Definition df-itgo 43191

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 43194. 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 43189	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11001	. . . 4 class ℂ
4	3	cpw 4550	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1540	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1540	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6481	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11003	. . . . . . 7 class 0
11	9, 10	wceq 1541	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26117	. . . . . . . . 9 class deg
13	8, 12	cfv 6481	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26116	. . . . . . . . 9 class coeff
15	8, 14	cfv 6481	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6481	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11004	. . . . . . 7 class 1
18	16, 17	wceq 1541	. . . . . 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 1540	. . . . . 6 class 𝑠
21		cply 26114	. . . . . 6 class Poly
22	20, 21	cfv 6481	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3056	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3395	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5172	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1541	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  43193  itgocn  43196