Theorem mpaaval 37721

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( a = A -> (degAA ` a ) = (degAA ` A ) )
2	1	eqeq2d 2632	. . . 4 |- ( a = A -> ( (deg ` p ) = (degAA ` a ) <-> (deg ` p ) = (degAA ` A ) ) )
3		fveq2 6191	. . . . 5 |- ( a = A -> ( p ` a ) = ( p ` A ) )
4	3	eqeq1d 2624	. . . 4 |- ( a = A -> ( ( p ` a ) = 0 <-> ( p ` A ) = 0 ) )
5	1	fveq2d 6195	. . . . 5 |- ( a = A -> ( (coeff ` p ) ` (degAA ` a ) ) = ( (coeff ` p ) ` (degAA ` A ) ) )
6	5	eqeq1d 2624	. . . 4 |- ( a = A -> ( ( (coeff ` p ) ` (degAA ` a ) ) = 1 <-> ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) )
7	2, 4, 6	3anbi123d 1399	. . 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 ) ) )
8	7	riotabidv 6613	. 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 37713	. 2 |- minPolyAA = ( a e. AA |-> ( iota_ p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` a ) /\ ( p ` a ) = 0 /\ ( (coeff ` p ) ` (degAA ` a ) ) = 1 ) ) )
10		riotaex 6615	. 2 |- ( iota_ p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) ) e. _V
11	8, 9, 10	fvmpt 6282	1 |- ( A e. AA -> (minPolyAA ` A ) = ( iota_ p e. (Poly ` QQ ) ( (deg ` p ) = (degAA ` A ) /\ ( p ` A ) = 0 /\ ( (coeff ` p ) ` (degAA ` A ) ) = 1 ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888   iota_crio 6610   0cc0 9936   1c1 9937   QQcq 11788  Polycply 23940  coeffccoe 23942  degcdgr 23943   AAcaa 24069  deg_AAcdgraa 37710  minPolyAAcmpaa 37711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-mpaa 37713

This theorem is referenced by:  mpaalem  37722