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Theorem elqaa 22803
Description: The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 22797 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
elqaa  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
Distinct variable group:    A, f

Proof of Theorem elqaa
Dummy variables  k  m  n  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elaa 22797 . . 3  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
2 zssq 11108 . . . . . 6  |-  ZZ  C_  QQ
3 qsscn 11112 . . . . . 6  |-  QQ  C_  CC
4 plyss 22681 . . . . . 6  |-  ( ( ZZ  C_  QQ  /\  QQ  C_  CC )  ->  (Poly `  ZZ )  C_  (Poly `  QQ ) )
52, 3, 4mp2an 670 . . . . 5  |-  (Poly `  ZZ )  C_  (Poly `  QQ )
6 ssdif 3553 . . . . 5  |-  ( (Poly `  ZZ )  C_  (Poly `  QQ )  ->  (
(Poly `  ZZ )  \  { 0p }
)  C_  ( (Poly `  QQ )  \  {
0p } ) )
7 ssrexv 3479 . . . . 5  |-  ( ( (Poly `  ZZ )  \  { 0p }
)  C_  ( (Poly `  QQ )  \  {
0p } )  ->  ( E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0  ->  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
85, 6, 7mp2b 10 . . . 4  |-  ( E. f  e.  ( (Poly `  ZZ )  \  {
0p } ) ( f `  A
)  =  0  ->  E. f  e.  (
(Poly `  QQ )  \  { 0p }
) ( f `  A )  =  0 )
98anim2i 567 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  {
0p } ) ( f `  A
)  =  0 )  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
101, 9sylbi 195 . 2  |-  ( A  e.  AA  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0p } ) ( f `  A
)  =  0 ) )
11 simpll 751 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  CC )
12 simplr 753 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  f  e.  ( (Poly `  QQ )  \  { 0p } ) )
13 simpr 459 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  (
f `  A )  =  0 )
14 eqid 2382 . . . 4  |-  (coeff `  f )  =  (coeff `  f )
15 fveq2 5774 . . . . . . . . . 10  |-  ( m  =  k  ->  (
(coeff `  f ) `  m )  =  ( (coeff `  f ) `  k ) )
1615oveq1d 6211 . . . . . . . . 9  |-  ( m  =  k  ->  (
( (coeff `  f
) `  m )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  j ) )
1716eleq1d 2451 . . . . . . . 8  |-  ( m  =  k  ->  (
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  j
)  e.  ZZ ) )
1817rabbidv 3026 . . . . . . 7  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { j  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  j
)  e.  ZZ }
)
19 oveq2 6204 . . . . . . . . 9  |-  ( j  =  n  ->  (
( (coeff `  f
) `  k )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  n ) )
2019eleq1d 2451 . . . . . . . 8  |-  ( j  =  n  ->  (
( ( (coeff `  f ) `  k
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  n
)  e.  ZZ ) )
2120cbvrabv 3033 . . . . . . 7  |-  { j  e.  NN  |  ( ( (coeff `  f
) `  k )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
2218, 21syl6eq 2439 . . . . . 6  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
)
2322supeq1d 7820 . . . . 5  |-  ( m  =  k  ->  sup ( { j  e.  NN  |  ( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  | 
( ( (coeff `  f ) `  k
)  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
2423cbvmptv 4458 . . . 4  |-  ( m  e.  NN0  |->  sup ( { j  e.  NN  |  ( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  `'  <  ) )  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( (coeff `  f ) `  k
)  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
25 eqid 2382 . . . 4  |-  (  seq 0 (  x.  , 
( m  e.  NN0  |->  sup ( { j  e.  NN  |  ( ( (coeff `  f ) `  m )  x.  j
)  e.  ZZ } ,  RR ,  `'  <  ) ) ) `  (deg `  f ) )  =  (  seq 0 (  x.  ,  ( m  e.  NN0  |->  sup ( { j  e.  NN  |  ( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  `'  <  ) ) ) `  (deg `  f ) )
2611, 12, 13, 14, 24, 25elqaalem3 22802 . . 3  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  AA )
2726r19.29an 2923 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0p } ) ( f `  A
)  =  0 )  ->  A  e.  AA )
2810, 27impbii 188 1  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736    \ cdif 3386    C_ wss 3389   {csn 3944    |-> cmpt 4425   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   supcsup 7815   CCcc 9401   RRcr 9402   0cc0 9403    x. cmul 9408    < clt 9539   NNcn 10452   NN0cn0 10712   ZZcz 10781   QQcq 11101    seqcseq 12010   0pc0p 22161  Polycply 22666  coeffccoe 22668  degcdgr 22669   AAcaa 22795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-0p 22162  df-ply 22670  df-coe 22672  df-dgr 22673  df-aa 22796
This theorem is referenced by:  qaa  22804  dgraalem  31262  dgraaub  31265  aaitgo  31279  aacllem  33550
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