Definition df-uc1p 26093

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 26099. (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 26088	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3440	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1540	. . . . . 6 class 𝑓
6	2	cv 1540	. . . . . . . 8 class 𝑟
7		cpl1 22117	. . . . . . . 8 class Poly1
8	6, 7	cfv 6492	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17359	. . . . . . 7 class 0g
10	8, 9	cfv 6492	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2932	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26015	. . . . . . . . 9 class deg1
13	6, 12	cfv 6492	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6492	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22118	. . . . . . . 8 class coe1
16	5, 15	cfv 6492	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6492	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20291	. . . . . . 7 class Unit
19	6, 18	cfv 6492	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2113	. . . . 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 17136	. . . . 5 class Base
23	8, 22	cfv 6492	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3399	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5179	. 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
26	1, 25	wceq 1541	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  26101