Definition df-uc1p 25648

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 25654. (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 25643	. 2 class Unic1p
2		vr	. . 3 setvar 𝑟
3		cvv 3474	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1540	. . . . . 6 class 𝑓
6	2	cv 1540	. . . . . . . 8 class 𝑟
7		cpl1 21700	. . . . . . . 8 class Poly1
8	6, 7	cfv 6543	. . . . . . 7 class (Poly1‘𝑟)
9		c0g 17384	. . . . . . 7 class 0g
10	8, 9	cfv 6543	. . . . . 6 class (0g‘(Poly1‘𝑟))
11	5, 10	wne 2940	. . . . 5 wff 𝑓 ≠ (0g‘(Poly1‘𝑟))
12		cdg1 25568	. . . . . . . . 9 class deg1
13	6, 12	cfv 6543	. . . . . . . 8 class ( deg1 ‘𝑟)
14	5, 13	cfv 6543	. . . . . . 7 class (( deg1 ‘𝑟)‘𝑓)
15		cco1 21701	. . . . . . . 8 class coe1
16	5, 15	cfv 6543	. . . . . . 7 class (coe1‘𝑓)
17	14, 16	cfv 6543	. . . . . 6 class ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓))
18		cui 20168	. . . . . . 7 class Unit
19	6, 18	cfv 6543	. . . . . 6 class (Unit‘𝑟)
20	17, 19	wcel 2106	. . . . 5 wff ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟)
21	11, 20	wa 396	. . . 4 wff (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22		cbs 17143	. . . . 5 class Base
23	8, 22	cfv 6543	. . . 4 class (Base‘(Poly1‘𝑟))
24	21, 4, 23	crab 3432	. . 3 class {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
25	2, 3, 24	cmpt 5231	. 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  25656