Definition df-itgo 42725

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 42728. 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 42723	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11138	. . . 4 class ℂ
4	3	cpw 4604	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1532	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1532	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6549	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11140	. . . . . . 7 class 0
11	9, 10	wceq 1533	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26166	. . . . . . . . 9 class deg
13	8, 12	cfv 6549	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26165	. . . . . . . . 9 class coeff
15	8, 14	cfv 6549	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6549	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11141	. . . . . . 7 class 1
18	16, 17	wceq 1533	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 394	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1532	. . . . . 6 class 𝑠
21		cply 26163	. . . . . 6 class Poly
22	20, 21	cfv 6549	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3059	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3418	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5232	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1533	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  42727  itgocn  42730