Definition df-uc1p 26064

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 26070. (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 26059	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3436	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1540	. . . . . 6 class 𝑓
6	2	cv 1540	. . . . . . . 8 class 𝑟
7		cpl1 22089	. . . . . . . 8 class Poly1
8	6, 7	cfv 6481	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17343	. . . . . . 7 class 0g
10	8, 9	cfv 6481	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2928	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 25986	. . . . . . . . 9 class deg1
13	6, 12	cfv 6481	. . . . . . . 8 class (deg1‘𝑟)
14	5, 13	cfv 6481	. . . . . . 7 class ((deg1‘𝑟)‘𝑓)
15		cco1 22090	. . . . . . . 8 class coe1
16	5, 15	cfv 6481	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6481	. . . . . 6 class ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓))
18		cui 20273	. . . . . . 7 class Unit
19	6, 18	cfv 6481	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2111	. . . . 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 6481	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3395	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5170	. 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  26072