Definition df-uc1p 26087

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 26093. (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 26082	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3459	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . 8 class 𝑟
7		cpl1 22110	. . . . . . . 8 class Poly1
8	6, 7	cfv 6530	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17451	. . . . . . 7 class 0g
10	8, 9	cfv 6530	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2932	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 26009	. . . . . . . . 9 class deg1
13	6, 12	cfv 6530	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6530	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22111	. . . . . . . 8 class coe1
16	5, 15	cfv 6530	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6530	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20313	. . . . . . 7 class Unit
19	6, 18	cfv 6530	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2108	. . . . 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 17226	. . . . 5 class Base
23	8, 22	cfv 6530	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3415	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5201	. 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  26095