Definition df-uc1p 26505

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 26511. (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 26500	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3454	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1569	. . . . . 6 class 𝑓
6	2	cv 1569	. . . . . . . 8 class 𝑟
7		cpl1 22550	. . . . . . . 8 class Poly1
8	6, 7	cfv 6566	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17660	. . . . . . 7 class 0g
10	8, 9	cfv 6566	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2959	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26427	. . . . . . . . 9 class deg1
13	6, 12	cfv 6566	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6566	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22551	. . . . . . . 8 class coe1
16	5, 15	cfv 6566	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6566	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20636	. . . . . . 7 class Unit
19	6, 18	cfv 6566	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2147	. . . . 5 wff ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟)
21	11, 20	wa 401	. . . 4 wff (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22		cbs 17431	. . . . 5 class Base
23	8, 22	cfv 6566	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3416	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5192	. 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
26	1, 25	wceq 1570	1 wff Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

This definition is used by:  uc1pval  26513