Definition df-itgo 40900

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 40903. 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 40898	. 2 class IntgOver
2		vs	. . 3 setvar 𝑠
3		cc 10800	. . . 4 class ℂ
4	3	cpw 4530	. . 3 class 𝒫 ℂ
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1538	. . . . . . . 8 class 𝑥
7		vp	. . . . . . . . 9 setvar 𝑝
8	7	cv 1538	. . . . . . . 8 class 𝑝
9	6, 8	cfv 6418	. . . . . . 7 class (𝑝‘𝑥)
10		cc0 10802	. . . . . . 7 class 0
11	9, 10	wceq 1539	. . . . . 6 wff (𝑝‘𝑥) = 0
12		cdgr 25253	. . . . . . . . 9 class deg
13	8, 12	cfv 6418	. . . . . . . 8 class (deg‘𝑝)
14		ccoe 25252	. . . . . . . . 9 class coeff
15	8, 14	cfv 6418	. . . . . . . 8 class (coeff‘𝑝)
16	13, 15	cfv 6418	. . . . . . 7 class ((coeff‘𝑝)‘(deg‘𝑝))
17		c1 10803	. . . . . . 7 class 1
18	16, 17	wceq 1539	. . . . . 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 1538	. . . . . 6 class 𝑠
21		cply 25250	. . . . . 6 class Poly
22	20, 21	cfv 6418	. . . . 5 class (Poly‘𝑠)
23	19, 7, 22	wrex 3064	. . . 4 wff ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)
24	23, 5, 3	crab 3067	. . 3 class {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}
25	2, 4, 24	cmpt 5153	. 2 class (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
26	1, 25	wceq 1539	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  40902  itgocn  40905