Theorem vmadivsum 25171

Description: The sum of the von Mangoldt function over n is asymptotic to log x + O(1). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)

Assertion

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

Distinct variable group:   x, n

Proof of Theorem vmadivsum

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 ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) e. _V )
6		ovexd 6680	. . . . 5 |- ( ( T. /\ x e. RR+ ) -> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) e. _V )
7		eqidd 2623	. . . . 5 |- ( T. -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) )
8		eqidd 2623	. . . . 5 |- ( T. -> ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) = ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) )
9	4, 5, 6, 7, 8	offval2 6914	. . . 4 |- ( T. -> ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) oF - ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) = ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) - ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) )
10	9	trud 1493	. . 3 |- ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) oF - ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) = ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) - ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) )
11		fzfid 12772	. . . . . . 7 |- ( x e. RR+ -> ( 1 ... ( |_ ` x ) ) e. Fin )
12		elfznn 12370	. . . . . . . . . 10 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
13	12	adantl 482	. . . . . . . . 9 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
14		vmacl 24844	. . . . . . . . 9 |- ( n e. NN -> (Λ ` n ) e. RR )
15	13, 14	syl 17	. . . . . . . 8 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. RR )
16	15, 13	nndivred 11069	. . . . . . 7 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) / n ) e. RR )
17	11, 16	fsumrecl 14465	. . . . . 6 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) e. RR )
18	17	recnd 10068	. . . . 5 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) e. CC )
19		relogcl 24322	. . . . . 6 |- ( x e. RR+ -> ( log ` x ) e. RR )
20	19	recnd 10068	. . . . 5 |- ( x e. RR+ -> ( log ` x ) e. CC )
21		rprege0 11847	. . . . . . . . . 10 |- ( x e. RR+ -> ( x e. RR /\ 0 <_ x ) )
22		flge0nn0 12621	. . . . . . . . . 10 |- ( ( x e. RR /\ 0 <_ x ) -> ( |_ ` x ) e. NN0 )
23		faccl 13070	. . . . . . . . . 10 |- ( ( |_ ` x ) e. NN0 -> ( ! ` ( |_ ` x ) ) e. NN )
24	21, 22, 23	3syl 18	. . . . . . . . 9 |- ( x e. RR+ -> ( ! ` ( |_ ` x ) ) e. NN )
25	24	nnrpd 11870	. . . . . . . 8 |- ( x e. RR+ -> ( ! ` ( |_ ` x ) ) e. RR+ )
26	25	relogcld 24369	. . . . . . 7 |- ( x e. RR+ -> ( log ` ( ! ` ( |_ ` x ) ) ) e. RR )
27		rerpdivcl 11861	. . . . . . 7 |- ( ( ( log ` ( ! ` ( |_ ` x ) ) ) e. RR /\ x e. RR+ ) -> ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) e. RR )
28	26, 27	mpancom 703	. . . . . 6 |- ( x e. RR+ -> ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) e. RR )
29	28	recnd 10068	. . . . 5 |- ( x e. RR+ -> ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) e. CC )
30	18, 20, 29	nnncan2d 10427	. . . 4 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) - ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( log ` x ) ) )
31	30	mpteq2ia 4740	. . 3 |- ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) - ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( log ` x ) ) )
32	10, 31	eqtri 2644	. 2 |- ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) oF - ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( log ` x ) ) )
33		1red 10055	. . . . 5 |- ( T. -> 1 e. RR )
34		chpo1ub 25169	. . . . . 6 |- ( x e. RR+ |-> ( (ψ ` x ) / x ) ) e. O(1)
35	34	a1i 11	. . . . 5 |- ( T. -> ( x e. RR+ |-> ( (ψ ` x ) / x ) ) e. O(1) )
36		rpre 11839	. . . . . . . . 9 |- ( x e. RR+ -> x e. RR )
37		chpcl 24850	. . . . . . . . 9 |- ( x e. RR -> (ψ ` x ) e. RR )
38	36, 37	syl 17	. . . . . . . 8 |- ( x e. RR+ -> (ψ ` x ) e. RR )
39		rerpdivcl 11861	. . . . . . . 8 |- ( ( (ψ ` x ) e. RR /\ x e. RR+ ) -> ( (ψ ` x ) / x ) e. RR )
40	38, 39	mpancom 703	. . . . . . 7 |- ( x e. RR+ -> ( (ψ ` x ) / x ) e. RR )
