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Definition df-uc1p 25029
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 25035. (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 25024 . 2 class Unic1p
2 vr . . 3 setvar 𝑟
3 cvv 3408 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1542 . . . . . 6 class 𝑓
62cv 1542 . . . . . . . 8 class 𝑟
7 cpl1 21098 . . . . . . . 8 class Poly1
86, 7cfv 6380 . . . . . . 7 class (Poly1𝑟)
9 c0g 16944 . . . . . . 7 class 0g
108, 9cfv 6380 . . . . . 6 class (0g‘(Poly1𝑟))
115, 10wne 2940 . . . . 5 wff 𝑓 ≠ (0g‘(Poly1𝑟))
12 cdg1 24949 . . . . . . . . 9 class deg1
136, 12cfv 6380 . . . . . . . 8 class ( deg1𝑟)
145, 13cfv 6380 . . . . . . 7 class (( deg1𝑟)‘𝑓)
15 cco1 21099 . . . . . . . 8 class coe1
165, 15cfv 6380 . . . . . . 7 class (coe1𝑓)
1714, 16cfv 6380 . . . . . 6 class ((coe1𝑓)‘(( deg1𝑟)‘𝑓))
18 cui 19657 . . . . . . 7 class Unit
196, 18cfv 6380 . . . . . 6 class (Unit‘𝑟)
2017, 19wcel 2110 . . . . 5 wff ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟)
2111, 20wa 399 . . . 4 wff (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22 cbs 16760 . . . . 5 class Base
238, 22cfv 6380 . . . 4 class (Base‘(Poly1𝑟))
2421, 4, 23crab 3065 . . 3 class {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
252, 3, 24cmpt 5135 . 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
261, 25wceq 1543 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  25037
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