Theorem mulog2sumlem3 25225

Description: Lemma for mulog2sum 25226. (Contributed by Mario Carneiro, 13-May-2016.)

Hypotheses

Ref	Expression
logdivsum.1	|- F = ( y e. RR+ |-> ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) - ( ( ( log ` y ) ^ 2 ) / 2 ) ) )
mulog2sumlem.1	|- ( ph -> F ~~> r L )

Assertion

Ref	Expression
mulog2sumlem3	|- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) - ( 2 x. ( log ` x ) ) ) ) e. O(1) )

Distinct variable groups:   i, n, x, y   x, F   n, L, x   ph, n, x

Allowed substitution hints:   ph( y, i)   F( y, i, n)   L( y, i)

Proof of Theorem mulog2sumlem3

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2cn 11091	. . . . . 6 |- 2 e. CC
2	1	a1i 11	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> 2 e. CC )
3		fzfid 12772	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin )
4		elfznn 12370	. . . . . . . . . . . 12 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
5	4	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
6		mucl 24867	. . . . . . . . . . 11 |- ( n e. NN -> ( mmu ` n ) e. ZZ )
7	5, 6	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` n ) e. ZZ )
8	7	zred 11482	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` n ) e. RR )
9	8, 5	nndivred 11069	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` n ) / n ) e. RR )
10	9	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` n ) / n ) e. CC )
11		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
12	4	nnrpd 11870	. . . . . . . . . . . 12 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. RR+ )
13		rpdivcl 11856	. . . . . . . . . . . 12 |- ( ( x e. RR+ /\ n e. RR+ ) -> ( x / n ) e. RR+ )
14	11, 12, 13	syl2an 494	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( x / n ) e. RR+ )
15	14	relogcld 24369	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` ( x / n ) ) e. RR )
16	15	recnd 10068	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( log ` ( x / n ) ) e. CC )
17	16	sqcld 13006	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( log ` ( x / n ) ) ^ 2 ) e. CC )
18	17	halfcld 11277	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) e. CC )
19	10, 18	mulcld 10060	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) e. CC )
20	3, 19	fsumcl 14464	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) e. CC )
21		relogcl 24322	. . . . . . 7 |- ( x e. RR+ -> ( log ` x ) e. RR )
22	21	adantl 482	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. RR )
23	22	recnd 10068	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. CC )
24	2, 20, 23	subdid 10486	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( 2 x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) = ( ( 2 x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) - ( 2 x. ( log ` x ) ) ) )
25	3, 2, 19	fsummulc2 14516	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( 2 x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( 2 x. ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) )
26	1	a1i 11	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 2 e. CC )
27	26, 10, 18	mul12d 10245	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 2 x. ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = ( ( ( mmu ` n ) / n ) x. ( 2 x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) )
28		2ne0 11113	. . . . . . . . . . 11 |- 2 =/= 0
29	28	a1i 11	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> 2 =/= 0 )
30	17, 26, 29	divcan2d 10803	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 2 x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) = ( ( log ` ( x / n ) ) ^ 2 ) )
31	30	oveq2d 6666	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( 2 x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) )
32	27, 31	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 2 x. ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) )
33	32	sumeq2dv 14433	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( 2 x. ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) )
34	25, 33	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( 2 x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) )
35	34	oveq1d 6665	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( ( 2 x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) ) - ( 2 x. ( log ` x ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) - ( 2 x. ( log ` x ) ) ) )
36	24, 35	eqtrd 2656	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( 2 x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) - ( 2 x. ( log ` x ) ) ) )
37	36	mpteq2dva 4744	. 2 |- ( ph -> ( x e. RR+ |-> ( 2 x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) - ( 2 x. ( log ` x ) ) ) ) )
38	20, 23	subcld 10392	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) e. CC )
39		rpssre 11843	. . . . 5 |- RR+ C_ RR
40		o1const 14350	. . . . 5 |- ( ( RR+ C_ RR /\ 2 e. CC ) -> ( x e. RR+ |-> 2 ) e. O(1) )
41	39, 1, 40	mp2an 708	. . . 4 |- ( x e. RR+ |-> 2 ) e. O(1)
42	41	a1i 11	. . 3 |- ( ph -> ( x e. RR+ |-> 2 ) e. O(1) )
43		emre 24732	. . . . . . . . . . . . 13 |- gamma e. RR
44	43	recni 10052	. . . . . . . . . . . 12 |- gamma e. CC
45		mulcl 10020	. . . . . . . . . . . 12 |- ( ( gamma e. CC /\ ( log ` ( x / n ) ) e. CC ) -> ( gamma x. ( log ` ( x / n ) ) ) e. CC )
46	44, 16, 45	sylancr 695	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( gamma x. ( log ` ( x / n ) ) ) e. CC )
47		mulog2sumlem.1	. . . . . . . . . . . . 13 |- ( ph -> F ~~> r L )
48		rlimcl 14234	. . . . . . . . . . . . 13 |- ( F ~~> r L -> L e. CC )
49	47, 48	syl 17	. . . . . . . . . . . 12 |- ( ph -> L e. CC )
50	49	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> L e. CC )
51	46, 50	subcld 10392	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( gamma x. ( log ` ( x / n ) ) ) - L ) e. CC )
52	18, 51	addcld 10059	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) e. CC )
53	10, 52	mulcld 10060	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) e. CC )
54	3, 53	fsumcl 14464	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) e. CC )
55	10, 51	mulcld 10060	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) e. CC )
56	3, 55	fsumcl 14464	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) e. CC )
57	54, 23, 56	sub32d 10424	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) )
58	3, 53, 55	fsumsub 14520	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) )
59	10, 52, 51	subdid 10486	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) - ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) )
