Definition df-uc1p 26035

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 26041. (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 26030	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3436	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1539	. . . . . 6 class 𝑓
6	2	cv 1539	. . . . . . . 8 class 𝑟
7		cpl1 22059	. . . . . . . 8 class Poly1
8	6, 7	cfv 6482	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17343	. . . . . . 7 class 0g
10	8, 9	cfv 6482	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2925	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 25957	. . . . . . . . 9 class deg1
13	6, 12	cfv 6482	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6482	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22060	. . . . . . . 8 class coe1
16	5, 15	cfv 6482	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6482	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20240	. . . . . . 7 class Unit
19	6, 18	cfv 6482	. . . . . 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 17120	. . . . 5 class Base
23	8, 22	cfv 6482	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3394	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5173	. 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  26043