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Theorem elqaa 23334
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 23325 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof shortened by AV, 3-Oct-2020.)
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 23325 . . 3  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  ZZ )  \  { 0p } ) ( f `
 A )  =  0 ) )
2 zssq 11305 . . . . . 6  |-  ZZ  C_  QQ
3 qsscn 11309 . . . . . 6  |-  QQ  C_  CC
4 plyss 23209 . . . . . 6  |-  ( ( ZZ  C_  QQ  /\  QQ  C_  CC )  ->  (Poly `  ZZ )  C_  (Poly `  QQ ) )
52, 3, 4mp2an 683 . . . . 5  |-  (Poly `  ZZ )  C_  (Poly `  QQ )
6 ssdif 3580 . . . . 5  |-  ( (Poly `  ZZ )  C_  (Poly `  QQ )  ->  (
(Poly `  ZZ )  \  { 0p }
)  C_  ( (Poly `  QQ )  \  {
0p } ) )
7 ssrexv 3506 . . . . 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 577 . . 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 200 . 2  |-  ( A  e.  AA  ->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0p } ) ( f `  A
)  =  0 ) )
11 simpll 765 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  CC )
12 simplr 767 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  f  e.  ( (Poly `  QQ )  \  { 0p } ) )
13 simpr 467 . . . 4  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  (
f `  A )  =  0 )
14 eqid 2462 . . . 4  |-  (coeff `  f )  =  (coeff `  f )
15 fveq2 5892 . . . . . . . . . 10  |-  ( m  =  k  ->  (
(coeff `  f ) `  m )  =  ( (coeff `  f ) `  k ) )
1615oveq1d 6335 . . . . . . . . 9  |-  ( m  =  k  ->  (
( (coeff `  f
) `  m )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  j ) )
1716eleq1d 2524 . . . . . . . 8  |-  ( m  =  k  ->  (
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  j
)  e.  ZZ ) )
1817rabbidv 3048 . . . . . . 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 6328 . . . . . . . . 9  |-  ( j  =  n  ->  (
( (coeff `  f
) `  k )  x.  j )  =  ( ( (coeff `  f
) `  k )  x.  n ) )
2019eleq1d 2524 . . . . . . . 8  |-  ( j  =  n  ->  (
( ( (coeff `  f ) `  k
)  x.  j )  e.  ZZ  <->  ( (
(coeff `  f ) `  k )  x.  n
)  e.  ZZ ) )
2120cbvrabv 3056 . . . . . . 7  |-  { j  e.  NN  |  ( ( (coeff `  f
) `  k )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
2218, 21syl6eq 2512 . . . . . 6  |-  ( m  =  k  ->  { j  e.  NN  |  ( ( (coeff `  f
) `  m )  x.  j )  e.  ZZ }  =  { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ }
)
2322infeq1d 8024 . . . . 5  |-  ( m  =  k  -> inf ( { j  e.  NN  | 
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  <  )  = inf ( { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ } ,  RR ,  <  )
)
2423cbvmptv 4511 . . . 4  |-  ( m  e.  NN0  |-> inf ( { j  e.  NN  | 
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  <  ) )  =  ( k  e. 
NN0  |-> inf ( { n  e.  NN  |  ( ( (coeff `  f ) `  k )  x.  n
)  e.  ZZ } ,  RR ,  <  )
)
25 eqid 2462 . . . 4  |-  (  seq 0 (  x.  , 
( m  e.  NN0  |-> inf ( { j  e.  NN  |  ( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  <  ) ) ) `  (deg `  f ) )  =  (  seq 0 (  x.  ,  ( m  e.  NN0  |-> inf ( { j  e.  NN  | 
( ( (coeff `  f ) `  m
)  x.  j )  e.  ZZ } ,  RR ,  <  ) ) ) `  (deg `  f ) )
2611, 12, 13, 14, 24, 25elqaalem3 23330 . . 3  |-  ( ( ( A  e.  CC  /\  f  e.  ( (Poly `  QQ )  \  {
0p } ) )  /\  ( f `
 A )  =  0 )  ->  A  e.  AA )
2726r19.29an 2943 . 2  |-  ( ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  {
0p } ) ( f `  A
)  =  0 )  ->  A  e.  AA )
2810, 27impbii 192 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750   {crab 2753    \ cdif 3413    C_ wss 3416   {csn 3980    |-> cmpt 4477   ` cfv 5605  (class class class)co 6320  infcinf 7986   CCcc 9568   RRcr 9569   0cc0 9570    x. cmul 9575    < clt 9706   NNcn 10642   NN0cn0 10903   ZZcz 10971   QQcq 11298    seqcseq 12251   0pc0p 22683  Polycply 23194  coeffccoe 23196  degcdgr 23197   AAcaa 23323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648  ax-addf 9649
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-inf 7988  df-oi 8056  df-card 8404  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-q 11299  df-rp 11337  df-fz 11820  df-fzo 11953  df-fl 12066  df-mod 12135  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-clim 13607  df-rlim 13608  df-sum 13808  df-0p 22684  df-ply 23198  df-coe 23200  df-dgr 23201  df-aa 23324
This theorem is referenced by:  qaa  23335  dgraalem  36053  dgraalemOLD  36054  dgraaub  36059  dgraaubOLD  36060  aaitgo  36074  aacllem  40909
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