Definition df-uc1p 26364

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 26370. (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 26359	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3453	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1569	. . . . . 6 class 𝑓
6	2	cv 1569	. . . . . . . 8 class 𝑟
7		cpl1 22408	. . . . . . . 8 class Poly1
8	6, 7	cfv 6537	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17530	. . . . . . 7 class 0g
10	8, 9	cfv 6537	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2957	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26286	. . . . . . . . 9 class deg1
13	6, 12	cfv 6537	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6537	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22409	. . . . . . . 8 class coe1
16	5, 15	cfv 6537	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6537	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20502	. . . . . . 7 class Unit
19	6, 18	cfv 6537	. . . . . 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 17307	. . . . 5 class Base
23	8, 22	cfv 6537	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3414	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5190	. 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  26372