Theorem mpaaval 44246

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 6885	. . . . 5 ⊢ (𝑎 = 𝐴 → (degAA‘𝑎) = (degAA‘𝐴))
2	1	eqeq2d 2771	. . . 4 ⊢ (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA‘𝑎) ↔ (deg‘𝑝) = (degAA‘𝐴)))
3		fveqeq2 6894	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0))
4		2fveq3 6890	. . . . 5 ⊢ (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA‘𝑎)) = ((coeff‘𝑝)‘(degAA‘𝐴)))
5	4	eqeq1d 2762	. . . 4 ⊢ (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA‘𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))
6	2, 3, 5	3anbi123d 1464	. . 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 7379	. 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 44238	. 2 ⊢ minPolyAA = (𝑎 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)))
9		riotaex 7381	. 2 ⊢ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ V
10	7, 8, 9	fvmpt 6993	1 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  ℩crio 7376  0cc0 11203  1c1 11204  ℚcq 13078  Polycply 26507  coeffccoe 26509  degcdgr 26510  𝔸caa 26642  deg_AAcdgraa 44235  minPolyAAcmpaa 44236

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-riota 7377  df-mpaa 44238

This theorem is used by:  mpaalem  44247