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

Definition df-risefac 15445
Description: Define the rising factorial function. This is the function (𝐴 · (𝐴 + 1) · ...(𝐴 + 𝑁)) for complex 𝐴 and nonnegative integers 𝑁. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
df-risefac RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
Distinct variable group:   𝑥,𝑛,𝑘

Detailed syntax breakdown of Definition df-risefac
StepHypRef Expression
1 crisefac 15444 . 2 class RiseFac
2 vx . . 3 setvar 𝑥
3 vn . . 3 setvar 𝑛
4 cc 10606 . . 3 class
5 cn0 11969 . . 3 class 0
6 cc0 10608 . . . . 5 class 0
73cv 1541 . . . . . 6 class 𝑛
8 c1 10609 . . . . . 6 class 1
9 cmin 10941 . . . . . 6 class
107, 8, 9co 7164 . . . . 5 class (𝑛 − 1)
11 cfz 12974 . . . . 5 class ...
126, 10, 11co 7164 . . . 4 class (0...(𝑛 − 1))
132cv 1541 . . . . 5 class 𝑥
14 vk . . . . . 6 setvar 𝑘
1514cv 1541 . . . . 5 class 𝑘
16 caddc 10611 . . . . 5 class +
1713, 15, 16co 7164 . . . 4 class (𝑥 + 𝑘)
1812, 17, 14cprod 15344 . . 3 class 𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘)
192, 3, 4, 5, 18cmpo 7166 . 2 class (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
201, 19wceq 1542 1 wff RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
Colors of variables: wff setvar class
This definition is referenced by:  risefacval  15447
  Copyright terms: Public domain W3C validator