Definition df-vma 24624

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 24618	. 2 class Λ
2		vx	. . 3 setvar 𝑥
3		cn 10897	. . 3 class ℕ
4		vs	. . . 4 setvar 𝑠
5		vp	. . . . . . 7 setvar 𝑝
6	5	cv 1474	. . . . . 6 class 𝑝
7	2	cv 1474	. . . . . 6 class 𝑥
8		cdvds 14821	. . . . . 6 class ∥
9	6, 7, 8	wbr 4583	. . . . 5 wff 𝑝 ∥ 𝑥
10		cprime 15223	. . . . 5 class ℙ
11	9, 5, 10	crab 2900	. . . 4 class {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}
12	4	cv 1474	. . . . . . 7 class 𝑠
13		chash 12979	. . . . . . 7 class #
14	12, 13	cfv 5804	. . . . . 6 class (#‘𝑠)
15		c1 9816	. . . . . 6 class 1
16	14, 15	wceq 1475	. . . . 5 wff (#‘𝑠) = 1
17	12	cuni 4372	. . . . . 6 class ∪ 𝑠
18		clog 24105	. . . . . 6 class log
19	17, 18	cfv 5804	. . . . 5 class (log‘∪ 𝑠)
20		cc0 9815	. . . . 5 class 0
21	16, 19, 20	cif 4036	. . . 4 class if((#‘𝑠) = 1, (log‘∪ 𝑠), 0)
22	4, 11, 21	csb 3499	. . 3 class ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0)
23	2, 3, 22	cmpt 4643	. 2 class (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0))
24	1, 23	wceq 1475	1 wff Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0))

This definition is referenced by:  vmaval  24639  vmaf  24645