Theorem mulog2sumlem1 25223

Description: Asymptotic formula for sum_ n <_ x , log ( x / n ) / n = ( 1 / 2 ) log ^ 2 ( x ) + gamma x. log x - L + O ( log x / x ), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-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 )
mulog2sumlem1.2	|- ( ph -> A e. RR+ )
mulog2sumlem1.3	|- ( ph -> _e <_ A )

Assertion

Ref	Expression
mulog2sumlem1	|- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) ) <_ ( 2 x. ( ( log ` A ) / A ) ) )

Distinct variable groups:   i, m, y, A   ph, m

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

Proof of Theorem mulog2sumlem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12772	. . . . . 6 |- ( ph -> ( 1 ... ( |_ ` A ) ) e. Fin )
2		mulog2sumlem1.2	. . . . . . . . 9 |- ( ph -> A e. RR+ )
3		elfznn 12370	. . . . . . . . . 10 |- ( m e. ( 1 ... ( |_ ` A ) ) -> m e. NN )
4	3	nnrpd 11870	. . . . . . . . 9 |- ( m e. ( 1 ... ( |_ ` A ) ) -> m e. RR+ )
5		rpdivcl 11856	. . . . . . . . 9 |- ( ( A e. RR+ /\ m e. RR+ ) -> ( A / m ) e. RR+ )
6	2, 4, 5	syl2an 494	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( A / m ) e. RR+ )
7	6	relogcld 24369	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` ( A / m ) ) e. RR )
8	3	adantl 482	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> m e. NN )
9	7, 8	nndivred 11069	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` ( A / m ) ) / m ) e. RR )
10	1, 9	fsumrecl 14465	. . . . 5 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) e. RR )
11	2	relogcld 24369	. . . . . . . 8 |- ( ph -> ( log ` A ) e. RR )
12	11	resqcld 13035	. . . . . . 7 |- ( ph -> ( ( log ` A ) ^ 2 ) e. RR )
13	12	rehalfcld 11279	. . . . . 6 |- ( ph -> ( ( ( log ` A ) ^ 2 ) / 2 ) e. RR )
14		emre 24732	. . . . . . . 8 |- gamma e. RR
15		remulcl 10021	. . . . . . . 8 |- ( ( gamma e. RR /\ ( log ` A ) e. RR ) -> ( gamma x. ( log ` A ) ) e. RR )
16	14, 11, 15	sylancr 695	. . . . . . 7 |- ( ph -> ( gamma x. ( log ` A ) ) e. RR )
17		rpsup 12665	. . . . . . . . 9 |- sup ( RR+ , RR* , < ) = +oo
18	17	a1i 11	. . . . . . . 8 |- ( ph -> sup ( RR+ , RR* , < ) = +oo )
19		logdivsum.1	. . . . . . . . . . . . 13 |- F = ( y e. RR+ |-> ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) - ( ( ( log ` y ) ^ 2 ) / 2 ) ) )
20	19	logdivsum 25222	. . . . . . . . . . . 12 |- ( F : RR+ --> RR /\ F e. dom ~~> r /\ ( ( F ~~> r L /\ A e. RR+ /\ _e <_ A ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( ( log ` A ) / A ) ) )
21	20	simp1i 1070	. . . . . . . . . . 11 |- F : RR+ --> RR
22	21	a1i 11	. . . . . . . . . 10 |- ( ph -> F : RR+ --> RR )
23	22	feqmptd 6249	. . . . . . . . 9 |- ( ph -> F = ( x e. RR+ |-> ( F ` x ) ) )
24		mulog2sumlem.1	. . . . . . . . 9 |- ( ph -> F ~~> r L )
25	23, 24	eqbrtrrd 4677	. . . . . . . 8 |- ( ph -> ( x e. RR+ |-> ( F ` x ) ) ~~> r L )
26	21	ffvelrni 6358	. . . . . . . . 9 |- ( x e. RR+ -> ( F ` x ) e. RR )
27	26	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) e. RR )
28	18, 25, 27	rlimrecl 14311	. . . . . . 7 |- ( ph -> L e. RR )
29	16, 28	resubcld 10458	. . . . . 6 |- ( ph -> ( ( gamma x. ( log ` A ) ) - L ) e. RR )
30	13, 29	readdcld 10069	. . . . 5 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) e. RR )
31	10, 30	resubcld 10458	. . . 4 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) e. RR )
32	31	recnd 10068	. . 3 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) e. CC )
33	32	abscld 14175	. 2 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) ) e. RR )
34		rerpdivcl 11861	. . . . . . . 8 |- ( ( ( log ` A ) e. RR /\ m e. RR+ ) -> ( ( log ` A ) / m ) e. RR )
35	11, 4, 34	syl2an 494	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` A ) / m ) e. RR )
36	35	recnd 10068	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` A ) / m ) e. CC )
37	1, 36	fsumcl 14464	. . . . 5 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) e. CC )
38	11	recnd 10068	. . . . . 6 |- ( ph -> ( log ` A ) e. CC )
39		readdcl 10019	. . . . . . . 8 |- ( ( ( log ` A ) e. RR /\ gamma e. RR ) -> ( ( log ` A ) + gamma ) e. RR )
40	11, 14, 39	sylancl 694	. . . . . . 7 |- ( ph -> ( ( log ` A ) + gamma ) e. RR )
41	40	recnd 10068	. . . . . 6 |- ( ph -> ( ( log ` A ) + gamma ) e. CC )
42	38, 41	mulcld 10060	. . . . 5 |- ( ph -> ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) e. CC )
43	37, 42	subcld 10392	. . . 4 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) e. CC )
