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Definition df-vma 24994
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 24988 . 2 class Λ
2 vx . . 3 setvar 𝑥
3 cn 11183 . . 3 class
4 vs . . . 4 setvar 𝑠
5 vp . . . . . . 7 setvar 𝑝
65cv 1619 . . . . . 6 class 𝑝
72cv 1619 . . . . . 6 class 𝑥
8 cdvds 15153 . . . . . 6 class
96, 7, 8wbr 4792 . . . . 5 wff 𝑝𝑥
10 cprime 15558 . . . . 5 class
119, 5, 10crab 3042 . . . 4 class {𝑝 ∈ ℙ ∣ 𝑝𝑥}
124cv 1619 . . . . . . 7 class 𝑠
13 chash 13282 . . . . . . 7 class
1412, 13cfv 6037 . . . . . 6 class (♯‘𝑠)
15 c1 10100 . . . . . 6 class 1
1614, 15wceq 1620 . . . . 5 wff (♯‘𝑠) = 1
1712cuni 4576 . . . . . 6 class 𝑠
18 clog 24471 . . . . . 6 class log
1917, 18cfv 6037 . . . . 5 class (log‘ 𝑠)
20 cc0 10099 . . . . 5 class 0
2116, 19, 20cif 4218 . . . 4 class if((♯‘𝑠) = 1, (log‘ 𝑠), 0)
224, 11, 21csb 3662 . . 3 class {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0)
232, 3, 22cmpt 4869 . 2 class (𝑥 ∈ ℕ ↦ {𝑝 ∈ ℙ ∣ 𝑝𝑥} / 𝑠if((♯‘𝑠) = 1, (log‘ 𝑠), 0))
241, 23wceq 1620 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  25009  vmaf  25015
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