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Definition df-dgraa 38231
Description: Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)
Assertion
Ref Expression
df-dgraa degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < ))
Distinct variable group:   𝑥,𝑑,𝑝

Detailed syntax breakdown of Definition df-dgraa
StepHypRef Expression
1 cdgraa 38229 . 2 class degAA
2 vx . . 3 setvar 𝑥
3 caa 24288 . . 3 class 𝔸
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1629 . . . . . . . . 9 class 𝑝
6 cdgr 24162 . . . . . . . . 9 class deg
75, 6cfv 6031 . . . . . . . 8 class (deg‘𝑝)
8 vd . . . . . . . . 9 setvar 𝑑
98cv 1629 . . . . . . . 8 class 𝑑
107, 9wceq 1630 . . . . . . 7 wff (deg‘𝑝) = 𝑑
112cv 1629 . . . . . . . . 9 class 𝑥
1211, 5cfv 6031 . . . . . . . 8 class (𝑝𝑥)
13 cc0 10137 . . . . . . . 8 class 0
1412, 13wceq 1630 . . . . . . 7 wff (𝑝𝑥) = 0
1510, 14wa 382 . . . . . 6 wff ((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)
16 cq 11990 . . . . . . . 8 class
17 cply 24159 . . . . . . . 8 class Poly
1816, 17cfv 6031 . . . . . . 7 class (Poly‘ℚ)
19 c0p 23655 . . . . . . . 8 class 0𝑝
2019csn 4314 . . . . . . 7 class {0𝑝}
2118, 20cdif 3718 . . . . . 6 class ((Poly‘ℚ) ∖ {0𝑝})
2215, 4, 21wrex 3061 . . . . 5 wff 𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)
23 cn 11221 . . . . 5 class
2422, 8, 23crab 3064 . . . 4 class {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}
25 cr 10136 . . . 4 class
26 clt 10275 . . . 4 class <
2724, 25, 26cinf 8502 . . 3 class inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < )
282, 3, 27cmpt 4861 . 2 class (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < ))
291, 28wceq 1630 1 wff degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < ))
Colors of variables: wff setvar class
This definition is referenced by:  dgraaval  38233  dgraaf  38236
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