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Theorem elqaa 21916
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 21910 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 21910 . . 3  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
2 zssq 11066 . . . . . 6  |-  ZZ  C_  QQ
3 qsscn 11070 . . . . . 6  |-  QQ  C_  CC
4 plyss 21795 . . . . . 6  |-  ( ( ZZ  C_  QQ  /\  QQ  C_  CC )  ->  (Poly `  ZZ )  C_  (Poly `  QQ ) )
52, 3, 4mp2an 672 . . . . 5  |-  (Poly `  ZZ )  C_  (Poly `  QQ )
6 ssdif 3594 . . . . 5  |-  ( (Poly `  ZZ )  C_  (Poly `  QQ )  ->  (
(Poly `  ZZ )  \  { 0p }
)  C_  ( (Poly `  QQ )  \  {
0p } ) )
7 ssrexv 3520 . . . . 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 569 . . 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 753 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  CC )
12 simplr 754 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  f  e.  ( (Poly `  QQ )  \  { 0p } ) )
13 simpr 461 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  (
f `  A )  =  0 )
14 eqid 2452 . . . . . 6  |-  (coeff `  f )  =  (coeff `  f )
15 fveq2 5794 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
(coeff `  f ) `  m )  =  ( (coeff `  f ) `  k ) )
1615oveq1d 6210 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( (coeff `  f
) `  m )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  j ) )
1716eleq1d 2521 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  j
)  e.  ZZ ) )
1817rabbidv 3064 . . . . . . . . 9  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { j  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  j
)  e.  ZZ }
)
19 oveq2 6203 . . . . . . . . . . 11  |-  ( j  =  n  ->  (
( (coeff `  f
) `  k )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  n ) )
2019eleq1d 2521 . . . . . . . . . 10  |-  ( j  =  n  ->  (
( ( (coeff `  f ) `  k
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  n
)  e.  ZZ ) )
2120cbvrabv 3071 . . . . . . . . 9  |-  { j  e.  NN  |  ( ( (coeff `  f
) `  k )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
2218, 21syl6eq 2509 . . . . . . . 8  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
)
2322supeq1d 7802 . . . . . . 7  |-  ( 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 4486 . . . . . 6  |-  ( 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 2452 . . . . . 6  |-  (  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 21915 . . . . 5  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  AA )
2726ex 434 . . . 4  |-  ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  ->  ( (
f `  A )  =  0  ->  A  e.  AA ) )
2827rexlimdva 2941 . . 3  |-  ( A  e.  CC  ->  ( E. f  e.  (
(Poly `  QQ )  \  { 0p }
) ( f `  A )  =  0  ->  A  e.  AA ) )
2928imp 429 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0p } ) ( f `  A
)  =  0 )  ->  A  e.  AA )
3010, 29impbii 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 369    = wceq 1370    e. wcel 1758   E.wrex 2797   {crab 2800    \ cdif 3428    C_ wss 3431   {csn 3980    |-> cmpt 4453   `'ccnv 4942   ` cfv 5521  (class class class)co 6195   supcsup 7796   CCcc 9386   RRcr 9387   0cc0 9388    x. cmul 9393    < clt 9524   NNcn 10428   NN0cn0 10685   ZZcz 10752   QQcq 11059    seqcseq 11918   0pc0p 21275  Polycply 21780  coeffccoe 21782  degcdgr 21783   AAcaa 21908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277  df-0p 21276  df-ply 21784  df-coe 21786  df-dgr 21787  df-aa 21909
This theorem is referenced by:  qaa  21917  dgraalem  29645  dgraaub  29648  aaitgo  29662
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