Definition df-itgo 43401

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 43404. 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 43399	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11024	. . . 4 class ℂ
4	3	cpw 4554	. . 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 6492	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11026	. . . . . . 7 class 0
11	9, 10	wceq 1541	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26148	. . . . . . . . 9 class deg
13	8, 12	cfv 6492	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26147	. . . . . . . . 9 class coeff
15	8, 14	cfv 6492	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6492	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11027	. . . . . . 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 26145	. . . . . 6 class Poly
22	20, 21	cfv 6492	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3060	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3399	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5179	. 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  43403  itgocn  43406