Definition df-itgo 43108

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 43111. 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 43106	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11119	. . . 4 class ℂ
4	3	cpw 4573	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1538	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1538	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6527	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11121	. . . . . . 7 class 0
11	9, 10	wceq 1539	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26129	. . . . . . . . 9 class deg
13	8, 12	cfv 6527	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26128	. . . . . . . . 9 class coeff
15	8, 14	cfv 6527	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6527	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11122	. . . . . . 7 class 1
18	16, 17	wceq 1539	. . . . . 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 1538	. . . . . 6 class 𝑠
21		cply 26126	. . . . . 6 class Poly
22	20, 21	cfv 6527	. . . . 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 3413	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5198	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1539	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  43110  itgocn  43113