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Definition df-uc1p 25305
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 25311. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Assertion
Ref Expression
df-uc1p Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
Distinct variable group:   𝑓,𝑟

Detailed syntax breakdown of Definition df-uc1p
StepHypRef Expression
1 cuc1p 25300 . 2 class Unic1p
2 vr . . 3 setvar 𝑟
3 cvv 3433 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1538 . . . . . 6 class 𝑓
62cv 1538 . . . . . . . 8 class 𝑟
7 cpl1 21357 . . . . . . . 8 class Poly1
86, 7cfv 6437 . . . . . . 7 class (Poly1𝑟)
9 c0g 17159 . . . . . . 7 class 0g
108, 9cfv 6437 . . . . . 6 class (0g‘(Poly1𝑟))
115, 10wne 2944 . . . . 5 wff 𝑓 ≠ (0g‘(Poly1𝑟))
12 cdg1 25225 . . . . . . . . 9 class deg1
136, 12cfv 6437 . . . . . . . 8 class ( deg1𝑟)
145, 13cfv 6437 . . . . . . 7 class (( deg1𝑟)‘𝑓)
15 cco1 21358 . . . . . . . 8 class coe1
165, 15cfv 6437 . . . . . . 7 class (coe1𝑓)
1714, 16cfv 6437 . . . . . 6 class ((coe1𝑓)‘(( deg1𝑟)‘𝑓))
18 cui 19890 . . . . . . 7 class Unit
196, 18cfv 6437 . . . . . 6 class (Unit‘𝑟)
2017, 19wcel 2107 . . . . 5 wff ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟)
2111, 20wa 396 . . . 4 wff (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22 cbs 16921 . . . . 5 class Base
238, 22cfv 6437 . . . 4 class (Base‘(Poly1𝑟))
2421, 4, 23crab 3069 . . 3 class {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
252, 3, 24cmpt 5158 . 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
261, 25wceq 1539 1 wff Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
Colors of variables: wff setvar class
This definition is referenced by:  uc1pval  25313
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