MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-fallfac Structured version   Visualization version   GIF version

Definition df-fallfac 15361
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 15358 . 2 class FallFac
2 vx . . 3 setvar 𝑥
3 vn . . 3 setvar 𝑛
4 cc 10533 . . 3 class
5 cn0 11894 . . 3 class 0
6 cc0 10535 . . . . 5 class 0
73cv 1537 . . . . . 6 class 𝑛
8 c1 10536 . . . . . 6 class 1
9 cmin 10868 . . . . . 6 class
107, 8, 9co 7149 . . . . 5 class (𝑛 − 1)
11 cfz 12894 . . . . 5 class ...
126, 10, 11co 7149 . . . 4 class (0...(𝑛 − 1))
132cv 1537 . . . . 5 class 𝑥
14 vk . . . . . 6 setvar 𝑘
1514cv 1537 . . . . 5 class 𝑘
1613, 15, 9co 7149 . . . 4 class (𝑥𝑘)
1712, 16, 14cprod 15259 . . 3 class 𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘)
182, 3, 4, 5, 17cmpo 7151 . 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
191, 18wceq 1538 1 wff FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
Colors of variables: wff setvar class
This definition is referenced by:  fallfacval  15363
  Copyright terms: Public domain W3C validator