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Theorem elqaa 20099
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 20093 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 )  \  { 0 p } ) ( 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 20093 . . 3  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
2 zssq 10506 . . . . . 6  |-  ZZ  C_  QQ
3 qsscn 10510 . . . . . 6  |-  QQ  C_  CC
4 plyss 19978 . . . . . 6  |-  ( ( ZZ  C_  QQ  /\  QQ  C_  CC )  ->  (Poly `  ZZ )  C_  (Poly `  QQ ) )
52, 3, 4mp2an 654 . . . . 5  |-  (Poly `  ZZ )  C_  (Poly `  QQ )
6 ssdif 3418 . . . . 5  |-  ( (Poly `  ZZ )  C_  (Poly `  QQ )  ->  (
(Poly `  ZZ )  \  { 0 p }
)  C_  ( (Poly `  QQ )  \  {
0 p } ) )
7 ssrexv 3344 . . . . 5  |-  ( ( (Poly `  ZZ )  \  { 0 p }
)  C_  ( (Poly `  QQ )  \  {
0 p } )  ->  ( E. f  e.  ( (Poly `  ZZ )  \  { 0 p } ) ( f `
 A )  =  0  ->  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
85, 6, 7mp2b 10 . . . 4  |-  ( E. f  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( f `  A
)  =  0  ->  E. f  e.  (
(Poly `  QQ )  \  { 0 p }
) ( f `  A )  =  0 )
98anim2i 553 . . 3  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  {
0 p } ) ( f `  A
)  =  0 )  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
101, 9sylbi 188 . 2  |-  ( A  e.  AA  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0 p } ) ( f `  A
)  =  0 ) )
11 simpll 731 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  CC )
12 simplr 732 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( f `
 A )  =  0 )  ->  f  e.  ( (Poly `  QQ )  \  { 0 p } ) )
13 simpr 448 . . . . . 6  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( f `
 A )  =  0 )  ->  (
f `  A )  =  0 )
14 eqid 2380 . . . . . 6  |-  (coeff `  f )  =  (coeff `  f )
15 fveq2 5661 . . . . . . . . . . . 12  |-  ( m  =  k  ->  (
(coeff `  f ) `  m )  =  ( (coeff `  f ) `  k ) )
1615oveq1d 6028 . . . . . . . . . . 11  |-  ( m  =  k  ->  (
( (coeff `  f
) `  m )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  j ) )
1716eleq1d 2446 . . . . . . . . . 10  |-  ( m  =  k  ->  (
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  j
)  e.  ZZ ) )
1817rabbidv 2884 . . . . . . . . 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 6021 . . . . . . . . . . 11  |-  ( j  =  n  ->  (
( (coeff `  f
) `  k )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  n ) )
2019eleq1d 2446 . . . . . . . . . 10  |-  ( j  =  n  ->  (
( ( (coeff `  f ) `  k
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  n
)  e.  ZZ ) )
2120cbvrabv 2891 . . . . . . . . 9  |-  { j  e.  NN  |  ( ( (coeff `  f
) `  k )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
2218, 21syl6eq 2428 . . . . . . . 8  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
)
2322supeq1d 7379 . . . . . . 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 4234 . . . . . 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 2380 . . . . . 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 20098 . . . . 5  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0 p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  AA )
2726ex 424 . . . 4  |-  ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0 p } ) )  ->  ( (
f `  A )  =  0  ->  A  e.  AA ) )
2827rexlimdva 2766 . . 3  |-  ( A  e.  CC  ->  ( E. f  e.  (
(Poly `  QQ )  \  { 0 p }
) ( f `  A )  =  0  ->  A  e.  AA ) )
2928imp 419 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0 p } ) ( f `  A
)  =  0 )  ->  A  e.  AA )
3010, 29impbii 181 1  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2643   {crab 2646    \ cdif 3253    C_ wss 3256   {csn 3750    e. cmpt 4200   `'ccnv 4810   ` cfv 5387  (class class class)co 6013   supcsup 7373   CCcc 8914   RRcr 8915   0cc0 8916    x. cmul 8921    < clt 9046   NNcn 9925   NN0cn0 10146   ZZcz 10207   QQcq 10499    seq cseq 11243   0 pc0p 19421  Polycply 19963  coeffccoe 19965  degcdgr 19966   AAcaa 20091
This theorem is referenced by:  qaa  20100  dgraalem  27012  dgraaub  27015  aaitgo  27029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203  df-sum 12400  df-0p 19422  df-ply 19967  df-coe 19969  df-dgr 19970  df-aa 20092
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