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Theorem aaliou2 22902
Description: Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
aaliou2  |-  ( A  e.  ( AA  i^i  RR )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
Distinct variable group:    A, k, x, p, q

Proof of Theorem aaliou2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elin 3673 . 2  |-  ( A  e.  ( AA  i^i  RR )  <->  ( A  e.  AA  /\  A  e.  RR ) )
2 elaa 22878 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. a  e.  ( (Poly `  ZZ )  \  { 0p } ) ( a `
 A )  =  0 ) )
3 eldifn 3613 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  -.  a  e.  { 0p } )
433ad2ant1 1015 . . . . . . . . . . 11  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  a  e.  { 0p } )
5 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  ( CC 
X.  { ( a `
 0 ) } ) )
6 fveq1 5847 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  ( CC  X.  { ( a ` 
0 ) } )  ->  ( a `  A )  =  ( ( CC  X.  {
( a `  0
) } ) `  A ) )
76adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( a `  A
)  =  ( ( CC  X.  { ( a `  0 ) } ) `  A
) )
8 simpl2 998 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( a `  A
)  =  0 )
9 simpl3 999 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  A  e.  RR )
109recnd 9611 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  A  e.  CC )
11 fvex 5858 . . . . . . . . . . . . . . . . . . 19  |-  ( a `
 0 )  e. 
_V
1211fvconst2 6103 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  CC  ->  (
( CC  X.  {
( a `  0
) } ) `  A )  =  ( a `  0 ) )
1310, 12syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( ( CC  X.  { ( a ` 
0 ) } ) `
 A )  =  ( a `  0
) )
147, 8, 133eqtr3rd 2504 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( a `  0
)  =  0 )
1514sneqd 4028 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  { ( a ` 
0 ) }  =  { 0 } )
1615xpeq2d 5012 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( CC  X.  {
( a `  0
) } )  =  ( CC  X.  {
0 } ) )
175, 16eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  ( CC 
X.  { 0 } ) )
18 df-0p 22243 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
1917, 18syl6eqr 2513 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  0p )
20 elsn 4030 . . . . . . . . . . . 12  |-  ( a  e.  { 0p }  <->  a  =  0p )
2119, 20sylibr 212 . . . . . . . . . . 11  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  e.  { 0p } )
224, 21mtand 657 . . . . . . . . . 10  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  a  =  ( CC  X.  { ( a `  0 ) } ) )
23 eldifi 3612 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
a  e.  (Poly `  ZZ ) )
24233ad2ant1 1015 . . . . . . . . . . 11  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  a  e.  (Poly `  ZZ ) )
25 0dgrb 22809 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  ZZ )  ->  ( (deg `  a )  =  0  <-> 
a  =  ( CC 
X.  { ( a `
 0 ) } ) ) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  ( (deg `  a )  =  0  <-> 
a  =  ( CC 
X.  { ( a `
 0 ) } ) ) )
2722, 26mtbird 299 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  (deg `  a
)  =  0 )
28 dgrcl 22796 . . . . . . . . . . 11  |-  ( a  e.  (Poly `  ZZ )  ->  (deg `  a
)  e.  NN0 )
2924, 28syl 16 . . . . . . . . . 10  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  (deg `  a
)  e.  NN0 )
30 elnn0 10793 . . . . . . . . . 10  |-  ( (deg
`  a )  e. 
