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Definition df-fallfac 14944
Description: Define the falling factorial function. This is the function (𝐴 · (𝐴 − 1) · ...(𝐴𝑁)) for complex 𝐴 and nonnegative integers 𝑁. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
df-fallfac FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
Distinct variable group:   𝑥,𝑛,𝑘

Detailed syntax breakdown of Definition df-fallfac
StepHypRef Expression
1 cfallfac 14941 . 2 class FallFac
2 vx . . 3 setvar 𝑥
3 vn . . 3 setvar 𝑛
4 cc 10136 . . 3 class
5 cn0 11494 . . 3 class 0
6 cc0 10138 . . . . 5 class 0
73cv 1630 . . . . . 6 class 𝑛
8 c1 10139 . . . . . 6 class 1
9 cmin 10468 . . . . . 6 class
107, 8, 9co 6793 . . . . 5 class (𝑛 − 1)
11 cfz 12533 . . . . 5 class ...
126, 10, 11co 6793 . . . 4 class (0...(𝑛 − 1))
132cv 1630 . . . . 5 class 𝑥
14 vk . . . . . 6 setvar 𝑘
1514cv 1630 . . . . 5 class 𝑘
1613, 15, 9co 6793 . . . 4 class (𝑥𝑘)
1712, 16, 14cprod 14842 . . 3 class 𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘)
182, 3, 4, 5, 17cmpt2 6795 . 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
191, 18wceq 1631 1 wff FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
Colors of variables: wff setvar class
This definition is referenced by:  fallfacval  14946
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