Theorem selberg 23599

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 5872	. . . . . . . . . . . . 13 |- ( n = d -> (Λ ` n ) = (Λ ` d ) )
2		oveq2 6303	. . . . . . . . . . . . . 14 |- ( n = d -> ( x / n ) = ( x / d ) )
3	2	fveq2d 5876	. . . . . . . . . . . . 13 |- ( n = d -> (ψ ` ( x / n ) ) = (ψ ` ( x / d ) ) )
4	1, 3	oveq12d 6313	. . . . . . . . . . . 12 |- ( n = d -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) )
5	4	cbvsumv 13498	. . . . . . . . . . 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 12063	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
7		elfznn 11726	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
8	7	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
9		vmacl 23258	. . . . . . . . . . . . . . . 16 |- ( d e. NN -> (Λ ` d ) e. RR )
10	8, 9	syl 16	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. RR )
11	10	recnd 9634	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. CC )
12		elfznn 11726	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 1 ... ( |_ ` ( x / d ) ) ) -> m e. NN )
13	12	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. NN )
14		vmacl 23258	. . . . . . . . . . . . . . . 16 |- ( m e. NN -> (Λ ` m ) e. RR )
15	13, 14	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. RR )
16	15	recnd 9634	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. CC )
17	6, 11, 16	fsummulc2 13579	. . . . . . . . . . . . 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 11267	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
19		rpdivcl 11254	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
20	18, 19	sylan2 474	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
21	20	rpred 11268	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR )
22		chpval 23262	. . . . . . . . . . . . . . 15 |- ( ( x / d ) e. RR -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
23	21, 22	syl 16	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / d ) ) = sum_ m e. ( 1 ... ( |_ ` ( x / d ) ) ) (Λ ` m ) )
24	23	oveq2d 6311	. . . . . . . . . . . . 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 10564	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. CC )
26	7	ad2antlr 726	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. NN )
27	26	nncnd 10564	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. CC )
28	26	nnne0d 10592	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d =/= 0 )
29	25, 27, 28	divcan3d 10337	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( d x. m ) / d ) = m )
30	29	fveq2d 5876	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` ( ( d x. m ) / d ) ) = (Λ ` m ) )
31	30	oveq2d 6311	. . . . . . . . . . . . . 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 13505	. . . . . . . . . . . . 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 2518	. . . . . . . . . . . 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 13505	. . . . . . . . . . 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 2520	. . . . . . . . . 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 6302	. . . . . . . . . . . . 13 |- ( n = ( d x. m ) -> ( n / d ) = ( ( d x. m ) / d ) )
37	36	fveq2d 5876	. . . . . . . . . . . 12 |- ( n = ( d x. m ) -> (Λ ` ( n / d ) ) = (Λ ` ( ( d x. m ) / d ) ) )
38	37	oveq2d 6311	. . . . . . . . . . 11 |- ( n = ( d x. m ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
39		rpre 11238	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
40		ssrab2 3590	. . . . . . . . . . . . . . . . 17 |- { y e. NN | y || n } C_ NN
41		simprr 756	. . . . . . . . . . . . . . . . 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 3507	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. NN )
43	42	anassrs 648	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> d e. NN )
44	43, 9	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` d ) e. RR )
45		elfznn 11726	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
46	45	adantl 466	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
47		dvdsdivcl 23323	. . . . . . . . . . . . . . . . 17 |- ( ( n e. NN /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
48	46, 47	sylan 471	. . . . . . . . . . . . . . . 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 3507	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. NN )
50		vmacl 23258	. . . . . . . . . . . . . . 15 |- ( ( n / d ) e. NN -> (Λ ` ( n / d ) ) e. RR )
51	49, 50	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> (Λ ` ( n / d ) ) e. RR )
52	44, 51	remulcld 9636	. . . . . . . . . . . . 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 9634	. . . . . . . . . . . 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 647	. . . . . . . . . . 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 23330	. . . . . . . . . 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 2511	. . . . . . . . 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 6310	. . . . . . . 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 12063	. . . . . . . . 9 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
59		vmacl 23258	. . . . . . . . . . . 12 |- ( n e. NN -> (Λ ` n ) e. RR )
60	46, 59	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. RR )
61	60	recnd 9634	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
62	45	nnrpd 11267	. . . . . . . . . . . . . 14 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
63		rpdivcl 11254	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. RR+ ) -> ( x / n ) e. RR+ )
