Theorem selberg 24386

Description: Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that sum_ n <_ x , Λ ( n ) log n + sum_ m x. n <_ x , Λ ( m )Λ ( n ) = 2 x log x + O ( x ). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)

Assertion

Ref	Expression
selberg	|- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)

Distinct variable group:   x, n

Proof of Theorem selberg

Dummy variables d m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5865	. . . . . . . . . . . . 13 |- ( n = d -> (Λ ` n ) = (Λ ` d ) )
2		oveq2 6298	. . . . . . . . . . . . . 14 |- ( n = d -> ( x / n ) = ( x / d ) )
3	2	fveq2d 5869	. . . . . . . . . . . . 13 |- ( n = d -> (ψ ` ( x / n ) ) = (ψ ` ( x / d ) ) )
4	1, 3	oveq12d 6308	. . . . . . . . . . . 12 |- ( n = d -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) )
5	4	cbvsumv 13762	. . . . . . . . . . 11 |- sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) ( (Λ ` d ) x. (ψ ` ( x / d ) ) )
6		fzfid 12186	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
7		elfznn 11828	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
8	7	adantl 468	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
9		vmacl 24045	. . . . . . . . . . . . . . . 16 |- ( d e. NN -> (Λ ` d ) e. RR )
10	8, 9	syl 17	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. RR )
11	10	recnd 9669	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. CC )
12		elfznn 11828	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 1 ... ( |_ ` ( x / d ) ) ) -> m e. NN )
13	12	adantl 468	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. NN )
14		vmacl 24045	. . . . . . . . . . . . . . . 16 |- ( m e. NN -> (Λ ` m ) e. RR )
15	13, 14	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. RR )
16	15	recnd 9669	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. CC )
17	6, 11, 16	fsummulc2 13845	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` m ) ) )
18	7	nnrpd 11339	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
19		rpdivcl 11325	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
20	18, 19	sylan2 477	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
21	20	rpred 11341	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR )
22		chpval 24049	. . . . . . . . . . . . . . 15 |- ( ( x / d ) e. RR -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
23	21, 22	syl 17	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
24	23	oveq2d 6306	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = ( (Λ ` d ) x. sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) ) )
25	13	nncnd 10625	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. CC )
26	7	ad2antlr 733	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. NN )
27	26	nncnd 10625	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. CC )
28	26	nnne0d 10654	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d =/= 0 )
29	25, 27, 28	divcan3d 10388	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( d x. m ) / d ) = m )
30	29	fveq2d 5869	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` ( ( d x. m ) / d ) ) = (Λ ` m ) )
31	30	oveq2d 6306	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) = ( (Λ ` d ) x. (Λ ` m ) ) )
32	31	sumeq2dv 13769	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` m ) ) )
33	17, 24, 32	3eqtr4d 2495	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
34	33	sumeq2dv 13769	. . . . . . . . . . 11 |- ( x e. RR+ -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
35	5, 34	syl5eq 2497	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
36		oveq1 6297	. . . . . . . . . . . . 13 |- ( n = ( d x. m ) -> ( n / d ) = ( ( d x. m ) / d ) )
37	36	fveq2d 5869	. . . . . . . . . . . 12 |- ( n = ( d x. m ) -> (Λ ` ( n / d ) ) = (Λ ` ( ( d x. m ) / d ) ) )
38	37	oveq2d 6306	. . . . . . . . . . 11 |- ( n = ( d x. m ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
39		rpre 11308	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
40		ssrab2 3514	. . . . . . . . . . . . . . . . 17 |- { y e. NN | y || n } C_ NN
41		simprr 766	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. { y e. NN | y || n } )
42	40, 41	sseldi 3430	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. NN )
43	42	anassrs 654	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> d e. NN )
44	43, 9	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` d ) e. RR )
45		elfznn 11828	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
46	45	adantl 468	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
47		dvdsdivcl 24110	. . . . . . . . . . . . . . . . 17 |- ( ( n e. NN /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
48	46, 47	sylan 474	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
49	40, 48	sseldi 3430	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. NN )
50		vmacl 24045	. . . . . . . . . . . . . . 15 |- ( ( n / d ) e. NN -> (Λ ` ( n / d ) ) e. RR )
51	49, 50	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` ( n / d ) ) e. RR )
52	44, 51	remulcld 9671	. . . . . . . . . . . . 13 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. RR )
53	52	recnd 9669	. . . . . . . . . . . 12 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
54	53	anasss 653	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
55	38, 39, 54	dvdsflsumcom 24117	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
56	35, 55	eqtr4d 2488	. . . . . . . . 9 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) )
57	56	oveq1d 6305	. . . . . . . 8 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
58		fzfid 12186	. . . . . . . . 9 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
59		vmacl 24045	. . . . . . . . . . . 12 |- ( n e. NN -> (Λ ` n ) e. RR )
60	46, 59	syl 17	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. RR )
61	60	recnd 9669	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
62	45	nnrpd 11339	. . . . . . . . . . . . . 14 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
63		rpdivcl 11325	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. RR+ ) -> ( x / n ) e. RR+ )
64	62, 63	sylan2 477	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ )
65	64	rpred 11341	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
66		chpcl 24051	. . . . . . . . . . . 12 |- ( ( x / n ) e. RR -> (ψ ` ( x / n ) ) e. RR )
67	65, 66	syl 17	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. RR )
