Definition df-vma 24824

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 e. NN |-> [_ { p e. Prime | p || x } / s ]_ if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 ) )

Distinct variable group:   s, p, x

Detailed syntax breakdown of Definition df-vma

Step	Hyp	Ref	Expression
1		cvma 24818	. 2 class Λ
2		vx	. . 3 setvar x
3		cn 11020	. . 3 class NN
4		vs	. . . 4 setvar s
5		vp	. . . . . . 7 setvar p
6	5	cv 1482	. . . . . 6 class p
7	2	cv 1482	. . . . . 6 class x
8		cdvds 14983	. . . . . 6 class ||
9	6, 7, 8	wbr 4653	. . . . 5 wff p || x
10		cprime 15385	. . . . 5 class Prime
11	9, 5, 10	crab 2916	. . . 4 class { p e. Prime | p || x }
12	4	cv 1482	. . . . . . 7 class s
13		chash 13117	. . . . . . 7 class #
14	12, 13	cfv 5888	. . . . . 6 class ( # ` s )
15		c1 9937	. . . . . 6 class 1
16	14, 15	wceq 1483	. . . . 5 wff ( # ` s ) = 1
17	12	cuni 4436	. . . . . 6 class U. s
18		clog 24301	. . . . . 6 class log
19	17, 18	cfv 5888	. . . . 5 class ( log ` U. s )
20		cc0 9936	. . . . 5 class 0
21	16, 19, 20	cif 4086	. . . 4 class if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 )
22	4, 11, 21	csb 3533	. . 3 class [_ { p e. Prime | p || x } / s ]_ if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 )
23	2, 3, 22	cmpt 4729	. 2 class ( x e. NN |-> [_ { p e. Prime | p || x } / s ]_ if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 ) )
24	1, 23	wceq 1483	1 wff Λ = ( x e. NN |-> [_ { p e. Prime | p || x } / s ]_ if ( ( # ` s ) = 1 , ( log ` U. s ) , 0 ) )

This definition is referenced by:  vmaval  24839  vmaf  24845