Definition df-itgo 43869

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 43872. 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 43867	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11099	. . . 4 class ℂ
4	3	cpw 4563	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1569	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1569	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6538	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11101	. . . . . . 7 class 0
11	9, 10	wceq 1570	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26325	. . . . . . . . 9 class deg
13	8, 12	cfv 6538	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26324	. . . . . . . . 9 class coeff
15	8, 14	cfv 6538	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6538	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11102	. . . . . . 7 class 1
18	16, 17	wceq 1570	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 400	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1569	. . . . . 6 class 𝑠
21		cply 26322	. . . . . 6 class Poly
22	20, 21	cfv 6538	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3089	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3416	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5193	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1570	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  43871  itgocn  43874