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Theorem aaliou 19645
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 19644 . 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 10310 . . . . 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 10295 . . . . . . . . . . . . . 14  |-  ( q  e.  NN  ->  q  e.  RR+ )
1110ad2antll 712 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  q  e.  RR+ )
123nnzd 10048 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
1312ad2antrr 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  N  e.  ZZ )
1411, 13rpexpcld 11199 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( q ^ N )  e.  RR+ )
159, 14rpdivcld 10339 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( (
a  /  2 )  /  ( q ^ N ) )  e.  RR+ )
1615rpred 10322 . . . . . . . . . 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 10339 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  ( q ^ N ) )  e.  RR+ )
1918rpred 10322 . . . . . . . . . 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 10252 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  q  e.  NN )  ->  ( p  /  q
)  e.  QQ )
22 qre 10253 . . . . . . . . . . . . . 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 9044 . . . . . . . . . . . . 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 8794 . . . . . . . . . . 11  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( A  -  ( p  / 
q ) )  e.  CC )
2726abscld 11848 . . . . . . . . . 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 10322 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR+ )  /\  (
p  e.  ZZ  /\  q  e.  NN )
)  ->  ( a  /  2 )  e.  RR )
30 rpre 10292 . . . . . . . . . . . . 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 10312 . . . . . . . . . . . . 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 10375 . . . . . . . . . . 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 8846 . . . . . . . . 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 2592 . . . . 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 2592 . . . 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 5764 . . . . . . . 8  |-  ( x  =  ( a  / 
2 )  ->  (
x  /  ( q ^ N ) )  =  ( ( a  /  2 )  / 
( q ^ N
) ) )
4443breq1d 3973 . . . . . . 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 2556 . . . . 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 2835 . . . 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 2638 . 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 2516   E.wrex 2517   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   RRcr 8669   0cc0 8670    < clt 8800    <_ cle 8801    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   ZZcz 9956   QQcq 10248   RR+crp 10286   ^cexp 11035   abscabs 11649  Polycply 19493  degcdgr 19496
This theorem is referenced by:  aaliou2  19647
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-rlim 11893  df-sum 12089  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-mulg 14419  df-subg 14545  df-cntz 14720  df-cmn 15018  df-mgp 15253  df-ring 15267  df-cring 15268  df-ur 15269  df-subrg 15470  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-cmp 17041  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-0p 18952  df-limc 19143  df-dv 19144  df-dvn 19145  df-cpn 19146  df-ply 19497  df-idp 19498  df-coe 19499  df-dgr 19500  df-quot 19598
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