44	43	abscld 14175	. . 3 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) e. RR )
45	8	nnrpd 11870	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> m e. RR+ )
46	45	relogcld 24369	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` m ) e. RR )
47	46, 8	nndivred 11069	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` m ) / m ) e. RR )
48	47	recnd 10068	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` m ) / m ) e. CC )
49	1, 48	fsumcl 14464	. . . . 5 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) e. CC )
50	13	recnd 10068	. . . . . 6 |- ( ph -> ( ( ( log ` A ) ^ 2 ) / 2 ) e. CC )
51	28	recnd 10068	. . . . . 6 |- ( ph -> L e. CC )
52	50, 51	addcld 10059	. . . . 5 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) e. CC )
53	49, 52	subcld 10392	. . . 4 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) e. CC )
54	53	abscld 14175	. . 3 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) e. RR )
55	44, 54	readdcld 10069	. 2 |- ( ph -> ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) + ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) e. RR )
56		2re 11090	. . 3 |- 2 e. RR
57	11, 2	rerpdivcld 11903	. . 3 |- ( ph -> ( ( log ` A ) / A ) e. RR )
58		remulcl 10021	. . 3 |- ( ( 2 e. RR /\ ( ( log ` A ) / A ) e. RR ) -> ( 2 x. ( ( log ` A ) / A ) ) e. RR )
59	56, 57, 58	sylancr 695	. 2 |- ( ph -> ( 2 x. ( ( log ` A ) / A ) ) e. RR )
60		relogdiv 24339	. . . . . . . . . . 11 |- ( ( A e. RR+ /\ m e. RR+ ) -> ( log ` ( A / m ) ) = ( ( log ` A ) - ( log ` m ) ) )
61	2, 4, 60	syl2an 494	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` ( A / m ) ) = ( ( log ` A ) - ( log ` m ) ) )
62	61	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` ( A / m ) ) / m ) = ( ( ( log ` A ) - ( log ` m ) ) / m ) )
63	38	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` A ) e. CC )
64	46	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` m ) e. CC )
65	45	rpcnne0d 11881	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( m e. CC /\ m =/= 0 ) )
66		divsubdir 10721	. . . . . . . . . 10 |- ( ( ( log ` A ) e. CC /\ ( log ` m ) e. CC /\ ( m e. CC /\ m =/= 0 ) ) -> ( ( ( log ` A ) - ( log ` m ) ) / m ) = ( ( ( log ` A ) / m ) - ( ( log ` m ) / m ) ) )
67	63, 64, 65, 66	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( ( log ` A ) - ( log ` m ) ) / m ) = ( ( ( log ` A ) / m ) - ( ( log ` m ) / m ) ) )
68	62, 67	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` ( A / m ) ) / m ) = ( ( ( log ` A ) / m ) - ( ( log ` m ) / m ) ) )
69	68	sumeq2dv 14433	. . . . . . 7 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( ( log ` A ) / m ) - ( ( log ` m ) / m ) ) )
70	1, 36, 48	fsumsub 14520	. . . . . . 7 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( ( log ` A ) / m ) - ( ( log ` m ) / m ) ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) ) )
71	69, 70	eqtrd 2656	. . . . . 6 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) ) )
72		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( ( log ` A ) e. RR /\ gamma e. RR ) -> ( ( log ` A ) x. gamma ) e. RR )
73	11, 14, 72	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> ( ( log ` A ) x. gamma ) e. RR )
74	13, 73	readdcld 10069	. . . . . . . . . . 11 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) e. RR )
75	74	recnd 10068	. . . . . . . . . 10 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) e. CC )
76	75, 50	pncand 10393	. . . . . . . . 9 |- ( ph -> ( ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) = ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) )
77	14	recni 10052	. . . . . . . . . . . . 13 |- gamma e. CC
78	77	a1i 11	. . . . . . . . . . . 12 |- ( ph -> gamma e. CC )
79	38, 38, 78	adddid 10064	. . . . . . . . . . 11 |- ( ph -> ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) = ( ( ( log ` A ) x. ( log ` A ) ) + ( ( log ` A ) x. gamma ) ) )
80	12	recnd 10068	. . . . . . . . . . . . . 14 |- ( ph -> ( ( log ` A ) ^ 2 ) e. CC )
81	80	2halvesd 11278	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) = ( ( log ` A ) ^ 2 ) )
82	38	sqvald 13005	. . . . . . . . . . . . 13 |- ( ph -> ( ( log ` A ) ^ 2 ) = ( ( log ` A ) x. ( log ` A ) ) )
83	81, 82	eqtrd 2656	. . . . . . . . . . . 12 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) = ( ( log ` A ) x. ( log ` A ) ) )
84	83	oveq1d 6665	. . . . . . . . . . 11 |- ( ph -> ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) + ( ( log ` A ) x. gamma ) ) = ( ( ( log ` A ) x. ( log ` A ) ) + ( ( log ` A ) x. gamma ) ) )
