Definition df-uc1p 26037

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 26043. (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 26032	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3447	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . 8 class 𝑟
7		cpl1 22061	. . . . . . . 8 class Poly1
8	6, 7	cfv 6511	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17402	. . . . . . 7 class 0g
10	8, 9	cfv 6511	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2925	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 25959	. . . . . . . . 9 class deg1
13	6, 12	cfv 6511	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6511	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22062	. . . . . . . 8 class coe1
16	5, 15	cfv 6511	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6511	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20264	. . . . . . 7 class Unit
19	6, 18	cfv 6511	. . . . . 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 17179	. . . . 5 class Base
23	8, 22	cfv 6511	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3405	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5188	. 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  26045