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Definition df-prmo 15671
Description: Define the primorial function on nonnegative integers as the product of all prime numbers less than or equal to the integer. For example, (#p‘10) = 2 · 3 · 5 · 7 = 210 (see ex-prmo 27187).

In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 24821, where the primorial function is defined by using the sequence builder (𝑃 = seq1( · , 𝐹)), the more specialized definition of a product of a series is used here. (Contributed by AV, 28-Aug-2020.)

Assertion
Ref Expression
df-prmo #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
Distinct variable group:   𝑘,𝑛

Detailed syntax breakdown of Definition df-prmo
StepHypRef Expression
1 cprmo 15670 . 2 class #p
2 vn . . 3 setvar 𝑛
3 cn0 11244 . . 3 class 0
4 c1 9889 . . . . 5 class 1
52cv 1479 . . . . 5 class 𝑛
6 cfz 12276 . . . . 5 class ...
74, 5, 6co 6610 . . . 4 class (1...𝑛)
8 vk . . . . . . 7 setvar 𝑘
98cv 1479 . . . . . 6 class 𝑘
10 cprime 15320 . . . . . 6 class
119, 10wcel 1987 . . . . 5 wff 𝑘 ∈ ℙ
1211, 9, 4cif 4063 . . . 4 class if(𝑘 ∈ ℙ, 𝑘, 1)
137, 12, 8cprod 14571 . . 3 class 𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1)
142, 3, 13cmpt 4678 . 2 class (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
151, 14wceq 1480 1 wff #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
Colors of variables: wff setvar class
This definition is referenced by:  prmoval  15672
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