Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mpaaval Unicode version

Theorem mpaaval 27356
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 5525 . . . . 5  |-  ( a  =  A  ->  (degAA `  a )  =  (degAA `  A ) )
21eqeq2d 2294 . . . 4  |-  ( a  =  A  ->  (
(deg `  p )  =  (degAA `  a )  <->  (deg `  p
)  =  (degAA `  A
) ) )
3 fveq2 5525 . . . . 5  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
43eqeq1d 2291 . . . 4  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
51fveq2d 5529 . . . . 5  |-  ( a  =  A  ->  (
(coeff `  p ) `  (degAA `  a ) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
65eqeq1d 2291 . . . 4  |-  ( a  =  A  ->  (
( (coeff `  p
) `  (degAA `  a
) )  =  1  <-> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
72, 4, 63anbi123d 1252 . . 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 6306 . 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 27348 . 2  |- minPolyAA  =  ( a  e.  AA  |->  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  a )  /\  ( p `  a
)  =  0  /\  ( (coeff `  p
) `  (degAA `  a
) )  =  1 ) ) )
10 riotaex 6308 . 2  |-  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  _V
118, 9, 10fvmpt 5602 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 934    = wceq 1623    e. wcel 1684   ` cfv 5255   iota_crio 6297   0cc0 8737   1c1 8738   QQcq 10316  Polycply 19566  coeffccoe 19568  degcdgr 19569   AAcaa 19694  degAAcdgraa 27345  minPolyAAcmpaa 27346
This theorem is referenced by:  mpaalem  27357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-riota 6304  df-mpaa 27348
  Copyright terms: Public domain W3C validator