41	40	recnd 10068	. . . . . 6 |- ( x e. RR+ -> ( (ψ ` x ) / x ) e. CC )
42	41	adantl 482	. . . . 5 |- ( ( T. /\ x e. RR+ ) -> ( (ψ ` x ) / x ) e. CC )
43	18, 29	subcld 10392	. . . . . 6 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) e. CC )
44	43	adantl 482	. . . . 5 |- ( ( T. /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) e. CC )
45	36	adantr 481	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> x e. RR )
46	16, 45	remulcld 10070	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( (Λ ` n ) / n ) x. x ) e. RR )
47		nndivre 11056	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ n e. NN ) -> ( x / n ) e. RR )
48	36, 12, 47	syl2an 494	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR )
49		reflcl 12597	. . . . . . . . . . . . 13 |- ( ( x / n ) e. RR -> ( |_ ` ( x / n ) ) e. RR )
50	48, 49	syl 17	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( |_ ` ( x / n ) ) e. RR )
51	15, 50	remulcld 10070	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) e. RR )
52	46, 51	resubcld 10458	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) e. RR )
53	48, 50	resubcld 10458	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( x / n ) - ( |_ ` ( x / n ) ) ) e. RR )
54		1red 10055	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 1 e. RR )
55		vmage0 24847	. . . . . . . . . . . . 13 |- ( n e. NN -> 0 <_ (Λ ` n ) )
56	13, 55	syl 17	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ (Λ ` n ) )
57		fracle1 12604	. . . . . . . . . . . . 13 |- ( ( x / n ) e. RR -> ( ( x / n ) - ( |_ ` ( x / n ) ) ) <_ 1 )
58	48, 57	syl 17	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( x / n ) - ( |_ ` ( x / n ) ) ) <_ 1 )
59	53, 54, 15, 56, 58	lemul2ad 10964	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( x / n ) - ( |_ ` ( x / n ) ) ) ) <_ ( (Λ ` n ) x. 1 ) )
60	15	recnd 10068	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> (Λ ` n ) e. CC )
61	48	recnd 10068	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. CC )
62	50	recnd 10068	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( |_ ` ( x / n ) ) e. CC )
63	60, 61, 62	subdid 10486	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( x / n ) - ( |_ ` ( x / n ) ) ) ) = ( ( (Λ ` n ) x. ( x / n ) ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
64		rpcn 11841	. . . . . . . . . . . . . . 15 |- ( x e. RR+ -> x e. CC )
65	64	adantr 481	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> x e. CC )
66	13	nnrpd 11870	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. RR+ )
67		rpcnne0 11850	. . . . . . . . . . . . . . 15 |- ( n e. RR+ -> ( n e. CC /\ n =/= 0 ) )
68	66, 67	syl 17	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( n e. CC /\ n =/= 0 ) )
69		div23 10704	. . . . . . . . . . . . . . 15 |- ( ( (Λ ` n ) e. CC /\ x e. CC /\ ( n e. CC /\ n =/= 0 ) ) -> ( ( (Λ ` n ) x. x ) / n ) = ( ( (Λ ` n ) / n ) x. x ) )
70		divass 10703	. . . . . . . . . . . . . . 15 |- ( ( (Λ ` n ) e. CC /\ x e. CC /\ ( n e. CC /\ n =/= 0 ) ) -> ( ( (Λ ` n ) x. x ) / n ) = ( (Λ ` n ) x. ( x / n ) ) )
71	69, 70	eqtr3d 2658	. . . . . . . . . . . . . 14 |- ( ( (Λ ` n ) e. CC /\ x e. CC /\ ( n e. CC /\ n =/= 0 ) ) -> ( ( (Λ ` n ) / n ) x. x ) = ( (Λ ` n ) x. ( x / n ) ) )
72	60, 65, 68, 71	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( (Λ ` n ) / n ) x. x ) = ( (Λ ` n ) x. ( x / n ) ) )
73	72	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) = ( ( (Λ ` n ) x. ( x / n ) ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
74	63, 73	eqtr4d 2659	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( ( x / n ) - ( |_ ` ( x / n ) ) ) ) = ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
75	60	mulid1d 10057	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. 1 ) = (Λ ` n ) )
76	59, 74, 75	3brtr3d 4684	. . . . . . . . . 10 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) <_ (Λ ` n ) )
77	11, 52, 15, 76	fsumle 14531	. . . . . . . . 9 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) <_ sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) )
78	16	recnd 10068	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) / n ) e. CC )
