Definition df-itgo 43111

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 43114. 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 43109	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11176	. . . 4 class ℂ
4	3	cpw 4622	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1536	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1536	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6568	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11178	. . . . . . 7 class 0
11	9, 10	wceq 1537	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26238	. . . . . . . . 9 class deg
13	8, 12	cfv 6568	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26237	. . . . . . . . 9 class coeff
15	8, 14	cfv 6568	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6568	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11179	. . . . . . 7 class 1
18	16, 17	wceq 1537	. . . . . 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 1536	. . . . . 6 class 𝑠
21		cply 26235	. . . . . 6 class Poly
22	20, 21	cfv 6568	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3076	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3443	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5249	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1537	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  43113  itgocn  43116