Definition df-itgo 43605

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 43608. 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 43603	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11027	. . . 4 class ℂ
4	3	cpw 4542	. . 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 6492	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11029	. . . . . . 7 class 0
11	9, 10	wceq 1542	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26162	. . . . . . . . 9 class deg
13	8, 12	cfv 6492	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26161	. . . . . . . . 9 class coeff
15	8, 14	cfv 6492	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6492	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11030	. . . . . . 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 26159	. . . . . 6 class Poly
22	20, 21	cfv 6492	. . . . 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 3390	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5167	. 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  43607  itgocn  43610