Theorem selberg 25237

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 6191	. . . . . . . . . . . . 13 |- ( n = d -> (Λ ` n ) = (Λ ` d ) )
2		oveq2 6658	. . . . . . . . . . . . . 14 |- ( n = d -> ( x / n ) = ( x / d ) )
3	2	fveq2d 6195	. . . . . . . . . . . . 13 |- ( n = d -> (ψ ` ( x / n ) ) = (ψ ` ( x / d ) ) )
4	1, 3	oveq12d 6668	. . . . . . . . . . . 12 |- ( n = d -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) )
5	4	cbvsumv 14426	. . . . . . . . . . 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 12772	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
7		elfznn 12370	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
8	7	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
9		vmacl 24844	. . . . . . . . . . . . . . . 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 10068	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. CC )
12		elfznn 12370	. . . . . . . . . . . . . . . . 17 |- ( m e. ( 1 ... ( |_ ` ( x / d ) ) ) -> m e. NN )
13	12	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. NN )
14		vmacl 24844	. . . . . . . . . . . . . . . 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 10068	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. CC )
17	6, 11, 16	fsummulc2 14516	. . . . . . . . . . . . 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 11870	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
19		rpdivcl 11856	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
20	18, 19	sylan2 491	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
21	20	rpred 11872	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR )
22		chpval 24848	. . . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . 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 11036	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> m e. CC )
26	7	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. NN )
27	26	nncnd 11036	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. CC )
28	26	nnne0d 11065	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d =/= 0 )
29	25, 27, 28	divcan3d 10806	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( d x. m ) / d ) = m )
30	29	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` ( ( d x. m ) / d ) ) = (Λ ` m ) )
31	30	oveq2d 6666	. . . . . . . . . . . . . 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 14433	. . . . . . . . . . . . 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 2666	. . . . . . . . . . . 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 14433	. . . . . . . . . . 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 2668	. . . . . . . . . 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 6657	. . . . . . . . . . . . 13 |- ( n = ( d x. m ) -> ( n / d ) = ( ( d x. m ) / d ) )
37	36	fveq2d 6195	. . . . . . . . . . . 12 |- ( n = ( d x. m ) -> (Λ ` ( n / d ) ) = (Λ ` ( ( d x. m ) / d ) ) )
38	37	oveq2d 6666	. . . . . . . . . . 11 |- ( n = ( d x. m ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
39		rpre 11839	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
40		ssrab2 3687	. . . . . . . . . . . . . . . . 17 |- { y e. NN | y || n } C_ NN
41		simprr 796	. . . . . . . . . . . . . . . . 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 3601	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. NN )
43	42	anassrs 680	. . . . . . . . . . . . . . 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 12370	. . . . . . . . . . . . . . . . . 18 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
46	45	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
47		dvdsdivcl 15038	. . . . . . . . . . . . . . . . 17 |- ( ( n e. NN /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. { y e. NN | y || n } )
48	46, 47	sylan 488	. . . . . . . . . . . . . . . 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 3601	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. NN )
50		vmacl 24844	. . . . . . . . . . . . . . 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 10070	. . . . . . . . . . . . 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 10068	. . . . . . . . . . . 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 679	. . . . . . . . . . 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 24914	. . . . . . . . . 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 2659	. . . . . . . . 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 6665	. . . . . . . 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 12772	. . . . . . . . 9 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
59		vmacl 24844	. . . . . . . . . . . 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 10068	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
62	45	nnrpd 11870	. . . . . . . . . . . . . 14 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
63		rpdivcl 11856	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. RR+ ) -> ( x / n ) e. RR+ )
64	62, 63	sylan2 491	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ )
65	64	rpred 11872	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
66		chpcl 24850	. . . . . . . . . . . 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 10068	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
69	61, 68	mulcld 10060	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
70	46	nnrpd 11870	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
71		relogcl 24322	. . . . . . . . . . . 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 10068	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
74	61, 73	mulcld 10060	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
75	58, 69, 74	fsumadd 14470	. . . . . . . 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 12772	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... n ) e. Fin )
77		dvdsssfz1 15040	. . . . . . . . . . . . 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 8180	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> { y e. NN | y || n } e. Fin )
81	80, 52	fsumrecl 14465	. . . . . . . . . 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 10068	. . . . . . . . 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 14470	. . . . . . . 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 2666	. . . . . . 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 10238	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) = ( (ψ ` ( x / n ) ) + ( log ` n ) ) )
86	85	oveq2d 6666	. . . . . . . . 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 10064	. . . . . . . . 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 2656	. . . . . . . 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 14433	. . . . . . 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 25232	. . . . . . . . 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 14433	. . . . . . 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 2666	. . . . . 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 6195	. . . . . . . . 9 |- ( n = ( d x. m ) -> ( log ` ( n / d ) ) = ( log ` ( ( d x. m ) / d ) ) )
95	94	oveq1d 6665	. . . . . . . 8 |- ( n = ( d x. m ) -> ( ( log ` ( n / d ) ) ^ 2 ) = ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) )
96	95	oveq2d 6666	. . . . . . 7 |- ( n = ( d x. m ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
97		mucl 24867	. . . . . . . . . 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 11483	. . . . . . . 8 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( mmu ` d ) e. CC )
100	62	ad2antrl 764	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> n e. RR+ )
101	42	nnrpd 11870	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. RR+ )
102	100, 101	rpdivcld 11889	. . . . . . . . . 10 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( n / d ) e. RR+ )
103		relogcl 24322	. . . . . . . . . . 11 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. RR )
104	103	recnd 10068	. . . . . . . . . 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 13006	. . . . . . . 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 10060	. . . . . . 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 24914	. . . . . 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 6195	. . . . . . . . . 10 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` ( ( d x. m ) / d ) ) = ( log ` m ) )
110	109	oveq1d 6665	. . . . . . . . 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 6666	. . . . . . . 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 14433	. . . . . . 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 14433	. . . . . 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 2660	. . . . 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 6665	. . . 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 6665	. . 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 4740	. 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 2622	. . 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 25235	. 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 2697	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 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   RR+crp 11832   ...cfz 12326   |_cfl 12591   ^cexp 12860   O(1)co1 14217   sum_csu 14416   || cdvds 14983   logclog 24301  Λcvma 24818  ψcchp 24819   mmucmu 24821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-o1 14221  df-lo1 14222  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-em 24719  df-vma 24824  df-chp 24825  df-mu 24827

This theorem is referenced by:  selbergb  25238  selberg2  25240  selbergs  25263