Definition df-uc1p 26358

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 26364. (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 26353	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3450	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1569	. . . . . 6 class 𝑓
6	2	cv 1569	. . . . . . . 8 class 𝑟
7		cpl1 22403	. . . . . . . 8 class Poly1
8	6, 7	cfv 6533	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17525	. . . . . . 7 class 0g
10	8, 9	cfv 6533	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2955	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26280	. . . . . . . . 9 class deg1
13	6, 12	cfv 6533	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6533	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22404	. . . . . . . 8 class coe1
16	5, 15	cfv 6533	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6533	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20497	. . . . . . 7 class Unit
19	6, 18	cfv 6533	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2145	. . . . 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 17302	. . . . 5 class Base
23	8, 22	cfv 6533	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3412	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5186	. 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  26366