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Definition df-risefac 15949
Description: Define the rising factorial function. This is the function (๐ด ยท (๐ด + 1) ยท ...(๐ด + ๐‘)) for complex ๐ด and nonnegative integers ๐‘. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
df-risefac RiseFac = (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ + ๐‘˜))
Distinct variable group:   ๐‘ฅ,๐‘›,๐‘˜

Detailed syntax breakdown of Definition df-risefac
StepHypRef Expression
1 crisefac 15948 . 2 class RiseFac
2 vx . . 3 setvar ๐‘ฅ
3 vn . . 3 setvar ๐‘›
4 cc 11107 . . 3 class โ„‚
5 cn0 12471 . . 3 class โ„•0
6 cc0 11109 . . . . 5 class 0
73cv 1540 . . . . . 6 class ๐‘›
8 c1 11110 . . . . . 6 class 1
9 cmin 11443 . . . . . 6 class โˆ’
107, 8, 9co 7408 . . . . 5 class (๐‘› โˆ’ 1)
11 cfz 13483 . . . . 5 class ...
126, 10, 11co 7408 . . . 4 class (0...(๐‘› โˆ’ 1))
132cv 1540 . . . . 5 class ๐‘ฅ
14 vk . . . . . 6 setvar ๐‘˜
1514cv 1540 . . . . 5 class ๐‘˜
16 caddc 11112 . . . . 5 class +
1713, 15, 16co 7408 . . . 4 class (๐‘ฅ + ๐‘˜)
1812, 17, 14cprod 15848 . . 3 class โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ + ๐‘˜)
192, 3, 4, 5, 18cmpo 7410 . 2 class (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ + ๐‘˜))
201, 19wceq 1541 1 wff RiseFac = (๐‘ฅ โˆˆ โ„‚, ๐‘› โˆˆ โ„•0 โ†ฆ โˆ๐‘˜ โˆˆ (0...(๐‘› โˆ’ 1))(๐‘ฅ + ๐‘˜))
Colors of variables: wff setvar class
This definition is referenced by:  risefacval  15951
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