60	18, 51	pncand 10393	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) - ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) = ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) )
61	60	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) - ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) )
62	59, 61	eqtr3d 2658	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) )
63	62	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) )
64	58, 63	eqtr3d 2658	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) )
65	64	oveq1d 6665	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) )
66	57, 65	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) )
67	66	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) )
68	54, 23	subcld 10392	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) e. CC )
69		logdivsum.1	. . . . . 6 |- F = ( y e. RR+ |-> ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) - ( ( ( log ` y ) ^ 2 ) / 2 ) ) )
70		eqid 2622	. . . . . 6 |- ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) = ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) )
71		eqid 2622	. . . . . 6 |- ( ( ( 1 / 2 ) + ( gamma + ( abs ` L ) ) ) + sum_ m e. ( 1 ... 2 ) ( ( log ` ( _e / m ) ) / m ) ) = ( ( ( 1 / 2 ) + ( gamma + ( abs ` L ) ) ) + sum_ m e. ( 1 ... 2 ) ( ( log ` ( _e / m ) ) / m ) )
72	69, 47, 70, 71	mulog2sumlem2 25224	. . . . 5 |- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) ) e. O(1) )
73	44	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> gamma e. CC )
74	10, 16	mulcld 10060	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC )
75	3, 73, 74	fsummulc2 14516	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) )
76	49	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> L e. CC )
77	3, 76, 10	fsummulc1 14517	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. L ) )
78	75, 77	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> ( ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. L ) ) )
79		mulcl 10020	. . . . . . . . . 10 |- ( ( gamma e. CC /\ ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC ) -> ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. CC )
80	44, 74, 79	sylancr 695	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. CC )
81	10, 50	mulcld 10060	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. L ) e. CC )
82	3, 80, 81	fsumsub 14520	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. L ) ) )
83	44	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> gamma e. CC )
84	83, 10, 16	mul12d 10245	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) = ( ( ( mmu ` n ) / n ) x. ( gamma x. ( log ` ( x / n ) ) ) ) )
85	84	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) = ( ( ( ( mmu ` n ) / n ) x. ( gamma x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) )
86	10, 46, 50	subdid 10486	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) = ( ( ( ( mmu ` n ) / n ) x. ( gamma x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) )
87	85, 86	eqtr4d 2659	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) = ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) )
88	87	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( gamma x. ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( ( ( mmu ` n ) / n ) x. L ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) )
89	78, 82, 88	3eqtr2d 2662	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) )
90	89	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( x e. RR+ |-> ( ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) ) ) = ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) )
91	3, 74	fsumcl 14464	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC )
92		mulcl 10020	. . . . . . . 8 |- ( ( gamma e. CC /\ sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) e. CC ) -> ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. CC )
93	44, 91, 92	sylancr 695	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. CC )
94	3, 10	fsumcl 14464	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) e. CC )
95	94, 76	mulcld 10060	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) e. CC )
96	44	a1i 11	. . . . . . . . 9 |- ( ph -> gamma e. CC )
97		o1const 14350	. . . . . . . . 9 |- ( ( RR+ C_ RR /\ gamma e. CC ) -> ( x e. RR+ |-> gamma ) e. O(1) )
98	39, 96, 97	sylancr 695	. . . . . . . 8 |- ( ph -> ( x e. RR+ |-> gamma ) e. O(1) )
99		mulogsum 25221	. . . . . . . . 9 |- ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. O(1)
100	99	a1i 11	. . . . . . . 8 |- ( ph -> ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) e. O(1) )
101	73, 91, 98, 100	o1mul2 14355	. . . . . . 7 |- ( ph -> ( x e. RR+ |-> ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) ) e. O(1) )
102		mudivsum 25219	. . . . . . . . 9 |- ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) ) e. O(1)
103	102	a1i 11	. . . . . . . 8 |- ( ph -> ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) ) e. O(1) )
104		o1const 14350	. . . . . . . . 9 |- ( ( RR+ C_ RR /\ L e. CC ) -> ( x e. RR+ |-> L ) e. O(1) )
105	39, 49, 104	sylancr 695	. . . . . . . 8 |- ( ph -> ( x e. RR+ |-> L ) e. O(1) )
106	94, 76, 103, 105	o1mul2 14355	. . . . . . 7 |- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) ) e. O(1) )
107	93, 95, 101, 106	o1sub2 14356	. . . . . 6 |- ( ph -> ( x e. RR+ |-> ( ( gamma x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( log ` ( x / n ) ) ) ) - ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( mmu ` n ) / n ) x. L ) ) ) e. O(1) )
108	90, 107	eqeltrrd 2702	. . . . 5 |- ( ph -> ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) e. O(1) )
109	68, 56, 72, 108	o1sub2 14356	. . . 4 |- ( ph -> ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) - ( log ` x ) ) - sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( gamma x. ( log ` ( x / n ) ) ) - L ) ) ) ) e. O(1) )
110	67, 109	eqeltrrd 2702	. . 3 |- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) e. O(1) )
111	2, 38, 42, 110	o1mul2 14355	. 2 |- ( ph -> ( x e. RR+ |-> ( 2 x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( ( log ` ( x / n ) ) ^ 2 ) / 2 ) ) - ( log ` x ) ) ) ) e. O(1) )
112	37, 111	eqeltrrd 2702	1 |- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( ( mmu ` n ) / n ) x. ( ( log ` ( x / n ) ) ^ 2 ) ) - ( 2 x. ( log ` x ) ) ) ) e. O(1) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   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   abscabs 13974   ~~> r crli 14216   O(1)co1 14217   sum_csu 14416   _eceu 14793   logclog 24301   gammacem 24718   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-mu 24827

This theorem is referenced by:  mulog2sum  25226