Definition df-uc1p 26171

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 26177. (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 26166	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3480	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . 8 class 𝑟
7		cpl1 22178	. . . . . . . 8 class Poly1
8	6, 7	cfv 6561	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17484	. . . . . . 7 class 0g
10	8, 9	cfv 6561	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2940	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26093	. . . . . . . . 9 class deg1
13	6, 12	cfv 6561	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6561	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22179	. . . . . . . 8 class coe1
16	5, 15	cfv 6561	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6561	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20355	. . . . . . 7 class Unit
19	6, 18	cfv 6561	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2108	. . . . 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 17247	. . . . 5 class Base
23	8, 22	cfv 6561	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3436	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5225	. 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
26	1, 25	wceq 1540	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  26179