Definition df-itgo 43141

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 43144. 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 43139	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11072	. . . 4 class ℂ
4	3	cpw 4565	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1539	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1539	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6513	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11074	. . . . . . 7 class 0
11	9, 10	wceq 1540	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26098	. . . . . . . . 9 class deg
13	8, 12	cfv 6513	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26097	. . . . . . . . 9 class coeff
15	8, 14	cfv 6513	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6513	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11075	. . . . . . 7 class 1
18	16, 17	wceq 1540	. . . . . 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 1539	. . . . . 6 class 𝑠
21		cply 26095	. . . . . 6 class Poly
22	20, 21	cfv 6513	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3054	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3408	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5190	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1540	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  43143  itgocn  43146