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Theorem aaliou 19550
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
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 19549 . 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 10257 . . . . 5  |-  ( a  e.  RR+  ->  ( a  /  2 )  e.  RR+ )
87adantl 454 . . . 4  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( a  /  2 )  e.  RR+ )
97ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR+ )
10 nnrp 10242 . . . . . . . . . . . . . 14  |-  ( q  e.  NN  ->  q  e.  RR+ )
1110ad2antll 712 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  q  e.  RR+ )
123nnzd 9995 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
1312ad2antrr 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  N  e.  ZZ )
1411, 13rpexpcld 11146 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( q ^ N )  e.  RR+ )
159, 14rpdivcld 10286 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR+ )
1615rpred 10269 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR )
17 simplr 734 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR+ )
1817, 14rpdivcld 10286 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR+ )
1918rpred 10269 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR )
204adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  RR+ )  ->  A  e.  RR )
21 znq 10199 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  QQ )
22 qre 10200 . . . . . . . . . . . . . 14  |-  ( ( p  /  q )  e.  QQ  ->  (
p  /  q )  e.  RR )
2321, 22syl 17 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  RR )
24 resubcl 8991 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( p  /  q
)  e.  RR )  ->  ( A  -  ( p  /  q
) )  e.  RR )
2520, 23, 24syl2an 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  RR )
2625recnd 8741 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  CC )
2726abscld 11795 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( abs `  ( A  -  (
p  /  q ) ) )  e.  RR )
2816, 19, 273jca 1137 . . . . . . . . 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 10269 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR )
30 rpre 10239 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  a  e.  RR )
3130ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR )
32 rphalflt 10259 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  ( a  /  2 )  < 
a )
3332ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  < 
a )
3429, 31, 14, 33ltdiv1dd 10322 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( a  /  (
q ^ N ) ) )
3534anim1i 554 . . . . . . . . . 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 425 . . . . . . . . 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 8793 . . . . . . . . 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 54 . . . . . . . 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 816 . . . . . . 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 632 . . . . . 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 2583 . . . . 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 2583 . . . 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 5717 . . . . . . . 8  |-  ( x  =  ( a  / 
2 )  ->  (
x  /  ( q ^ N ) )  =  ( ( a  /  2 )  / 
( q ^ N
) ) )
4443breq1d 3930 . . . . . . 7  |-  ( x  =  ( a  / 
2 )  ->  (
( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) )  <->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
4544orbi2d 685 . . . . . 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 2547 . . . . 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 ) ) ) ) ) )
4746rcla4ev 2821 . . . 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 1359 . . 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 2629 . 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 16 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 6    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   RRcr 8616   0cc0 8617    < clt 8747    <_ cle 8748    - cmin 8917    / cdiv 9303   NNcn 9626   2c2 9675   ZZcz 9903   QQcq 10195   RR+crp 10233   ^cexp 10982   abscabs 11596  Polycply 19398  degcdgr 19401
This theorem is referenced by:  aaliou2  19552
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-rlim 11840  df-sum 12036  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-mulg 14327  df-subg 14453  df-cntz 14628  df-cmn 14926  df-mgp 15161  df-ring 15175  df-cring 15176  df-ur 15177  df-subrg 15378  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-lp 16700  df-perf 16701  df-cn 16789  df-cnp 16790  df-haus 16875  df-cmp 16946  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cncf 18214  df-0p 18857  df-limc 19048  df-dv 19049  df-dvn 19050  df-cpn 19051  df-ply 19402  df-idp 19403  df-coe 19404  df-dgr 19405  df-quot 19503
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