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Definition df-ply 24788
Description: Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
df-ply Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
Distinct variable group:   𝑓,𝑎,𝑘,𝑛,𝑥,𝑧

Detailed syntax breakdown of Definition df-ply
StepHypRef Expression
1 cply 24784 . 2 class Poly
2 vx . . 3 setvar 𝑥
3 cc 10528 . . . 4 class
43cpw 4500 . . 3 class 𝒫 ℂ
5 vf . . . . . . . 8 setvar 𝑓
65cv 1537 . . . . . . 7 class 𝑓
7 vz . . . . . . . 8 setvar 𝑧
8 cc0 10530 . . . . . . . . . 10 class 0
9 vn . . . . . . . . . . 11 setvar 𝑛
109cv 1537 . . . . . . . . . 10 class 𝑛
11 cfz 12889 . . . . . . . . . 10 class ...
128, 10, 11co 7139 . . . . . . . . 9 class (0...𝑛)
13 vk . . . . . . . . . . . 12 setvar 𝑘
1413cv 1537 . . . . . . . . . . 11 class 𝑘
15 va . . . . . . . . . . . 12 setvar 𝑎
1615cv 1537 . . . . . . . . . . 11 class 𝑎
1714, 16cfv 6328 . . . . . . . . . 10 class (𝑎𝑘)
187cv 1537 . . . . . . . . . . 11 class 𝑧
19 cexp 13429 . . . . . . . . . . 11 class
2018, 14, 19co 7139 . . . . . . . . . 10 class (𝑧𝑘)
21 cmul 10535 . . . . . . . . . 10 class ·
2217, 20, 21co 7139 . . . . . . . . 9 class ((𝑎𝑘) · (𝑧𝑘))
2312, 22, 13csu 15037 . . . . . . . 8 class Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))
247, 3, 23cmpt 5113 . . . . . . 7 class (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
256, 24wceq 1538 . . . . . 6 wff 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
262cv 1537 . . . . . . . 8 class 𝑥
278csn 4528 . . . . . . . 8 class {0}
2826, 27cun 3882 . . . . . . 7 class (𝑥 ∪ {0})
29 cn0 11889 . . . . . . 7 class 0
30 cmap 8393 . . . . . . 7 class m
3128, 29, 30co 7139 . . . . . 6 class ((𝑥 ∪ {0}) ↑m0)
3225, 15, 31wrex 3110 . . . . 5 wff 𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
3332, 9, 29wrex 3110 . . . 4 wff 𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
3433, 5cab 2779 . . 3 class {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))}
352, 4, 34cmpt 5113 . 2 class (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
361, 35wceq 1538 1 wff Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
Colors of variables: wff setvar class
This definition is referenced by:  plyval  24793  plybss  24794
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