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Theorem aaliou 20257
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 20256 . 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 10638 . . . . 5  |-  ( a  e.  RR+  ->  ( a  /  2 )  e.  RR+ )
87adantl 454 . . . 4  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( a  /  2 )  e.  RR+ )
97ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR+ )
10 nnrp 10623 . . . . . . . . . . . . . 14  |-  ( q  e.  NN  ->  q  e.  RR+ )
1110ad2antll 711 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  q  e.  RR+ )
123nnzd 10376 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
1312ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  N  e.  ZZ )
1411, 13rpexpcld 11548 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( q ^ N )  e.  RR+ )
159, 14rpdivcld 10667 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR+ )
1615rpred 10650 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR )
17 simplr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR+ )
1817, 14rpdivcld 10667 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR+ )
1918rpred 10650 . . . . . . . . . 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 10580 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  QQ )
22 qre 10581 . . . . . . . . . . . . . 14  |-  ( ( p  /  q )  e.  QQ  ->  (
p  /  q )  e.  RR )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  RR )
24 resubcl 9367 . . . . . . . . . . . . 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 9116 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  CC )
2726abscld 12240 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( abs `  ( A  -  (
p  /  q ) ) )  e.  RR )
2816, 19, 273jca 1135 . . . . . . . . 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 10650 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR )
30 rpre 10620 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  a  e.  RR )
3130ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR )
32 rphalflt 10640 . . . . . . . . . . . . 13  |-  ( a  e.  RR+  ->  ( a  /  2 )  < 
a )
3332ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  < 
a )
3429, 31, 14, 33ltdiv1dd 10703 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( a  /  (
q ^ N ) ) )
3534anim1i 553 . . . . . . . . . 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 9168 . . . . . . . . 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 55 . . . . . . . 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 815 . . . . . . 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 631 . . . . . 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 2786 . . . . 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 2786 . . . 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 6090 . . . . . . . 8  |-  ( x  =  ( a  / 
2 )  ->  (
x  /  ( q ^ N ) )  =  ( ( a  /  2 )  / 
( q ^ N
) ) )
4443breq1d 4224 . . . . . . 7  |-  ( x  =  ( a  / 
2 )  ->  (
( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) )  <->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
4544orbi2d 684 . . . . . 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 2749 . . . . 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 3054 . . . 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 1373 . . 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 2832 . 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 15 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 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   RRcr 8991   0cc0 8992    < clt 9122    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   2c2 10051   ZZcz 10284   QQcq 10576   RR+crp 10614   ^cexp 11384   abscabs 12041  Polycply 20105  degcdgr 20108
This theorem is referenced by:  aaliou2  20259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-mulg 14817  df-subg 14943  df-cntz 15118  df-cmn 15416  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-subrg 15868  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-0p 19564  df-limc 19755  df-dv 19756  df-dvn 19757  df-cpn 19758  df-ply 20109  df-idp 20110  df-coe 20111  df-dgr 20112  df-quot 20210
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