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Definition df-ply 23692
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 23688 . 2 class Poly
2 vx . . 3 setvar 𝑥
3 cc 9790 . . . 4 class
43cpw 4107 . . 3 class 𝒫 ℂ
5 vf . . . . . . . 8 setvar 𝑓
65cv 1473 . . . . . . 7 class 𝑓
7 vz . . . . . . . 8 setvar 𝑧
8 cc0 9792 . . . . . . . . . 10 class 0
9 vn . . . . . . . . . . 11 setvar 𝑛
109cv 1473 . . . . . . . . . 10 class 𝑛
11 cfz 12154 . . . . . . . . . 10 class ...
128, 10, 11co 6526 . . . . . . . . 9 class (0...𝑛)
13 vk . . . . . . . . . . . 12 setvar 𝑘
1413cv 1473 . . . . . . . . . . 11 class 𝑘
15 va . . . . . . . . . . . 12 setvar 𝑎
1615cv 1473 . . . . . . . . . . 11 class 𝑎
1714, 16cfv 5789 . . . . . . . . . 10 class (𝑎𝑘)
187cv 1473 . . . . . . . . . . 11 class 𝑧
19 cexp 12679 . . . . . . . . . . 11 class
2018, 14, 19co 6526 . . . . . . . . . 10 class (𝑧𝑘)
21 cmul 9797 . . . . . . . . . 10 class ·
2217, 20, 21co 6526 . . . . . . . . 9 class ((𝑎𝑘) · (𝑧𝑘))
2312, 22, 13csu 14212 . . . . . . . 8 class Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))
247, 3, 23cmpt 4637 . . . . . . 7 class (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
256, 24wceq 1474 . . . . . 6 wff 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
262cv 1473 . . . . . . . 8 class 𝑥
278csn 4124 . . . . . . . 8 class {0}
2826, 27cun 3537 . . . . . . 7 class (𝑥 ∪ {0})
29 cn0 11141 . . . . . . 7 class 0
30 cmap 7721 . . . . . . 7 class 𝑚
3128, 29, 30co 6526 . . . . . 6 class ((𝑥 ∪ {0}) ↑𝑚0)
3225, 15, 31wrex 2896 . . . . 5 wff 𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
3332, 9, 29wrex 2896 . . . 4 wff 𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))
3433, 5cab 2595 . . 3 class {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))}
352, 4, 34cmpt 4637 . 2 class (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
361, 35wceq 1474 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  23697  plybss  23698
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