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Definition df-fwddifn 31438
 Description: Define the nth forward difference operator. This works out to be the forward difference operator iterated 𝑛 times. (Contributed by Scott Fenton, 28-May-2020.)
Assertion
Ref Expression
df-fwddifn n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
Distinct variable group:   𝑓,𝑛,𝑥,𝑦,𝑘

Detailed syntax breakdown of Definition df-fwddifn
StepHypRef Expression
1 cfwddifn 31437 . 2 class n
2 vn . . 3 setvar 𝑛
3 vf . . 3 setvar 𝑓
4 cn0 11169 . . 3 class 0
5 cc 9813 . . . 4 class
6 cpm 7745 . . . 4 class pm
75, 5, 6co 6549 . . 3 class (ℂ ↑pm ℂ)
8 vx . . . 4 setvar 𝑥
9 vy . . . . . . . . 9 setvar 𝑦
109cv 1474 . . . . . . . 8 class 𝑦
11 vk . . . . . . . . 9 setvar 𝑘
1211cv 1474 . . . . . . . 8 class 𝑘
13 caddc 9818 . . . . . . . 8 class +
1410, 12, 13co 6549 . . . . . . 7 class (𝑦 + 𝑘)
153cv 1474 . . . . . . . 8 class 𝑓
1615cdm 5038 . . . . . . 7 class dom 𝑓
1714, 16wcel 1977 . . . . . 6 wff (𝑦 + 𝑘) ∈ dom 𝑓
18 cc0 9815 . . . . . . 7 class 0
192cv 1474 . . . . . . 7 class 𝑛
20 cfz 12197 . . . . . . 7 class ...
2118, 19, 20co 6549 . . . . . 6 class (0...𝑛)
2217, 11, 21wral 2896 . . . . 5 wff 𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓
2322, 9, 5crab 2900 . . . 4 class {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓}
24 cbc 12951 . . . . . . 7 class C
2519, 12, 24co 6549 . . . . . 6 class (𝑛C𝑘)
26 c1 9816 . . . . . . . . 9 class 1
2726cneg 10146 . . . . . . . 8 class -1
28 cmin 10145 . . . . . . . . 9 class
2919, 12, 28co 6549 . . . . . . . 8 class (𝑛𝑘)
30 cexp 12722 . . . . . . . 8 class
3127, 29, 30co 6549 . . . . . . 7 class (-1↑(𝑛𝑘))
328cv 1474 . . . . . . . . 9 class 𝑥
3332, 12, 13co 6549 . . . . . . . 8 class (𝑥 + 𝑘)
3433, 15cfv 5804 . . . . . . 7 class (𝑓‘(𝑥 + 𝑘))
35 cmul 9820 . . . . . . 7 class ·
3631, 34, 35co 6549 . . . . . 6 class ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))
3725, 36, 35co 6549 . . . . 5 class ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))
3821, 37, 11csu 14264 . . . 4 class Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))
398, 23, 38cmpt 4643 . . 3 class (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))))
402, 3, 4, 7, 39cmpt2 6551 . 2 class (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
411, 40wceq 1475 1 wff n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
 Colors of variables: wff setvar class This definition is referenced by:  fwddifnval  31440
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