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Theorem mpaaval 41664
Description: Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
mpaaval (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
Distinct variable group:   𝐴,𝑝

Proof of Theorem mpaaval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6878 . . . . 5 (𝑎 = 𝐴 → (degAA𝑎) = (degAA𝐴))
21eqeq2d 2742 . . . 4 (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA𝑎) ↔ (deg‘𝑝) = (degAA𝐴)))
3 fveqeq2 6887 . . . 4 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
4 2fveq3 6883 . . . . 5 (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA𝑎)) = ((coeff‘𝑝)‘(degAA𝐴)))
54eqeq1d 2733 . . . 4 (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
62, 3, 53anbi123d 1436 . . 3 (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
76riotabidv 7351 . 2 (𝑎 = 𝐴 → (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1)) = (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
8 df-mpaa 41656 . 2 minPolyAA = (𝑎 ∈ 𝔸 ↦ (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1)))
9 riotaex 7353 . 2 (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ V
107, 8, 9fvmpt 6984 1 (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6532  crio 7348  0cc0 11092  1c1 11093  cq 12914  Polycply 25627  coeffccoe 25629  degcdgr 25630  𝔸caa 25756  degAAcdgraa 41653  minPolyAAcmpaa 41654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-riota 7349  df-mpaa 41656
This theorem is referenced by:  mpaalem  41665
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