Theorem mpaaval 43609

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 6830	. . . . 5 ⊢ (𝑎 = 𝐴 → (degAA‘𝑎) = (degAA‘𝐴))
2	1	eqeq2d 2752	. . . 4 ⊢ (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA‘𝑎) ↔ (deg‘𝑝) = (degAA‘𝐴)))
3		fveqeq2 6839	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0))
4		2fveq3 6835	. . . . 5 ⊢ (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA‘𝑎)) = ((coeff‘𝑝)‘(degAA‘𝐴)))
5	4	eqeq1d 2743	. . . 4 ⊢ (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA‘𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
6	2, 3, 5	3anbi123d 1445	. . 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 7318	. 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 43601	. 2 ⊢ minPolyAA = (𝑎 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)))
9		riotaex 7320	. 2 ⊢ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ V
10	7, 8, 9	fvmpt 6938	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 1093   = wceq 1548   ∈ wcel 2121  ‘cfv 6488  ℩crio 7315  0cc0 11034  1c1 11035  ℚcq 12893  Polycply 26170  coeffccoe 26172  degcdgr 26173  𝔸caa 26301  deg_AAcdgraa 43598  minPolyAAcmpaa 43599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6444  df-fun 6490  df-fv 6496  df-riota 7316  df-mpaa 43601

This theorem is referenced by:  mpaalem  43610