Definition df-itgo 43513

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 43516. 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 43511	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11036	. . . 4 class ℂ
4	3	cpw 4556	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1541	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1541	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6500	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11038	. . . . . . 7 class 0
11	9, 10	wceq 1542	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26160	. . . . . . . . 9 class deg
13	8, 12	cfv 6500	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26159	. . . . . . . . 9 class coeff
15	8, 14	cfv 6500	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6500	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11039	. . . . . . 7 class 1
18	16, 17	wceq 1542	. . . . . 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 1541	. . . . . 6 class 𝑠
21		cply 26157	. . . . . 6 class Poly
22	20, 21	cfv 6500	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3062	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3401	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5181	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1542	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  43515  itgocn  43518