Theorem selberg 22777

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 5686	. . . . . . . . . . . . 13 |- ( n = d -> (Λ ` n ) = (Λ ` d ) )
2		oveq2 6094	. . . . . . . . . . . . . 14 |- ( n = d -> ( x / n ) = ( x / d ) )
3	2	fveq2d 5690	. . . . . . . . . . . . 13 |- ( n = d -> (ψ ` ( x / n ) ) = (ψ ` ( x / d ) ) )
4	1, 3	oveq12d 6104	. . . . . . . . . . . 12 |- ( n = d -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) = ( (Λ ` d ) x. (ψ ` ( x / d ) ) ) )
5	4	cbvsumv 13165	. . . . . . . . . . 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 11787	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
7		elfznn 11470	. . . . . . . . . . . . . . . . 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 22436	. . . . . . . . . . . . . . . 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 9404	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` d ) e. CC )
12		elfznn 11470	. . . . . . . . . . . . . . . . 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 22436	. . . . . . . . . . . . . . . 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 9404	. . . . . . . . . . . . . 14 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` m ) e. CC )
17	6, 11, 16	fsummulc2 13243	. . . . . . . . . . . . 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 11018	. . . . . . . . . . . . . . . . 17 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
19		rpdivcl 11005	. . . . . . . . . . . . . . . . 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 11019	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR )
22		chpval 22440	. . . . . . . . . . . . . . 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 6102	. . . . . . . . . . . . 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 10330	. . . . . . . . . . . . . . . . 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 10330	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d e. CC )
28	26	nnne0d 10358	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> d =/= 0 )
29	25, 27, 28	divcan3d 10104	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( d x. m ) / d ) = m )
30	29	fveq2d 5690	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> (Λ ` ( ( d x. m ) / d ) ) = (Λ ` m ) )
31	30	oveq2d 6102	. . . . . . . . . . . . . 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 13172	. . . . . . . . . . . . 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 2480	. . . . . . . . . . . 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 13172	. . . . . . . . . . 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 2482	. . . . . . . . . 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 6093	. . . . . . . . . . . . 13 |- ( n = ( d x. m ) -> ( n / d ) = ( ( d x. m ) / d ) )
37	36	fveq2d 5690	. . . . . . . . . . . 12 |- ( n = ( d x. m ) -> (Λ ` ( n / d ) ) = (Λ ` ( ( d x. m ) / d ) ) )
38	37	oveq2d 6102	. . . . . . . . . . 11 |- ( n = ( d x. m ) -> ( (Λ ` d ) x. (Λ ` ( n / d ) ) ) = ( (Λ ` d ) x. (Λ ` ( ( d x. m ) / d ) ) ) )
39		rpre 10989	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
40		ssrab2 3432	. . . . . . . . . . . . . . . . 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 3349	. . . . . . . . . . . . . . . 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 11470	. . . . . . . . . . . . . . . . . 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 22501	. . . . . . . . . . . . . . . . 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 3349	. . . . . . . . . . . . . . 15 |- ( ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ d e. { y e. NN | y || n } ) -> ( n / d ) e. NN )
50		vmacl 22436	. . . . . . . . . . . . . . 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 9406	. . . . . . . . . . . . 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 9404	. . . . . . . . . . . 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 22508	. . . . . . . . . 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 2473	. . . . . . . . 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 6101	. . . . . . . 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 11787	. . . . . . . . 9 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
59		vmacl 22436	. . . . . . . . . . . 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 9404	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
62	45	nnrpd 11018	. . . . . . . . . . . . . 14 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
63		rpdivcl 11005	. . . . . . . . . . . . . 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 11019	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
66		chpcl 22442	. . . . . . . . . . . 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 9404	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
69	61, 68	mulcld 9398	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
70	46	nnrpd 11018	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
71		relogcl 22007	. . . . . . . . . . . 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 9404	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
74	61, 73	mulcld 9398	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
75	58, 69, 74	fsumadd 13207	. . . . . . . 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 11787	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... n ) e. Fin )
77		sgmss 22424	. . . . . . . . . . . . 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 7525	. . . . . . . . . . . 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 13203	. . . . . . . . . 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 9404	. . . . . . . . 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 13207	. . . . . . . 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 2480	. . . . . . 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 9563	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) = ( (ψ ` ( x / n ) ) + ( log ` n ) ) )
86	85	oveq2d 6102	. . . . . . . . 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 9402	. . . . . . . . 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 2470	. . . . . . . 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 13172	. . . . . . 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 22772	. . . . . . . . 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 13172	. . . . . . 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 2480	. . . . . 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 5690	. . . . . . . . 9 |- ( n = ( d x. m ) -> ( log ` ( n / d ) ) = ( log ` ( ( d x. m ) / d ) ) )
95	94	oveq1d 6101	. . . . . . . 8 |- ( n = ( d x. m ) -> ( ( log ` ( n / d ) ) ^ 2 ) = ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) )
96	95	oveq2d 6102	. . . . . . 7 |- ( n = ( d x. m ) -> ( ( mmu ` d ) x. ( ( log ` ( n / d ) ) ^ 2 ) ) = ( ( mmu ` d ) x. ( ( log ` ( ( d x. m ) / d ) ) ^ 2 ) ) )
97		mucl 22459	. . . . . . . . . 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 10740	. . . . . . . 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 11018	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> d e. RR+ )
102	100, 101	rpdivcld 11036	. . . . . . . . . 10 |- ( ( x e. RR+ /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ d e. { y e. NN | y || n } ) ) -> ( n / d ) e. RR+ )
103		relogcl 22007	. . . . . . . . . . 11 |- ( ( n / d ) e. RR+ -> ( log ` ( n / d ) ) e. RR )
104	103	recnd 9404	. . . . . . . . . 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 11998	. . . . . . . 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 9398	. . . . . . 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 22508	. . . . . 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 5690	. . . . . . . . . 10 |- ( ( ( x e. RR+ /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ m e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` ( ( d x. m ) / d ) ) = ( log ` m ) )
110	109	oveq1d 6101	. . . . . . . . 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 6102	. . . . . . . 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 13172	. . . . . . 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 13172	. . . . . 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 2474	. . . . 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 6101	. . . 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 6101	. . 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 4369	. 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 2438	. . 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 22775	. 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 2508	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 1369   e. wcel 1756   {crab 2714   C_ wss 3323   class class class wbr 4287   e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   1c1 9275   + caddc 9277   x. cmul 9279   - cmin 9587   / cdiv 9985   NNcn 10314   2c2 10363   ZZcz 10638   RR+crp 10983   ...cfz 11429   |_cfl 11632   ^cexp 11857   O(1)co1 12956   sum_csu 13155   || cdivides 13527   logclog 21986  Λcvma 22409  ψcchp 22410   mmucmu 22412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-o1 12960  df-lo1 12961  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-cxp 21989  df-em 22366  df-vma 22415  df-chp 22416  df-mu 22418

This theorem is referenced by:  selbergb  22778  selberg2  22780  selbergs  22803