Definition df-itgo 44426

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 44429. 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 44424	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11237	. . . 4 class ℂ
4	3	cpw 4560	. . 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 6566	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11239	. . . . . . 7 class 0
11	9, 10	wceq 1570	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26560	. . . . . . . . 9 class deg
13	8, 12	cfv 6566	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26559	. . . . . . . . 9 class coeff
15	8, 14	cfv 6566	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6566	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11240	. . . . . . 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 26557	. . . . . 6 class Poly
22	20, 21	cfv 6566	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3090	. . . 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 5192	. 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  44428  itgocn  44431