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Theorem mpaaval 29513
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 5696 . . . . 5  |-  ( a  =  A  ->  (degAA `  a )  =  (degAA `  A ) )
21eqeq2d 2454 . . . 4  |-  ( a  =  A  ->  (
(deg `  p )  =  (degAA `  a )  <->  (deg `  p
)  =  (degAA `  A
) ) )
3 fveq2 5696 . . . . 5  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
43eqeq1d 2451 . . . 4  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
51fveq2d 5700 . . . . 5  |-  ( a  =  A  ->  (
(coeff `  p ) `  (degAA `  a ) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
65eqeq1d 2451 . . . 4  |-  ( a  =  A  ->  (
( (coeff `  p
) `  (degAA `  a
) )  =  1  <-> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
72, 4, 63anbi123d 1289 . . 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 6059 . 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 29505 . 2  |- minPolyAA  =  ( a  e.  AA  |->  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  a )  /\  ( p `  a
)  =  0  /\  ( (coeff `  p
) `  (degAA `  a
) )  =  1 ) ) )
10 riotaex 6061 . 2  |-  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  _V
118, 9, 10fvmpt 5779 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423   iota_crio 6056   0cc0 9287   1c1 9288   QQcq 10958  Polycply 21657  coeffccoe 21659  degcdgr 21660   AAcaa 21785  degAAcdgraa 29502  minPolyAAcmpaa 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-riota 6057  df-mpaa 29505
This theorem is referenced by:  mpaalem  29514
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