Definition df-uc1p 26110

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 26116. (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 26105	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3430	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1541	. . . . . 6 class 𝑓
6	2	cv 1541	. . . . . . . 8 class 𝑟
7		cpl1 22153	. . . . . . . 8 class Poly1
8	6, 7	cfv 6493	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17396	. . . . . . 7 class 0g
10	8, 9	cfv 6493	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2933	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26032	. . . . . . . . 9 class deg1
13	6, 12	cfv 6493	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6493	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22154	. . . . . . . 8 class coe1
16	5, 15	cfv 6493	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6493	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20329	. . . . . . 7 class Unit
19	6, 18	cfv 6493	. . . . . 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 17173	. . . . 5 class Base
23	8, 22	cfv 6493	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3390	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5167	. 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  26118