Definition df-uc1p 26044

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 26050. (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 26039	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3450	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . 8 class 𝑟
7		cpl1 22068	. . . . . . . 8 class Poly1
8	6, 7	cfv 6514	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17409	. . . . . . 7 class 0g
10	8, 9	cfv 6514	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2926	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 25966	. . . . . . . . 9 class deg1
13	6, 12	cfv 6514	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6514	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22069	. . . . . . . 8 class coe1
16	5, 15	cfv 6514	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6514	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20271	. . . . . . 7 class Unit
19	6, 18	cfv 6514	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2109	. . . . 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 17186	. . . . 5 class Base
23	8, 22	cfv 6514	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3408	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5191	. 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  26052