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Definition df-dgraa 41512
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 = (π‘₯ ∈ 𝔸 ↦ inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)}, ℝ, < ))
Distinct variable group:   π‘₯,𝑑,𝑝

Detailed syntax breakdown of Definition df-dgraa
StepHypRef Expression
1 cdgraa 41510 . 2 class degAA
2 vx . . 3 setvar π‘₯
3 caa 25690 . . 3 class 𝔸
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1541 . . . . . . . . 9 class 𝑝
6 cdgr 25564 . . . . . . . . 9 class deg
75, 6cfv 6497 . . . . . . . 8 class (degβ€˜π‘)
8 vd . . . . . . . . 9 setvar 𝑑
98cv 1541 . . . . . . . 8 class 𝑑
107, 9wceq 1542 . . . . . . 7 wff (degβ€˜π‘) = 𝑑
112cv 1541 . . . . . . . . 9 class π‘₯
1211, 5cfv 6497 . . . . . . . 8 class (π‘β€˜π‘₯)
13 cc0 11056 . . . . . . . 8 class 0
1412, 13wceq 1542 . . . . . . 7 wff (π‘β€˜π‘₯) = 0
1510, 14wa 397 . . . . . 6 wff ((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)
16 cq 12878 . . . . . . . 8 class β„š
17 cply 25561 . . . . . . . 8 class Poly
1816, 17cfv 6497 . . . . . . 7 class (Polyβ€˜β„š)
19 c0p 25049 . . . . . . . 8 class 0𝑝
2019csn 4587 . . . . . . 7 class {0𝑝}
2118, 20cdif 3908 . . . . . 6 class ((Polyβ€˜β„š) βˆ– {0𝑝})
2215, 4, 21wrex 3070 . . . . 5 wff βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)
23 cn 12158 . . . . 5 class β„•
2422, 8, 23crab 3406 . . . 4 class {𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)}
25 cr 11055 . . . 4 class ℝ
26 clt 11194 . . . 4 class <
2724, 25, 26cinf 9382 . . 3 class inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)}, ℝ, < )
282, 3, 27cmpt 5189 . 2 class (π‘₯ ∈ 𝔸 ↦ inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)}, ℝ, < ))
291, 28wceq 1542 1 wff degAA = (π‘₯ ∈ 𝔸 ↦ inf({𝑑 ∈ β„• ∣ βˆƒπ‘ ∈ ((Polyβ€˜β„š) βˆ– {0𝑝})((degβ€˜π‘) = 𝑑 ∧ (π‘β€˜π‘₯) = 0)}, ℝ, < ))
Colors of variables: wff setvar class
This definition is referenced by:  dgraaval  41514  dgraaf  41517
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