Definition df-itgo 41891

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 41894. 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 41889	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11107	. . . 4 class ℂ
4	3	cpw 4602	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1540	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1540	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6543	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11109	. . . . . . 7 class 0
11	9, 10	wceq 1541	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 25700	. . . . . . . . 9 class deg
13	8, 12	cfv 6543	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 25699	. . . . . . . . 9 class coeff
15	8, 14	cfv 6543	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6543	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11110	. . . . . . 7 class 1
18	16, 17	wceq 1541	. . . . . 6 wff ((coeff‘𝑝)‘(deg‘𝑝)) = 1
19	11, 18	wa 396	. . . . 5 wff ((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
20	2	cv 1540	. . . . . 6 class 𝑠
21		cply 25697	. . . . . 6 class Poly
22	20, 21	cfv 6543	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3070	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3432	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5231	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1541	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  41893  itgocn  41896