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Theorem aaliou2 21825
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 3558 . 2  |-  ( A  e.  ( AA  i^i  RR )  <->  ( A  e.  AA  /\  A  e.  RR ) )
2 elaa 21801 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. a  e.  ( (Poly `  ZZ )  \  { 0p } ) ( a `
 A )  =  0 ) )
3 eldifn 3498 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  ZZ )  \  { 0p } )  ->  -.  a  e.  { 0p } )
433ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  a  e.  { 0p } )
5 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  ( CC 
X.  { ( a `
 0 ) } ) )
6 fveq1 5709 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  ( CC  X.  { ( a ` 
0 ) } )  ->  ( a `  A )  =  ( ( CC  X.  {
( a `  0
) } ) `  A ) )
76adantl 466 . . . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( a `  A
)  =  0 )
9 simpl3 993 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  A  e.  RR )
109recnd 9431 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  A  e.  CC )
11 fvex 5720 . . . . . . . . . . . . . . . . . . 19  |-  ( a `
 0 )  e. 
_V
1211fvconst2 5952 . . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
( a `  0
)  =  0 )
1514sneqd 3908 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  ->  { ( a ` 
0 ) }  =  { 0 } )
1615xpeq2d 4883 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  ( CC 
X.  { 0 } ) )
18 df-0p 21167 . . . . . . . . . . . . 13  |-  0p  =  ( CC  X.  { 0 } )
1917, 18syl6eqr 2493 . . . . . . . . . . . 12  |-  ( ( ( a  e.  ( (Poly `  ZZ )  \  { 0p }
)  /\  ( a `  A )  =  0  /\  A  e.  RR )  /\  a  =  ( CC  X.  { ( a `  0 ) } ) )  -> 
a  =  0p )
20 elsn 3910 . . . . . . . . . . . 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 659 . . . . . . . . . 10  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  a  =  ( CC  X.  { ( a `  0 ) } ) )
23 eldifi 3497 . . . . . . . . . . . 12  |-  ( a  e.  ( (Poly `  ZZ )  \  { 0p } )  -> 
a  e.  (Poly `  ZZ ) )
24233ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  a  e.  (Poly `  ZZ ) )
25 0dgrb 21733 . . . . . . . . . . 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 301 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  -.  (deg `  a
)  =  0 )
28 dgrcl 21720 . . . . . . . . . . 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 10600 . . . . . . . . . 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 383 . . . . . . . . 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 2443 . . . . . . . . 9  |-  (deg `  a )  =  (deg
`  a )
35 simp3 990 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  A  e.  RR )
36 simp2 989 . . . . . . . . 9  |-  ( ( a  e.  ( (Poly `  ZZ )  \  {
0p } )  /\  ( a `  A )  =  0  /\  A  e.  RR )  ->  ( a `  A )  =  0 )
3734, 24, 33, 35, 36aaliou 21823 . . . . . . . 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 6118 . . . . . . . . . . . . . 14  |-  ( k  =  (deg `  a
)  ->  ( q ^ k )  =  ( q ^ (deg `  a ) ) )
3938oveq2d 6126 . . . . . . . . . . . . 13  |-  ( k  =  (deg `  a
)  ->  ( x  /  ( q ^
k ) )  =  ( x  /  (
q ^ (deg `  a ) ) ) )
4039breq1d 4321 . . . . . . . . . . . 12  |-  ( k  =  (deg `  a
)  ->  ( (
x  /  ( q ^ k ) )  <  ( abs `  ( A  -  ( p  /  q ) ) )  <->  ( x  / 
( q ^ (deg `  a ) ) )  <  ( abs `  ( A  -  ( p  /  q ) ) ) ) )
4140orbi2d 701 . . . . . . . . . . 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 2776 . . . . . . . . . 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 2755 . . . . . . . . 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 3092 . . . . . . . 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 661 . . . . . . 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 1186 . . . . . 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 2854 . . . . 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 466 . . . 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 429 . 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 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2734   E.wrex 2735    \ cdif 3344    i^i cin 3346   {csn 3896   class class class wbr 4311    X. cxp 4857   ` cfv 5437  (class class class)co 6110   CCcc 9299   RRcr 9300   0cc0 9301    < clt 9437    - cmin 9614    / cdiv 10012   NNcn 10341   NN0cn0 10598   ZZcz 10665   RR+crp 11010   ^cexp 11884   abscabs 12742   0pc0p 21166  Polycply 21671  degcdgr 21674   AAcaa 21799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379  ax-addf 9380  ax-mulf 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-supp 6710  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-ixp 7283  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fsupp 7640  df-fi 7680  df-sup 7710  df-oi 7743  df-card 8128  df-cda 8356  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ioo 11323  df-ico 11325  df-icc 11326  df-fz 11457  df-fzo 11568  df-fl 11661  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-rlim 12986  df-sum 13183  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-starv 14272  df-sca 14273  df-vsca 14274  df-ip 14275  df-tset 14276  df-ple 14277  df-ds 14279  df-unif 14280  df-hom 14281  df-cco 14282  df-rest 14380  df-topn 14381  df-0g 14399  df-gsum 14400  df-topgen 14401  df-pt 14402  df-prds 14405  df-xrs 14459  df-qtop 14464  df-imas 14465  df-xps 14467  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-submnd 15484  df-grp 15564  df-minusg 15565  df-mulg 15567  df-subg 15697  df-cntz 15854  df-cmn 16298  df-mgp 16611  df-ur 16623  df-rng 16666  df-cring 16667  df-subrg 16882  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-fbas 17833  df-fg 17834  df-cnfld 17838  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-cld 18642  df-ntr 18643  df-cls 18644  df-nei 18721  df-lp 18759  df-perf 18760  df-cn 18850  df-cnp 18851  df-haus 18938  df-cmp 19009  df-tx 19154  df-hmeo 19347  df-fil 19438  df-fm 19530  df-flim 19531  df-flf 19532  df-xms 19914  df-ms 19915  df-tms 19916  df-cncf 20473  df-0p 21167  df-limc 21360  df-dv 21361  df-dvn 21362  df-cpn 21363  df-ply 21675  df-idp 21676  df-coe 21677  df-dgr 21678  df-quot 21776  df-aa 21800
This theorem is referenced by:  aaliou2b  21826
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