Metamath Proof Explorer


Definition df-mpaa

Description: Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA . (Contributed by Stefan O'Rear, 25-Nov-2014)

Ref Expression
Assertion df-mpaa minPolyAA = ( 𝑥 ∈ 𝔸 ↦ ( 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA𝑥 ) ∧ ( 𝑝𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cmpaa minPolyAA
1 vx 𝑥
2 caa 𝔸
3 vp 𝑝
4 cply Poly
5 cq
6 5 4 cfv ( Poly ‘ ℚ )
7 cdgr deg
8 3 cv 𝑝
9 8 7 cfv ( deg ‘ 𝑝 )
10 cdgraa degAA
11 1 cv 𝑥
12 11 10 cfv ( degAA𝑥 )
13 9 12 wceq ( deg ‘ 𝑝 ) = ( degAA𝑥 )
14 11 8 cfv ( 𝑝𝑥 )
15 cc0 0
16 14 15 wceq ( 𝑝𝑥 ) = 0
17 ccoe coeff
18 8 17 cfv ( coeff ‘ 𝑝 )
19 12 18 cfv ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) )
20 c1 1
21 19 20 wceq ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1
22 13 16 21 w3a ( ( deg ‘ 𝑝 ) = ( degAA𝑥 ) ∧ ( 𝑝𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1 )
23 22 3 6 crio ( 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA𝑥 ) ∧ ( 𝑝𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1 ) )
24 1 2 23 cmpt ( 𝑥 ∈ 𝔸 ↦ ( 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA𝑥 ) ∧ ( 𝑝𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1 ) ) )
25 0 24 wceq minPolyAA = ( 𝑥 ∈ 𝔸 ↦ ( 𝑝 ∈ ( Poly ‘ ℚ ) ( ( deg ‘ 𝑝 ) = ( degAA𝑥 ) ∧ ( 𝑝𝑥 ) = 0 ∧ ( ( coeff ‘ 𝑝 ) ‘ ( degAA𝑥 ) ) = 1 ) ) )