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Theorem mpaaval 27335
Description: Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaval  |-  ( A  e.  AA  ->  (minPolyAA `  A )  =  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) ) )
Distinct variable group:    A, p

Proof of Theorem mpaaval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . . 5  |-  ( a  =  A  ->  (degAA `  a )  =  (degAA `  A ) )
21eqeq2d 2449 . . . 4  |-  ( a  =  A  ->  (
(deg `  p )  =  (degAA `  a )  <->  (deg `  p
)  =  (degAA `  A
) ) )
3 fveq2 5730 . . . . 5  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
43eqeq1d 2446 . . . 4  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
51fveq2d 5734 . . . . 5  |-  ( a  =  A  ->  (
(coeff `  p ) `  (degAA `  a ) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
65eqeq1d 2446 . . . 4  |-  ( a  =  A  ->  (
( (coeff `  p
) `  (degAA `  a
) )  =  1  <-> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
72, 4, 63anbi123d 1255 . . 3  |-  ( a  =  A  ->  (
( (deg `  p
)  =  (degAA `  a
)  /\  ( p `  a )  =  0  /\  ( (coeff `  p ) `  (degAA `  a ) )  =  1 )  <->  ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) ) )
87riotabidv 6553 . 2  |-  ( a  =  A  ->  ( iota_ 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 ) ) )
9 df-mpaa 27327 . 2  |- minPolyAA  =  ( a  e.  AA  |->  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  a )  /\  ( p `  a
)  =  0  /\  ( (coeff `  p
) `  (degAA `  a
) )  =  1 ) ) )
10 riotaex 6555 . 2  |-  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  _V
118, 9, 10fvmpt 5808 1  |-  ( A  e.  AA  ->  (minPolyAA `  A )  =  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0  /\  ( (coeff `  p
) `  (degAA `  A
) )  =  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456   iota_crio 6544   0cc0 8992   1c1 8993   QQcq 10576  Polycply 20105  coeffccoe 20107  degcdgr 20108   AAcaa 20233  degAAcdgraa 27324  minPolyAAcmpaa 27325
This theorem is referenced by:  mpaalem  27336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-riota 6551  df-mpaa 27327
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