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Definition df-vma 24741
Description: Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere, see definition in [ApostolNT] p. 32. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
df-vma Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))
Distinct variable group:   𝑠,𝑝,𝑥

Detailed syntax breakdown of Definition df-vma
StepHypRef Expression
1 cvma 24735 . 2 class Λ
2 vx . . 3 setvar 𝑥
3 cn 10972 . . 3 class
4 vs . . . 4 setvar 𝑠
5 vp . . . . . . 7 setvar 𝑝
65cv 1479 . . . . . 6 class 𝑝
72cv 1479 . . . . . 6 class 𝑥
8 cdvds 14918 . . . . . 6 class
96, 7, 8wbr 4618 . . . . 5 wff 𝑝𝑥
10 cprime 15320 . . . . 5 class
119, 5, 10crab 2911 . . . 4 class {𝑝 ∈ ℙ ∣ 𝑝𝑥}
124cv 1479 . . . . . . 7 class 𝑠
13 chash 13065 . . . . . . 7 class #
1412, 13cfv 5852 . . . . . 6 class (#‘𝑠)
15 c1 9889 . . . . . 6 class 1
1614, 15wceq 1480 . . . . 5 wff (#‘𝑠) = 1
1712cuni 4407 . . . . . 6 class 𝑠
18 clog 24222 . . . . . 6 class log
1917, 18cfv 5852 . . . . 5 class (log‘ 𝑠)
20 cc0 9888 . . . . 5 class 0
2116, 19, 20cif 4063 . . . 4 class if((#‘𝑠) = 1, (log‘ 𝑠), 0)
224, 11, 21csb 3518 . . 3 class {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0)
232, 3, 22cmpt 4678 . 2 class (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))
241, 23wceq 1480 1 wff Λ = (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((#‘𝑠) = 1, (log‘ 𝑠), 0))
Colors of variables: wff setvar class
This definition is referenced by:  vmaval  24756  vmaf  24762
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