79	11, 64, 78	fsummulc1 14517	. . . . . . . . . . 11 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) / n ) x. x ) )
80		logfac2 24942	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 <_ x ) -> ( log ` ( ! ` ( |_ ` x ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) )
81	21, 80	syl 17	. . . . . . . . . . 11 |- ( x e. RR+ -> ( log ` ( ! ` ( |_ ` x ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) )
82	79, 81	oveq12d 6668	. . . . . . . . . 10 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) / n ) x. x ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
83	46	recnd 10068	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( (Λ ` n ) / n ) x. x ) e. CC )
84	51	recnd 10068	. . . . . . . . . . 11 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) e. CC )
85	11, 83, 84	fsumsub 14520	. . . . . . . . . 10 |- ( x e. RR+ -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` n ) / n ) x. x ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
86	82, 85	eqtr4d 2659	. . . . . . . . 9 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
87		chpval 24848	. . . . . . . . . 10 |- ( x e. RR -> (ψ ` x ) = sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) )
88	36, 87	syl 17	. . . . . . . . 9 |- ( x e. RR+ -> (ψ ` x ) = sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) )
89	77, 86, 88	3brtr4d 4685	. . . . . . . 8 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) <_ (ψ ` x ) )
90	17, 36	remulcld 10070	. . . . . . . . . 10 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) e. RR )
91	90, 26	resubcld 10458	. . . . . . . . 9 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) e. RR )
92		rpregt0 11846	. . . . . . . . 9 |- ( x e. RR+ -> ( x e. RR /\ 0 < x ) )
93		lediv1 10888	. . . . . . . . 9 |- ( ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) e. RR /\ (ψ ` x ) e. RR /\ ( x e. RR /\ 0 < x ) ) -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) <_ (ψ ` x ) <-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) <_ ( (ψ ` x ) / x ) ) )
94	91, 38, 92, 93	syl3anc 1326	. . . . . . . 8 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) <_ (ψ ` x ) <-> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) <_ ( (ψ ` x ) / x ) ) )
95	89, 94	mpbid 222	. . . . . . 7 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) <_ ( (ψ ` x ) / x ) )
96	90	recnd 10068	. . . . . . . . . . 11 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) e. CC )
97	26	recnd 10068	. . . . . . . . . . 11 |- ( x e. RR+ -> ( log ` ( ! ` ( |_ ` x ) ) ) e. CC )
98		rpcnne0 11850	. . . . . . . . . . 11 |- ( x e. RR+ -> ( x e. CC /\ x =/= 0 ) )
99		divsubdir 10721	. . . . . . . . . . 11 |- ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) e. CC /\ ( log ` ( ! ` ( |_ ` x ) ) ) e. CC /\ ( x e. CC /\ x =/= 0 ) ) -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) / x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) )
100	96, 97, 98, 99	syl3anc 1326	. . . . . . . . . 10 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) / x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) )
101		rpne0 11848	. . . . . . . . . . . 12 |- ( x e. RR+ -> x =/= 0 )
102	18, 64, 101	divcan4d 10807	. . . . . . . . . . 11 |- ( x e. RR+ -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) / x ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) )
103	102	oveq1d 6665	. . . . . . . . . 10 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) / x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) )
104	100, 103	eqtr2d 2657	. . . . . . . . 9 |- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) )
105	104	fveq2d 6195	. . . . . . . 8 |- ( x e. RR+ -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) = ( abs ` ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) ) )
106		rerpdivcl 11861	. . . . . . . . . 10 |- ( ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) e. RR /\ x e. RR+ ) -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) e. RR )
107	91, 106	mpancom 703	. . . . . . . . 9 |- ( x e. RR+ -> ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) e. RR )
108		flle 12600	. . . . . . . . . . . . . . . 16 |- ( ( x / n ) e. RR -> ( |_ ` ( x / n ) ) <_ ( x / n ) )
