Metamath Proof Explorer


Definition df-fallfac

Description: Define the falling factorial function. This is the function ( A x. ( A - 1 ) x. ... ( A - N ) ) for complex A and nonnegative integers N . (Contributed by Scott Fenton, 5-Jan-2018)

Ref Expression
Assertion df-fallfac FallFac = ( 𝑥 ∈ ℂ , 𝑛 ∈ ℕ0 ↦ ∏ 𝑘 ∈ ( 0 ... ( 𝑛 − 1 ) ) ( 𝑥𝑘 ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cfallfac FallFac
1 vx 𝑥
2 cc
3 vn 𝑛
4 cn0 0
5 vk 𝑘
6 cc0 0
7 cfz ...
8 3 cv 𝑛
9 cmin
10 c1 1
11 8 10 9 co ( 𝑛 − 1 )
12 6 11 7 co ( 0 ... ( 𝑛 − 1 ) )
13 1 cv 𝑥
14 5 cv 𝑘
15 13 14 9 co ( 𝑥𝑘 )
16 12 15 5 cprod 𝑘 ∈ ( 0 ... ( 𝑛 − 1 ) ) ( 𝑥𝑘 )
17 1 3 2 4 16 cmpo ( 𝑥 ∈ ℂ , 𝑛 ∈ ℕ0 ↦ ∏ 𝑘 ∈ ( 0 ... ( 𝑛 − 1 ) ) ( 𝑥𝑘 ) )
18 0 17 wceq FallFac = ( 𝑥 ∈ ℂ , 𝑛 ∈ ℕ0 ↦ ∏ 𝑘 ∈ ( 0 ... ( 𝑛 − 1 ) ) ( 𝑥𝑘 ) )