Definition df-itgo 41255

Description: A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 41258. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use Monic. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion

Ref	Expression
df-itgo	⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})

Distinct variable group:   𝑥,𝑝,𝑠

Detailed syntax breakdown of Definition df-itgo

Step	Hyp	Ref	Expression
1		citgo 41253	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 10970	. . . 4 class ℂ
4	3	cpw 4547	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1539	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1539	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6479	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 10972	. . . . . . 7 class 0
11	9, 10	wceq 1540	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 25454	. . . . . . . . 9 class deg
13	8, 12	cfv 6479	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 25453	. . . . . . . . 9 class coeff
15	8, 14	cfv 6479	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6479	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 10973	. . . . . . 7 class 1
18	16, 17	wceq 1540	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 396	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1539	. . . . . 6 class 𝑠
21		cply 25451	. . . . . 6 class Poly
22	20, 21	cfv 6479	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3070	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3403	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5175	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1540	1 wff IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})

This definition is referenced by:  itgoval  41257  itgocn  41260