Theorem mpaaval 40892

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 6756	. . . . 5 ⊢ (𝑎 = 𝐴 → (degAA‘𝑎) = (degAA‘𝐴))
2	1	eqeq2d 2749	. . . 4 ⊢ (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA‘𝑎) ↔ (deg‘𝑝) = (degAA‘𝐴)))
3		fveqeq2 6765	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0))
4		2fveq3 6761	. . . . 5 ⊢ (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA‘𝑎)) = ((coeff‘𝑝)‘(degAA‘𝐴)))
5	4	eqeq1d 2740	. . . 4 ⊢ (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA‘𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
6	2, 3, 5	3anbi123d 1434	. . 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 7214	. 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 40884	. 2 ⊢ minPolyAA = (𝑎 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)))
9		riotaex 7216	. 2 ⊢ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ V
10	7, 8, 9	fvmpt 6857	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 1085   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  ℩crio 7211  0cc0 10802  1c1 10803  ℚcq 12617  Polycply 25250  coeffccoe 25252  degcdgr 25253  𝔸caa 25379  deg_AAcdgraa 40881  minPolyAAcmpaa 40882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-riota 7212  df-mpaa 40884

This theorem is referenced by:  mpaalem  40893