109	48, 108	syl 17	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( |_ ` ( x / n ) ) <_ ( x / n ) )
110	48, 50	subge0d 10617	. . . . . . . . . . . . . . 15 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 0 <_ ( ( x / n ) - ( |_ ` ( x / n ) ) ) <-> ( |_ ` ( x / n ) ) <_ ( x / n ) ) )
111	109, 110	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( ( x / n ) - ( |_ ` ( x / n ) ) ) )
112	15, 53, 56, 111	mulge0d 10604	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( (Λ ` n ) x. ( ( x / n ) - ( |_ ` ( x / n ) ) ) ) )
113	112, 74	breqtrd 4679	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
114	11, 52, 113	fsumge0 14527	. . . . . . . . . . 11 |- ( x e. RR+ -> 0 <_ sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( (Λ ` n ) / n ) x. x ) - ( (Λ ` n ) x. ( |_ ` ( x / n ) ) ) ) )
115	114, 86	breqtrrd 4681	. . . . . . . . . 10 |- ( x e. RR+ -> 0 <_ ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) )
116		divge0 10892	. . . . . . . . . 10 |- ( ( ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) e. RR /\ 0 <_ ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) ) /\ ( x e. RR /\ 0 < x ) ) -> 0 <_ ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) )
117	91, 115, 92, 116	syl21anc 1325	. . . . . . . . 9 |- ( x e. RR+ -> 0 <_ ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) )
118	107, 117	absidd 14161	. . . . . . . 8 |- ( x e. RR+ -> ( abs ` ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) )
119	105, 118	eqtrd 2656	. . . . . . 7 |- ( x e. RR+ -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) = ( ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) x. x ) - ( log ` ( ! ` ( |_ ` x ) ) ) ) / x ) )
120		chpge0 24852	. . . . . . . . . 10 |- ( x e. RR -> 0 <_ (ψ ` x ) )
121	36, 120	syl 17	. . . . . . . . 9 |- ( x e. RR+ -> 0 <_ (ψ ` x ) )
122		divge0 10892	. . . . . . . . 9 |- ( ( ( (ψ ` x ) e. RR /\ 0 <_ (ψ ` x ) ) /\ ( x e. RR /\ 0 < x ) ) -> 0 <_ ( (ψ ` x ) / x ) )
123	38, 121, 92, 122	syl21anc 1325	. . . . . . . 8 |- ( x e. RR+ -> 0 <_ ( (ψ ` x ) / x ) )
124	40, 123	absidd 14161	. . . . . . 7 |- ( x e. RR+ -> ( abs ` ( (ψ ` x ) / x ) ) = ( (ψ ` x ) / x ) )
125	95, 119, 124	3brtr4d 4685	. . . . . 6 |- ( x e. RR+ -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) <_ ( abs ` ( (ψ ` x ) / x ) ) )
126	125	ad2antrl 764	. . . . 5 |- ( ( T. /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) <_ ( abs ` ( (ψ ` x ) / x ) ) )
127	33, 35, 42, 44, 126	o1le 14383	. . . 4 |- ( T. -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1) )
128	127	trud 1493	. . 3 |- ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1)
129		logfacrlim 24949	. . . 4 |- ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ~~> r 1
130		rlimo1 14347	. . . 4 |- ( ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ~~> r 1 -> ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1) )
131	129, 130	ax-mp 5	. . 3 |- ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1)
132		o1sub 14346	. . 3 |- ( ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1) /\ ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) e. O(1) ) -> ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) oF - ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) e. O(1) )
133	128, 131, 132	mp2an 708	. 2 |- ( ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) oF - ( x e. RR+ |-> ( ( log ` x ) - ( ( log ` ( ! ` ( |_ ` x ) ) ) / x ) ) ) ) e. O(1)
134	32, 133	eqeltrri 2698	1 |- ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( (Λ ` n ) / n ) - ( log ` x ) ) ) e. O(1)

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   RR+crp 11832   ...cfz 12326   |_cfl 12591   !cfa 13060   abscabs 13974   ~~> r crli 14216   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-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-cht 24823  df-vma 24824  df-chp 24825  df-ppi 24826

This theorem is referenced by:  vmadivsumb  25172  rpvmasumlem  25176  vmalogdivsum2  25227  vmalogdivsum  25228  2vmadivsumlem  25229  selberg3lem1  25246  selberg4lem1  25249  pntrsumo1  25254  selbergr  25257