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

Theorem mpaaval 35716
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 5881 . . . . 5  |-  ( a  =  A  ->  (degAA `  a )  =  (degAA `  A ) )
21eqeq2d 2443 . . . 4  |-  ( a  =  A  ->  (
(deg `  p )  =  (degAA `  a )  <->  (deg `  p
)  =  (degAA `  A
) ) )
3 fveq2 5881 . . . . 5  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
43eqeq1d 2431 . . . 4  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
51fveq2d 5885 . . . . 5  |-  ( a  =  A  ->  (
(coeff `  p ) `  (degAA `  a ) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
65eqeq1d 2431 . . . 4  |-  ( a  =  A  ->  (
( (coeff `  p
) `  (degAA `  a
) )  =  1  <-> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
72, 4, 63anbi123d 1335 . . 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 6269 . 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 35708 . 2  |- minPolyAA  =  ( a  e.  AA  |->  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  a )  /\  ( p `  a
)  =  0  /\  ( (coeff `  p
) `  (degAA `  a
) )  =  1 ) ) )
10 riotaex 6271 . 2  |-  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  _V
118, 9, 10fvmpt 5964 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 982    = wceq 1437    e. wcel 1870   ` cfv 5601   iota_crio 6266   0cc0 9538   1c1 9539   QQcq 11264  Polycply 23006  coeffccoe 23008  degcdgr 23009   AAcaa 23135  degAAcdgraa 35705  minPolyAAcmpaa 35706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-mpaa 35708
This theorem is referenced by:  mpaalem  35717
  Copyright terms: Public domain W3C validator