Definition df-itgo 43148

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 43151. 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 43146	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11066	. . . 4 class ℂ
4	3	cpw 4563	. . 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 6511	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11068	. . . . . . 7 class 0
11	9, 10	wceq 1540	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26092	. . . . . . . . 9 class deg
13	8, 12	cfv 6511	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26091	. . . . . . . . 9 class coeff
15	8, 14	cfv 6511	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6511	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11069	. . . . . . 7 class 1
18	16, 17	wceq 1540	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 395	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1539	. . . . . 6 class 𝑠
21		cply 26089	. . . . . 6 class Poly
22	20, 21	cfv 6511	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3053	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3405	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5188	. 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  43150  itgocn  43153