68	67	recnd 9669	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
69	61, 68	mulcld 9663	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
70	46	nnrpd 11339	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
71		relogcl 23525	. . . . . . . . . . . 12 |- ( n e. RR+ -> ( log ` n ) e. RR )
72	70, 71	syl 17	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. RR )
73	72	recnd 9669	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
74	61, 73	mulcld 9663	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
75	58, 69, 74	fsumadd 13805	. . . . . . . 8 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
76		fzfid 12186	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... n ) e. Fin )
77		sgmss 24033	. . . . . . . . . . . . 13 |- ( n e. NN -> { y e. NN | y || n } C_ ( 1 ... n ) )
78	46, 77	syl 17	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } C_ ( 1 ... n ) )
79		ssfi 7792	. . . . . . . . . . . 12 |- ( ( ( 1 ... n ) e. Fin /\ { y e. NN | y || n } C_ ( 1 ... n ) ) -> { y e. NN | y || n } e. Fin )
80	76, 78, 79	syl2anc 667	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } e. Fin )
81	80, 52	fsumrecl 13800	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. RR )
82	81	recnd 9669	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) e. CC )
83	58, 82, 74	fsumadd 13805	. . . . . . . 8 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) ) )
84	57, 75, 83	3eqtr4d 2495	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
85	73, 68	addcomd 9835	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) = ( (ψ ` ( x / n ) ) + ( log ` n ) ) )
86	85	oveq2d 6306	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( (Λ ` n ) x. ( (ψ ` ( x / n ) ) + ( log ` n ) ) ) )
87	61, 68, 73	adddid 9667	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( (ψ ` ( x / n ) ) + ( log ` n ) ) ) = ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
88	86, 87	eqtrd 2485	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
89	88	sumeq2dv 13769	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
90		logsqvma2 24381	. . . . . . . . 9 |- ( n e. NN -> sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
91	46, 90	syl 17	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
92	91	sumeq2dv 13769	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( sum_ d e. { y e. NN | y || n } ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) + ( (Λ ` n ) x. ( log ` n ) ) ) )
93	84, 89, 92	3eqtr4d 2495	. . . . . 6 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) )
94	36	fveq2d 5869	. . . . . . . . 9 |- ( n = ( d x. m ) -> ( log ` ( n / d ) ) = ( log ` ( ( d x. m ) / d ) ) )
95	94	oveq1d 6305	. . . . . . . 8 |- ( n = ( d x. m ) -> ( ( log ` ( n / d ) ) ^ 2 ) = ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) )
96	95	oveq2d 6306	. . . . . . 7 |- ( n = ( d x. m ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
97		mucl 24068	. . . . . . . . . 10 |- ( d e. NN -> ( mmu ` d ) e. ZZ )
98	42, 97	syl 17	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. ZZ )
99	98	zcnd 11041	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. CC )
100	62	ad2antrl 734	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> n e. RR+ )
101	42	nnrpd 11339	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. RR+ )
102	100, 101	rpdivcld 11358	. . . . . . . . . 10 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( n / d ) e. RR+ )
103		relogcl 23525	. . . . . . . . . . 11 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. RR )
104	103	recnd 9669	. . . . . . . . . 10 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. CC )
105	102, 104	syl 17	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( log ` ( n / d ) ) e. CC )
106	105	sqcld 12414	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( ( log ` ( n / d ) ) ^ 2 ) e. CC )
107	99, 106	mulcld 9663	. . . . . . 7 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) e. CC )
108	96, 39, 107	dvdsflsumcom 24117	. . . . . 6 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ d e. { y e. NN | y || n } ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
109	29	fveq2d 5869	. . . . . . . . . 10 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` ( ( d x. m ) / d ) ) = ( log ` m ) )
110	109	oveq1d 6305	. . . . . . . . 9 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) = ( ( log ` m ) ^ 2 ) )
111	110	oveq2d 6306	. . . . . . . 8 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
112	111	sumeq2dv 13769	. . . . . . 7 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
113	112	sumeq2dv 13769	. . . . . 6 |- ( x e. RR+ -> sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
114	93, 108, 113	3eqtrd 2489	. . . . 5 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) )
115	114	oveq1d 6305	. . . 4 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) )
116	115	oveq1d 6305	. . 3 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) = ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
117	116	mpteq2ia 4485	. 2 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) = ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
118		eqid 2451	. . 3 |- ( ( ( ( log ` ( x / d ) ) ^ 2 ) + ( 2 - ( 2 x. ( log ` ( x / d ) ) ) ) ) / d ) = ( ( ( ( log ` ( x / d ) ) ^ 2 ) + ( 2 - ( 2 x. ( log ` ( x / d ) ) ) ) ) / d )
119	118	selberglem2 24384	. 2 |- ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( mmu ` d ) x. ( ( log ` m ) ^ 2 ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)
120	117, 119	eqeltri 2525	1 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)

Syntax hints:   /\ wa 371   = wceq 1444   e. wcel 1887   {crab 2741   C_ wss 3404   class class class wbr 4402   |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   Fincfn 7569   CCcc 9537   RRcr 9538   1c1 9540   + caddc 9542   x. cmul 9544   - cmin 9860   / cdiv 10269   NNcn 10609   2c2 10659   ZZcz 10937   RR+crp 11302   ...cfz 11784   |_cfl 12026   ^cexp 12272   O(1)co1 13550   sum_csu 13752   || cdvds 14305   logclog 23504  Λcvma 24018  ψcchp 24019   mmucmu 24021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-o1 13554  df-lo1 13555  df-sum 13753  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-em 23918  df-vma 24024  df-chp 24025  df-mu 24027

This theorem is referenced by:  selbergb  24387  selberg2  24389  selbergs  24412