Theorem selberg2 25240

Description: Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)

Assertion

Ref	Expression
selberg2	|- ( x e. RR+ |-> ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)

Distinct variable group:   x, n

Proof of Theorem selberg2

Step	Hyp	Ref	Expression
1		reex 10027	. . . . . . 7 |- RR e. _V
2		rpssre 11843	. . . . . . 7 |- RR+ C_ RR
3	1, 2	ssexi 4803	. . . . . 6 |- RR+ e. _V
4	3	a1i 11	. . . . 5 |- ( T. -> RR+ e. _V )
5		ovexd 6680	. . . . 5 |- ( ( T. /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) e. _V )
6		ovexd 6680	. . . . 5 |- ( ( T. /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) e. _V )
7		eqidd 2623	. . . . 5 |- ( T. -> ( 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_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) )
8		eqidd 2623	. . . . 5 |- ( T. -> ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) = ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) )
9	4, 5, 6, 7, 8	offval2 6914	. . . 4 |- ( T. -> ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) oF - ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) )
10	9	trud 1493	. . 3 |- ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) oF - ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) )
11		fzfid 12772	. . . . . . . 8 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
12		elfznn 12370	. . . . . . . . . . . 12 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
13	12	adantl 482	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
14		vmacl 24844	. . . . . . . . . . 11 |- ( n e. NN -> (Λ ` n ) e. RR )
15	13, 14	syl 17	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. RR )
16	15	recnd 10068	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
17	13	nnrpd 11870	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
18		relogcl 24322	. . . . . . . . . . . 12 |- ( n e. RR+ -> ( log ` n ) e. RR )
19	17, 18	syl 17	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. RR )
20	19	recnd 10068	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` n ) e. CC )
21		rpre 11839	. . . . . . . . . . . . 13 |- ( x e. RR+ -> x e. RR )
22		nndivre 11056	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ n e. NN ) -> ( x / n ) e. RR )
23	21, 12, 22	syl2an 494	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
24		chpcl 24850	. . . . . . . . . . . 12 |- ( ( x / n ) e. RR -> (ψ ` ( x / n ) ) e. RR )
25	23, 24	syl 17	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. RR )
26	25	recnd 10068	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (ψ ` ( x / n ) ) e. CC )
27	20, 26	addcld 10059	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` n ) + (ψ ` ( x / n ) ) ) e. CC )
28	16, 27	mulcld 10060	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) e. CC )
29	11, 28	fsumcl 14464	. . . . . . 7 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) e. CC )
30		rpcn 11841	. . . . . . 7 |- ( x e. RR+ -> x e. CC )
31		rpne0 11848	. . . . . . 7 |- ( x e. RR+ -> x =/= 0 )
32	29, 30, 31	divcld 10801	. . . . . 6 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) e. CC )
33		2cn 11091	. . . . . . 7 |- 2 e. CC
34		relogcl 24322	. . . . . . . 8 |- ( x e. RR+ -> ( log ` x ) e. RR )
35	34	recnd 10068	. . . . . . 7 |- ( x e. RR+ -> ( log ` x ) e. CC )
36		mulcl 10020	. . . . . . 7 |- ( ( 2 e. CC /\ ( log ` x ) e. CC ) -> ( 2 x. ( log ` x ) ) e. CC )
37	33, 35, 36	sylancr 695	. . . . . 6 |- ( x e. RR+ -> ( 2 x. ( log ` x ) ) e. CC )
38	16, 20	mulcld 10060	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( log ` n ) ) e. CC )
39	11, 38	fsumcl 14464	. . . . . . . 8 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) e. CC )
40		chpcl 24850	. . . . . . . . . . 11 |- ( x e. RR -> (ψ ` x ) e. RR )
41	21, 40	syl 17	. . . . . . . . . 10 |- ( x e. RR+ -> (ψ ` x ) e. RR )
42	41	recnd 10068	. . . . . . . . 9 |- ( x e. RR+ -> (ψ ` x ) e. CC )
43	42, 35	mulcld 10060	. . . . . . . 8 |- ( x e. RR+ -> ( (ψ ` x ) x. ( log ` x ) ) e. CC )
44	39, 43	subcld 10392	. . . . . . 7 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) e. CC )
45	44, 30, 31	divcld 10801	. . . . . 6 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) e. CC )
46	32, 37, 45	sub32d 10424	. . . . 5 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) - ( 2 x. ( log ` x ) ) ) )
47		rpcnne0 11850	. . . . . . . 8 |- ( x e. RR+ -> ( x e. CC /\ x =/= 0 ) )
48		divsubdir 10721	. . . . . . . 8 |- ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) e. CC /\ ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) e. CC /\ ( x e. CC /\ x =/= 0 ) ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) / x ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) )
49	29, 44, 47, 48	syl3anc 1326	. . . . . . 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. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) / x ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) )
50	16, 20, 26	adddid 10064	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( ( (Λ ` n ) x. ( log ` n ) ) + ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
51	50	sumeq2dv 14433	. . . . . . . . . . 11 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. ( log ` n ) ) + ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
52	16, 26	mulcld 10060	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
53	11, 38, 52	fsumadd 14470	. . . . . . . . . . 11 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) x. ( log ` n ) ) + ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
54	51, 53	eqtrd 2656	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
55	54	oveq1d 6665	. . . . . . . . 9 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (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 ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) )
56	11, 52	fsumcl 14464	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) e. CC )
57	39, 56, 43	pnncand 10431	. . . . . . . . 9 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (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 ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (ψ ` x ) x. ( log ` x ) ) ) )
58	56, 43	addcomd 10238	. . . . . . . . 9 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) + ( (ψ ` x ) x. ( log ` x ) ) ) = ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
59	55, 57, 58	3eqtrd 2660	. . . . . . . 8 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) = ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) )
60	59	oveq1d 6665	. . . . . . 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. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) ) / x ) = ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) )
61	49, 60	eqtr3d 2658	. . . . . 6 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) = ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) )
62	61	oveq1d 6665	. . . . 5 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) - ( 2 x. ( log ` x ) ) ) = ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
63	46, 62	eqtrd 2656	. . . 4 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) = ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
64	63	mpteq2ia 4740	. . 3 |- ( x e. RR+ |-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) - ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
65	10, 64	eqtri 2644	. 2 |- ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) oF - ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) )
66		selberg 25237	. . 3 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)
67		selberg2lem 25239	. . 3 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) e. O(1)
68		o1sub 14346	. . 3 |- ( ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1) /\ ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) e. O(1) ) -> ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) oF - ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) e. O(1) )
69	66, 67, 68	mp2an 708	. 2 |- ( ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( ( log ` n ) + (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) oF - ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( log ` n ) ) - ( (ψ ` x ) x. ( log ` x ) ) ) / x ) ) ) e. O(1)
70	65, 69	eqeltrri 2698	1 |- ( x e. RR+ |-> ( ( ( ( (ψ ` x ) x. ( log ` x ) ) + sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. (ψ ` ( x / n ) ) ) ) / x ) - ( 2 x. ( log ` x ) ) ) ) e. O(1)

Syntax hints:   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   ...cfz 12326   |_cfl 12591   O(1)co1 14217   sum_csu 14416   logclog 24301  Λcvma 24818  ψcchp 24819

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-cht 24823  df-vma 24824  df-chp 24825  df-ppi 24826  df-mu 24827

This theorem is referenced by:  selberg2b  25241  selberg3  25248  selberg4  25250  selbergr  25257