q )  \/  ( x  / 
( q ^ N
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) )  <-> 
( A  =  ( p  /  q )  \/  ( ( a  /  2 )  / 
( q ^ N
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) ) )
46452ralbidv 2587 . . . . 5  |-  ( x  =  ( a  / 
2 )  ->  ( A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) )  <->  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( ( a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
4746rspcev 2886 . . . 4  |-  ( ( ( a  /  2
)  e.  RR+  /\  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( ( a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
488, 42, 47ee12an 1353 . . 3  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( a  /  ( q ^ N ) )  <_ 
( abs `  ( A  -  ( p  /  q ) ) ) )  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
4948rexlimdva 2669 . 2  |-  ( ph  ->  ( E. a  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( a  /  (
q ^ N ) )  <_  ( abs `  ( A  -  (
p  /  q ) ) ) )  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
506, 49mpd 14 1  |-  ( ph  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   RRcr 8738   0cc0 8739    < clt 8869    <_ cle 8870    - cmin 9039    / cdiv 9425   NNcn 9748   2c2 9797   ZZcz 10026   QQcq 10318   RR+crp 10356   ^cexp 11106   abscabs 11721  Polycply 19568  degcdgr 19571
This theorem is referenced by:  aaliou2  19722
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  ax-addf 8818  ax-mulf 8819
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-iin 3910  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-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  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-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  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-clim 11964  df-rlim 11965  df-sum 12161  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-mulg 14494  df-subg 14620  df-cntz 14795  df-cmn 15093  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-subrg 15545  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-cmp 17116  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-0p 19027  df-limc 19218  df-dv 19219  df-dvn 19220  df-cpn 19221  df-ply 19572  df-idp 19573  df-coe 19574  df-dgr 19575  df-quot 19673
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