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) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))

Distinct variable group:   𝐴,𝑝

Proof of Theorem mpaaval

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . . 5 ⊢ (𝑎 = 𝐴 → (degAA‘𝑎) = (degAA‘𝐴))
2	1	eqeq2d 2740	. . . 4 ⊢ (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA‘𝑎) ↔ (deg‘𝑝) = (degAA‘𝐴)))
3		fveqeq2 6867	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0))
4		2fveq3 6863	. . . . 5 ⊢ (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA‘𝑎)) = ((coeff‘𝑝)‘(degAA‘𝐴)))
5	4	eqeq1d 2731	. . . 4 ⊢ (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA‘𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
6	2, 3, 5	3anbi123d 1438	. . 3 ⊢ (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))
7	6	riotabidv 7346	. 2 ⊢ (𝑎 = 𝐴 → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))
8		df-mpaa 43132	. 2 ⊢ minPolyAA = (𝑎 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)))
9		riotaex 7348	. 2 ⊢ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ V
10	7, 8, 9	fvmpt 6968	1 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  ℩crio 7343  0cc0 11068  1c1 11069  ℚcq 12907  Polycply 26089  coeffccoe 26091  degcdgr 26092  𝔸caa 26222  deg_AAcdgraa 43129  minPolyAAcmpaa 43130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-mpaa 43132

This theorem is referenced by:  mpaalem  43141