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Theorem mpaaval 43140
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 6907 . . . . 5 (𝑎 = 𝐴 → (degAA𝑎) = (degAA𝐴))
21eqeq2d 2746 . . . 4 (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA𝑎) ↔ (deg‘𝑝) = (degAA𝐴)))
3 fveqeq2 6916 . . . 4 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
4 2fveq3 6912 . . . . 5 (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA𝑎)) = ((coeff‘𝑝)‘(degAA𝐴)))
54eqeq1d 2737 . . . 4 (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
62, 3, 53anbi123d 1435 . . 3 (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
76riotabidv 7390 . 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 43132 . 2 minPolyAA = (𝑎 ∈ 𝔸 ↦ (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1)))
9 riotaex 7392 . 2 (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ V
107, 8, 9fvmpt 7016 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 1086   = wceq 1537  wcel 2106  cfv 6563  crio 7387  0cc0 11153  1c1 11154  cq 12988  Polycply 26238  coeffccoe 26240  degcdgr 26241  𝔸caa 26371  degAAcdgraa 43129  minPolyAAcmpaa 43130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-mpaa 43132
This theorem is referenced by:  mpaalem  43141
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