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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  cq 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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