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Theorem mpaaval 30733
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 5866 . . . . 5  |-  ( a  =  A  ->  (degAA `  a )  =  (degAA `  A ) )
21eqeq2d 2481 . . . 4  |-  ( a  =  A  ->  (
(deg `  p )  =  (degAA `  a )  <->  (deg `  p
)  =  (degAA `  A
) ) )
3 fveq2 5866 . . . . 5  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
43eqeq1d 2469 . . . 4  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
51fveq2d 5870 . . . . 5  |-  ( a  =  A  ->  (
(coeff `  p ) `  (degAA `  a ) )  =  ( (coeff `  p ) `  (degAA `  A ) ) )
65eqeq1d 2469 . . . 4  |-  ( a  =  A  ->  (
( (coeff `  p
) `  (degAA `  a
) )  =  1  <-> 
( (coeff `  p
) `  (degAA `  A
) )  =  1 ) )
72, 4, 63anbi123d 1299 . . 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 6247 . 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 30725 . 2  |- minPolyAA  =  ( a  e.  AA  |->  (
iota_ p  e.  (Poly `  QQ ) ( (deg
`  p )  =  (degAA `  a )  /\  ( p `  a
)  =  0  /\  ( (coeff `  p
) `  (degAA `  a
) )  =  1 ) ) )
10 riotaex 6249 . 2  |-  ( iota_ p  e.  (Poly `  QQ ) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0  /\  (
(coeff `  p ) `  (degAA `  A ) )  =  1 ) )  e.  _V
118, 9, 10fvmpt 5950 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 973    = wceq 1379    e. wcel 1767   ` cfv 5588   iota_crio 6244   0cc0 9492   1c1 9493   QQcq 11182  Polycply 22344  coeffccoe 22346  degcdgr 22347   AAcaa 22472  degAAcdgraa 30722  minPolyAAcmpaa 30723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6245  df-mpaa 30725
This theorem is referenced by:  mpaalem  30734
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