NN0 
<->  ( (deg `  a
)  e.  NN  \/  (deg `  a )  =  0 ) )
3129, 30sylib 196 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  ( (deg `  a )  e.  NN  \/  (deg `  a )  =  0 ) )
32 orel2 381 . . . . . . . . 9  |-  ( -.  (deg `  a )  =  0  ->  (
( (deg `  a
)  e.  NN  \/  (deg `  a )  =  0 )  ->  (deg `  a )  e.  NN ) )
3327, 31, 32sylc 60 . . . . . . . 8  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  (deg `  a
)  e.  NN )
34 eqid 2454 . . . . . . . . 9  |-  (deg `  a )  =  (deg
`  a )
35 simp3 996 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  A  e.  RR )
36 simp2 995 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  ( a `  A )  =  0 )
3734, 24, 33, 35, 36aaliou 22900 . . . . . . . 8  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ (deg `  a ) ) )  <  ( abs `  ( A  -  ( p  /  q ) ) ) ) )
38 oveq2 6278 . . . . . . . . . . . . . 14  |-  ( k  =  (deg `  a
)  ->  ( q ^ k )  =  ( q ^ (deg `  a ) ) )
3938oveq2d 6286 . . . . . . . . . . . . 13  |-  ( k  =  (deg `  a
)  ->  ( x  /  ( q ^
k ) )  =  ( x  /  (
q ^ (deg `  a ) ) ) )
4039breq1d 4449 . . . . . . . . . . . 12  |-  ( k  =  (deg `  a
)  ->  ( (
x  /  ( q ^ k ) )  <  ( abs `  ( A  -  ( p  /  q ) ) )  <->  ( x  / 
( q ^ (deg `  a ) ) )  <  ( abs `  ( A  -  ( p  /  q ) ) ) ) )
4140orbi2d 699 . . . . . . . . . . 11  |-  ( k  =  (deg `  a
)  ->  ( ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) )  <->  ( A  =  ( p  / 
q )  \/  (
x  /  ( q ^ (deg `  a
) ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
42412ralbidv 2898 . . . . . . . . . 10  |-  ( k  =  (deg `  a
)  ->  ( A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) )  <->  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
(deg `  a )
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) ) )
4342rexbidv 2965 . . . . . . . . 9  |-  ( k  =  (deg `  a
)  ->  ( E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) )  <->  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
(deg `  a )
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) ) )
4443rspcev 3207 . . . . . . . 8  |-  ( ( (deg `  a )  e.  NN  /\  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
(deg `  a )
) )  <  ( abs `  ( A  -  ( p  /  q
) ) ) ) )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
4533, 37, 44syl2anc 659 . . . . . . 7  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
46453exp 1193 . . . . . 6  |-  ( a  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
( ( a `  A )  =  0  ->  ( A  e.  RR  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) ) ) )
4746rexlimiv 2940 . . . . 5  |-  ( E. a  e.  ( (Poly `  ZZ )  \  {
0p } ) ( a `  A
)  =  0  -> 
( A  e.  RR  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
4847adantl 464 . . . 4  |-  ( ( A  e.  CC  /\  E. a  e.  ( (Poly `  ZZ )  \  {
0p } ) ( a `  A
)  =  0 )  ->  ( A  e.  RR  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) ) )
492, 48sylbi 195 . . 3  |-  ( A  e.  AA  ->  ( A  e.  RR  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) ) )
5049imp 427 . 2  |-  ( ( A  e.  AA  /\  A  e.  RR )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q
)  \/  ( x  /  ( q ^
k ) )  < 
( abs `  ( A  -  ( p  /  q ) ) ) ) )
511, 50sylbi 195 1  |-  ( A  e.  ( AA  i^i  RR )  ->  E. k  e.  NN  E. x  e.  RR+  A. p  e.  ZZ  A. q  e.  NN  ( A  =  ( p  /  q )  \/  ( x  /  (
q ^ k ) )  <  ( abs `  ( A  -  (
p  /  q ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    \ cdif 3458    i^i cin 3460   {csn 4016   class class class wbr 4439    X. cxp 4986   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    < clt 9617    - cmin 9796    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   RR+crp 11221   ^cexp 12148   abscabs 13149   0pc0p 22242  Polycply 22747  degcdgr 22750   AAcaa 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-mulg 16259  df-subg 16397  df-cntz 16554  df-cmn 16999  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-subrg 17622  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-cmp 20054  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-0p 22243  df-limc 22436  df-dv 22437  df-dvn 22438  df-cpn 22439  df-ply 22751  df-idp 22752  df-coe 22753  df-dgr 22754  df-quot 22853  df-aa 22877
This theorem is referenced by:  aaliou2b  22903
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