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Theorem mpaaval 40490
 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 6658 . . . . 5 (𝑎 = 𝐴 → (degAA𝑎) = (degAA𝐴))
21eqeq2d 2769 . . . 4 (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA𝑎) ↔ (deg‘𝑝) = (degAA𝐴)))
3 fveqeq2 6667 . . . 4 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
4 2fveq3 6663 . . . . 5 (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA𝑎)) = ((coeff‘𝑝)‘(degAA𝐴)))
54eqeq1d 2760 . . . 4 (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
62, 3, 53anbi123d 1433 . . 3 (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
76riotabidv 7110 . 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 40482 . 2 minPolyAA = (𝑎 ∈ 𝔸 ↦ (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1)))
9 riotaex 7112 . 2 (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ V
107, 8, 9fvmpt 6759 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 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  ℩crio 7107  0cc0 10575  1c1 10576  ℚcq 12388  Polycply 24880  coeffccoe 24882  degcdgr 24883  𝔸caa 25009  degAAcdgraa 40479  minPolyAAcmpaa 40480 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-riota 7108  df-mpaa 40482 This theorem is referenced by:  mpaalem  40491
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