Definition df-uc1p 26105

Description: Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 26111. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Assertion

Ref	Expression
df-uc1p	⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

Distinct variable group:   𝑓,𝑟

Detailed syntax breakdown of Definition df-uc1p

Step	Hyp	Ref	Expression
1		cuc1p 26100	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3442	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1541	. . . . . 6 class 𝑓
6	2	cv 1541	. . . . . . . 8 class 𝑟
7		cpl1 22129	. . . . . . . 8 class Poly1
8	6, 7	cfv 6500	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17371	. . . . . . 7 class 0g
10	8, 9	cfv 6500	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2933	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26027	. . . . . . . . 9 class deg1
13	6, 12	cfv 6500	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6500	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22130	. . . . . . . 8 class coe1
16	5, 15	cfv 6500	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6500	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20303	. . . . . . 7 class Unit
19	6, 18	cfv 6500	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2114	. . . . 5 wff ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟)
21	11, 20	wa 395	. . . 4 wff (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22		cbs 17148	. . . . 5 class Base
23	8, 22	cfv 6500	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3401	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5181	. 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
26	1, 25	wceq 1542	1 wff Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

This definition is referenced by:  uc1pval  26113