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Theorem aaliou 22712
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. This is Metamath 100 proof #18. (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 22711 . 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 11255 . . . . 5  |-  ( a  e.  RR+  ->  ( a  /  2 )  e.  RR+ )
87adantl 466 . . . 4  |-  ( (
ph  /\  a  e.  RR+ )  ->  ( a  /  2 )  e.  RR+ )
97ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR+ )
10 nnrp 11240 . . . . . . . . . . . 12  |-  ( q  e.  NN  ->  q  e.  RR+ )
1110ad2antll 728 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  q  e.  RR+ )
123nnzd 10975 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ZZ )
1312ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  N  e.  ZZ )
1411, 13rpexpcld 12315 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( q ^ N )  e.  RR+ )
159, 14rpdivcld 11284 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR+ )
1615rpred 11267 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR )
17 simplr 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR+ )
1817, 14rpdivcld 11284 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR+ )
1918rpred 11267 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR )
204adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  RR+ )  ->  A  e.  RR )
21 znq 11197 . . . . . . . . . . . 12  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  QQ )
22 qre 11198 . . . . . . . . . . . 12  |-  ( ( p  /  q )  e.  QQ  ->  (
p  /  q )  e.  RR )
2321, 22syl 16 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  RR )
24 resubcl 9888 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( p  /  q
)  e.  RR )  ->  ( A  -  ( p  /  q
) )  e.  RR )
2520, 23, 24syl2an 477 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  RR )
2625recnd 9625 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  CC )
2726abscld 13249 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( abs `  ( A  -  (
p  /  q ) ) )  e.  RR )
2816, 19, 273jca 1177 . . . . . . 7  |-  ( ( ( 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 11267 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR )
30 rpre 11237 . . . . . . . . . . 11  |-  ( a  e.  RR+  ->  a  e.  RR )
3130ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  a  e.  RR )
32 rphalflt 11257 . . . . . . . . . . 11  |-  ( a  e.  RR+  ->  ( a  /  2 )  < 
a )
3332ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  < 
a )
3429, 31, 14, 33ltdiv1dd 11320 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( a  /  (
q ^ N ) ) )
3534anim1i 568 . . . . . . . 8  |-  ( ( ( ( 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 434 . . . . . . 7  |-  ( ( ( 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 9679 . . . . . . 7  |-  ( ( ( ( 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 56 . . . . . 6  |-  ( ( ( 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 840 . . . . 5  |-  ( ( ( 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
) ) ) ) ) )
4039ralimdvva 2854 . . . 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 ) ) ) ) ) )
41 oveq1 6288 . . . . . . . 8  |-  ( x  =  ( a  / 
2 )  ->  (
x  /  ( q ^ N ) )  =  ( ( a  /  2 )  / 
( q ^ N
) ) )
4241breq1d 4447 . . . . . . 7  |-  ( x  =  ( a  / 
2 )  ->  (
( x  /  (
q ^ N ) )  <  ( abs `  ( A  -  (
p  /  q ) ) )  <->  ( (
a  /  2 )  /  ( q ^ N ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
4342orbi2d 701 . . . . . 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
) ) ) ) ) )
44432ralbidv 2887 . . . . 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 ) ) ) ) ) )
4544rspcev 3196 . . . 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 ) ) ) ) )
468, 40, 45syl6an 545 . . 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 ) ) ) ) ) )
4746rexlimdva 2935 . 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 ) ) ) ) ) )
486, 47mpd 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   NNcn 10543   2c2 10592   ZZcz 10871   QQcq 11193   RR+crp 11231   ^cexp 12148   abscabs 13049  Polycply 22559  degcdgr 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-grp 16036  df-minusg 16037  df-mulg 16039  df-subg 16177  df-cntz 16334  df-cmn 16779  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-subrg 17406  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-cmp 19865  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-0p 22055  df-limc 22248  df-dv 22249  df-dvn 22250  df-cpn 22251  df-ply 22563  df-idp 22564  df-coe 22565  df-dgr 22566  df-quot 22665
This theorem is referenced by:  aaliou2  22714
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