64	62, 63	sylan2 474	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ )
65	64	rpred 11268	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
66		chpcl 23264	. . . . . . . . . . . 12 |- ( ( x / n ) e. RR -> (ψ ` ( x / n ) ) e. RR )
67	65, 66	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. RR )
68	67	recnd 9634	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
69	61, 68	mulcld 9628	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
70	46	nnrpd 11267	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
71		relogcl 22829	. . . . . . . . . . . 12 |- ( n e. RR+ -> ( log ` n ) e. RR )
72	70, 71	syl 16	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. RR )
73	72	recnd 9634	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
74	61, 73	mulcld 9628	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
75	58, 69, 74	fsumadd 13541	. . . . . . . 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 12063	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... n ) e. Fin )
77		sgmss 23246	. . . . . . . . . . . . 13 |- ( n e. NN -> { y e. NN | y || n } C_ ( 1 ... n ) )
78	46, 77	syl 16	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } C_ ( 1 ... n ) )
79		ssfi 7752	. . . . . . . . . . . 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 661	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } e. Fin )
81	80, 52	fsumrecl 13536	. . . . . . . . . 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 9634	. . . . . . . . 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 13541	. . . . . . . 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 2518	. . . . . . 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 9793	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) = ( (ψ ` ( x / n ) ) + ( log ` n ) ) )
86	85	oveq2d 6311	. . . . . . . . 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 9632	. . . . . . . . 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 2508	. . . . . . . 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 13505	. . . . . . 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 23594	. . . . . . . . 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 16	. . . . . . . 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 13505	. . . . . . 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 2518	. . . . . 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 5876	. . . . . . . . 9 |- ( n = ( d x. m ) -> ( log ` ( n / d ) ) = ( log ` ( ( d x. m ) / d ) ) )
95	94	oveq1d 6310	. . . . . . . 8 |- ( n = ( d x. m ) -> ( ( log ` ( n / d ) ) ^ 2 ) = ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) )
96	95	oveq2d 6311	. . . . . . 7 |- ( n = ( d x. m ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
97		mucl 23281	. . . . . . . . . 10 |- ( d e. NN -> ( mmu ` d ) e. ZZ )
98	42, 97	syl 16	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. ZZ )
99	98	zcnd 10979	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. CC )
100	62	ad2antrl 727	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> n e. RR+ )
101	42	nnrpd 11267	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. RR+ )
102	100, 101	rpdivcld 11285	. . . . . . . . . 10 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( n / d ) e. RR+ )
103		relogcl 22829	. . . . . . . . . . 11 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. RR )
104	103	recnd 9634	. . . . . . . . . 10 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. CC )
105	102, 104	syl 16	. . . . . . . . 9 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( log ` ( n / d ) ) e. CC )
106	105	sqcld 12288	. . . . . . . 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 9628	. . . . . . 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 23330	. . . . . 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 5876	. . . . . . . . . 10 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` ( ( d x. m ) / d ) ) = ( log ` m ) )
110	109	oveq1d 6310	. . . . . . . . 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 6311	. . . . . . . 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 13505	. . . . . . 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 13505	. . . . . 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 2512	. . . . 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 6310	. . . 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 6310	. . 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 4535	. 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 2467	. . 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 23597	. 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 2551	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 369   = wceq 1379   e. wcel 1767   {crab 2821   C_ wss 3481   class class class wbr 4453   |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   RRcr 9503   1c1 9505   + caddc 9507   x. cmul 9509   - cmin 9817   / cdiv 10218   NNcn 10548   2c2 10597   ZZcz 10876   RR+crp 11232   ...cfz 11684   |_cfl 11907   ^cexp 12146   O(1)co1 13289   sum_csu 13488   || cdivides 13864   logclog 22808  Λcvma 23231  ψcchp 23232   mmucmu 23234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-o1 13293  df-lo1 13294  df-sum 13489  df-ef 13682  df-e 13683  df-sin 13684  df-cos 13685  df-pi 13687  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810  df-cxp 22811  df-em 23188  df-vma 23237  df-chp 23238  df-mu 23240

This theorem is referenced by:  selbergb  23600  selberg2  23602  selbergs  23625