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Definition df-dgraa 40504
 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 40502 . 2 class degAA
2 vx . . 3 setvar 𝑥
3 caa 25023 . . 3 class 𝔸
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1537 . . . . . . . . 9 class 𝑝
6 cdgr 24897 . . . . . . . . 9 class deg
75, 6cfv 6340 . . . . . . . 8 class (deg‘𝑝)
8 vd . . . . . . . . 9 setvar 𝑑
98cv 1537 . . . . . . . 8 class 𝑑
107, 9wceq 1538 . . . . . . 7 wff (deg‘𝑝) = 𝑑
112cv 1537 . . . . . . . . 9 class 𝑥
1211, 5cfv 6340 . . . . . . . 8 class (𝑝𝑥)
13 cc0 10588 . . . . . . . 8 class 0
1412, 13wceq 1538 . . . . . . 7 wff (𝑝𝑥) = 0
1510, 14wa 399 . . . . . 6 wff ((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)
16 cq 12401 . . . . . . . 8 class
17 cply 24894 . . . . . . . 8 class Poly
1816, 17cfv 6340 . . . . . . 7 class (Poly‘ℚ)
19 c0p 24383 . . . . . . . 8 class 0𝑝
2019csn 4525 . . . . . . 7 class {0𝑝}
2118, 20cdif 3857 . . . . . 6 class ((Poly‘ℚ) ∖ {0𝑝})
2215, 4, 21wrex 3071 . . . . 5 wff 𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)
23 cn 11687 . . . . 5 class
2422, 8, 23crab 3074 . . . 4 class {𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}
25 cr 10587 . . . 4 class
26 clt 10726 . . . 4 class <
2724, 25, 26cinf 8951 . . 3 class inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < )
282, 3, 27cmpt 5116 . 2 class (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < ))
291, 28wceq 1538 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  40506  dgraaf  40509
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