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Theorem mpaaval 38038
 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 6229 . . . . 5 (𝑎 = 𝐴 → (degAA𝑎) = (degAA𝐴))
21eqeq2d 2661 . . . 4 (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA𝑎) ↔ (deg‘𝑝) = (degAA𝐴)))
3 fveq2 6229 . . . . 5 (𝑎 = 𝐴 → (𝑝𝑎) = (𝑝𝐴))
43eqeq1d 2653 . . . 4 (𝑎 = 𝐴 → ((𝑝𝑎) = 0 ↔ (𝑝𝐴) = 0))
51fveq2d 6233 . . . . 5 (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA𝑎)) = ((coeff‘𝑝)‘(degAA𝐴)))
65eqeq1d 2653 . . . 4 (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA𝐴)) = 1))
72, 4, 63anbi123d 1439 . . 3 (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))
87riotabidv 6653 . 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)))
9 df-mpaa 38030 . 2 minPolyAA = (𝑎 ∈ 𝔸 ↦ (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑎) ∧ (𝑝𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑎)) = 1)))
10 riotaex 6655 . 2 (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)) ∈ V
118, 9, 10fvmpt 6321 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 1054   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  ℩crio 6650  0cc0 9974  1c1 9975  ℚcq 11826  Polycply 23985  coeffccoe 23987  degcdgr 23988  𝔸caa 24114  degAAcdgraa 38027  minPolyAAcmpaa 38028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-mpaa 38030 This theorem is referenced by:  mpaalem  38039
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