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Definition df-uc1p 25480
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 25486. (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 25475 . 2 class Unic1p
2 vr . . 3 setvar 𝑟
3 cvv 3443 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1540 . . . . . 6 class 𝑓
62cv 1540 . . . . . . . 8 class 𝑟
7 cpl1 21532 . . . . . . . 8 class Poly1
86, 7cfv 6493 . . . . . . 7 class (Poly1𝑟)
9 c0g 17313 . . . . . . 7 class 0g
108, 9cfv 6493 . . . . . 6 class (0g‘(Poly1𝑟))
115, 10wne 2941 . . . . 5 wff 𝑓 ≠ (0g‘(Poly1𝑟))
12 cdg1 25400 . . . . . . . . 9 class deg1
136, 12cfv 6493 . . . . . . . 8 class ( deg1𝑟)
145, 13cfv 6493 . . . . . . 7 class (( deg1𝑟)‘𝑓)
15 cco1 21533 . . . . . . . 8 class coe1
165, 15cfv 6493 . . . . . . 7 class (coe1𝑓)
1714, 16cfv 6493 . . . . . 6 class ((coe1𝑓)‘(( deg1𝑟)‘𝑓))
18 cui 20053 . . . . . . 7 class Unit
196, 18cfv 6493 . . . . . 6 class (Unit‘𝑟)
2017, 19wcel 2106 . . . . 5 wff ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟)
2111, 20wa 396 . . . 4 wff (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))
22 cbs 17075 . . . . 5 class Base
238, 22cfv 6493 . . . 4 class (Base‘(Poly1𝑟))
2421, 4, 23crab 3405 . . 3 class {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))}
252, 3, 24cmpt 5186 . 2 class (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
261, 25wceq 1541 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  25488
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