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Theorem mpaalem 27325
Description: Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaalem  |-  ( A  e.  AA  ->  (
(minPolyAA `  A )  e.  (Poly `  QQ )  /\  ( (deg `  (minPolyAA `  A ) )  =  (degAA `  A )  /\  ( (minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `
 (degAA `  A ) )  =  1 ) ) )

Proof of Theorem mpaalem
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 mpaaval 27324 . . 3  |-  ( A  e.  AA  ->  (minPolyAA `  A )  =  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) ) )
2 mpaaeu 27323 . . . 4  |-  ( A  e.  AA  ->  E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )
3 riotacl2 6555 . . . 4  |-  ( E! p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 )  -> 
( iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )  e.  {
p  e.  (Poly `  QQ )  |  (
(deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) } )
42, 3syl 16 . . 3  |-  ( A  e.  AA  ->  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  { p  e.  (Poly `  QQ )  |  ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) } )
51, 4eqeltrd 2509 . 2  |-  ( A  e.  AA  ->  (minPolyAA `  A )  e.  {
p  e.  (Poly `  QQ )  |  (
(deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) } )
6 fveq2 5720 . . . . 5  |-  ( p  =  (minPolyAA `  A
)  ->  (deg `  p
)  =  (deg `  (minPolyAA `  A ) ) )
76eqeq1d 2443 . . . 4  |-  ( p  =  (minPolyAA `  A
)  ->  ( (deg `  p )  =  (degAA `  A )  <->  (deg `  (minPolyAA `  A ) )  =  (degAA `  A ) ) )
8 fveq1 5719 . . . . 5  |-  ( p  =  (minPolyAA `  A
)  ->  ( p `  A )  =  ( (minPolyAA `  A ) `  A ) )
98eqeq1d 2443 . . . 4  |-  ( p  =  (minPolyAA `  A
)  ->  ( (
p `  A )  =  0  <->  ( (minPolyAA `  A ) `  A
)  =  0 ) )
10 fveq2 5720 . . . . . 6  |-  ( p  =  (minPolyAA `  A
)  ->  (coeff `  p
)  =  (coeff `  (minPolyAA `  A ) ) )
1110fveq1d 5722 . . . . 5  |-  ( p  =  (minPolyAA `  A
)  ->  ( (coeff `  p ) `  (degAA `  A ) )  =  ( (coeff `  (minPolyAA `  A ) ) `  (degAA `  A ) ) )
1211eqeq1d 2443 . . . 4  |-  ( p  =  (minPolyAA `  A
)  ->  ( (
(coeff `  p ) `  (degAA `  A ) )  =  1  <->  ( (coeff `  (minPolyAA `  A )
) `  (degAA `  A
) )  =  1 ) )
137, 9, 123anbi123d 1254 . . 3  |-  ( p  =  (minPolyAA `  A
)  ->  ( (
(deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 )  <->  ( (deg `  (minPolyAA `  A ) )  =  (degAA `  A )  /\  ( (minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `
 (degAA `  A ) )  =  1 ) ) )
1413elrab 3084 . 2  |-  ( (minPolyAA `  A )  e.  {
p  e.  (Poly `  QQ )  |  (
(deg `  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) }  <->  ( (minPolyAA `  A )  e.  (Poly `  QQ )  /\  (
(deg `  (minPolyAA `  A
) )  =  (degAA `  A )  /\  (
(minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `
 (degAA `  A ) )  =  1 ) ) )
155, 14sylib 189 1  |-  ( A  e.  AA  ->  (
(minPolyAA `  A )  e.  (Poly `  QQ )  /\  ( (deg `  (minPolyAA `  A ) )  =  (degAA `  A )  /\  ( (minPolyAA `  A ) `  A )  =  0  /\  ( (coeff `  (minPolyAA `  A ) ) `
 (degAA `  A ) )  =  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E!wreu 2699   {crab 2701   ` cfv 5446   iota_crio 6534   0cc0 8982   1c1 8983   QQcq 10566  Polycply 20095  coeffccoe 20097  degcdgr 20098   AAcaa 20223  degAAcdgraa 27313  minPolyAAcmpaa 27314
This theorem is referenced by:  mpaacl  27326  mpaadgr  27327  mpaaroot  27328  mpaamn  27329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102  df-aa 20224  df-dgraa 27315  df-mpaa 27316
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