Definition df-vma 25683

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 ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))

Distinct variable group:   𝑠,𝑝,𝑥

Detailed syntax breakdown of Definition df-vma

Step	Hyp	Ref	Expression
1		cvma 25677	. 2 class Λ
2		vx	. . 3 setvar 𝑥
3		cn 11625	. . 3 class ℕ
4		vs	. . . 4 setvar 𝑠
5		vp	. . . . . . 7 setvar 𝑝
6	5	cv 1537	. . . . . 6 class 𝑝
7	2	cv 1537	. . . . . 6 class 𝑥
8		cdvds 15599	. . . . . 6 class ∥
9	6, 7, 8	wbr 5030	. . . . 5 wff 𝑝 ∥ 𝑥
10		cprime 16005	. . . . 5 class ℙ
11	9, 5, 10	crab 3110	. . . 4 class {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}
12	4	cv 1537	. . . . . . 7 class 𝑠
13		chash 13686	. . . . . . 7 class ♯
14	12, 13	cfv 6324	. . . . . 6 class (♯‘𝑠)
15		c1 10527	. . . . . 6 class 1
16	14, 15	wceq 1538	. . . . 5 wff (♯‘𝑠) = 1
17	12	cuni 4800	. . . . . 6 class ∪ 𝑠
18		clog 25146	. . . . . 6 class log
19	17, 18	cfv 6324	. . . . 5 class (log‘∪ 𝑠)
20		cc0 10526	. . . . 5 class 0
21	16, 19, 20	cif 4425	. . . 4 class if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0)
22	4, 11, 21	csb 3828	. . 3 class ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0)
23	2, 3, 22	cmpt 5110	. 2 class (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))
24	1, 23	wceq 1538	1 wff Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0))

This definition is referenced by:  vmaval  25698  vmaf  25704