Definition df-itgo 43919

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 43922. 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 43917	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 11109	. . . 4 class ℂ
4	3	cpw 4564	. . 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 6540	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 11111	. . . . . . 7 class 0
11	9, 10	wceq 1570	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 26375	. . . . . . . . 9 class deg
13	8, 12	cfv 6540	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 26374	. . . . . . . . 9 class coeff
15	8, 14	cfv 6540	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6540	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 11112	. . . . . . 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 26372	. . . . . 6 class Poly
22	20, 21	cfv 6540	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3091	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3418	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5194	. 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  43921  itgocn  43924