Definition df-itgo 44000

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 44003. 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 43998	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11122	. . . 4 class ℂ
4	3	cpw 4557	. . 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 6533	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11124	. . . . . . 7 class 0
11	9, 10	wceq 1570	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26412	. . . . . . . . 9 class deg
13	8, 12	cfv 6533	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26411	. . . . . . . . 9 class coeff
15	8, 14	cfv 6533	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6533	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11125	. . . . . . 7 class 1
18	16, 17	wceq 1570	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 401	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1569	. . . . . 6 class 𝑠
21		cply 26409	. . . . . 6 class Poly
22	20, 21	cfv 6533	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3086	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3412	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5186	. 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 used by:  itgoval  44002  itgocn  44005