85	73	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> ( ( log ` A ) x. gamma ) e. CC )
86	50, 50, 85	add32d 10263	. . . . . . . . . . 11 |- ( ph -> ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) + ( ( log ` A ) x. gamma ) ) = ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
87	79, 84, 86	3eqtr2d 2662	. . . . . . . . . 10 |- ( ph -> ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) = ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
88	87	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) = ( ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) + ( ( ( log ` A ) ^ 2 ) / 2 ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
89		mulcom 10022	. . . . . . . . . . 11 |- ( ( gamma e. CC /\ ( log ` A ) e. CC ) -> ( gamma x. ( log ` A ) ) = ( ( log ` A ) x. gamma ) )
90	77, 38, 89	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( gamma x. ( log ` A ) ) = ( ( log ` A ) x. gamma ) )
91	90	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( gamma x. ( log ` A ) ) ) = ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( log ` A ) x. gamma ) ) )
92	76, 88, 91	3eqtr4rd 2667	. . . . . . . 8 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( gamma x. ( log ` A ) ) ) = ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
93	92	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( gamma x. ( log ` A ) ) ) - L ) = ( ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) - L ) )
94	90, 85	eqeltrd 2701	. . . . . . . 8 |- ( ph -> ( gamma x. ( log ` A ) ) e. CC )
95	50, 94, 51	addsubassd 10412	. . . . . . 7 |- ( ph -> ( ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( gamma x. ( log ` A ) ) ) - L ) = ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) )
96	42, 50, 51	subsub4d 10423	. . . . . . 7 |- ( ph -> ( ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) - L ) = ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) )
97	93, 95, 96	3eqtr3d 2664	. . . . . 6 |- ( ph -> ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) = ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) )
98	71, 97	oveq12d 6668	. . . . 5 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) = ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) ) - ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) )
99	37, 49, 42, 52	sub4d 10441	. . . . 5 |- ( ph -> ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) ) - ( ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) = ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) - ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) )
100	98, 99	eqtrd 2656	. . . 4 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) = ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) - ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) )
101	100	fveq2d 6195	. . 3 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) ) = ( abs ` ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) - ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) )
102	43, 53	abs2dif2d 14197	. . 3 |- ( ph -> ( abs ` ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) - ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) <_ ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) + ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) )
103	101, 102	eqbrtrd 4675	. 2 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) ) <_ ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) + ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) )
104		harmonicbnd4 24737	. . . . . . 7 |- ( A e. RR+ -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) <_ ( 1 / A ) )
105	2, 104	syl 17	. . . . . 6 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) <_ ( 1 / A ) )
106	8	nnrecred 11066	. . . . . . . . . . 11 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 / m ) e. RR )
107	1, 106	fsumrecl 14465	. . . . . . . . . 10 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) e. RR )
108	107, 40	resubcld 10458	. . . . . . . . 9 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) e. RR )
109	108	recnd 10068	. . . . . . . 8 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) e. CC )
110	109	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) e. RR )
111	2	rprecred 11883	. . . . . . 7 |- ( ph -> ( 1 / A ) e. RR )
112		0red 10041	. . . . . . . 8 |- ( ph -> 0 e. RR )
113		1red 10055	. . . . . . . 8 |- ( ph -> 1 e. RR )
114		0lt1 10550	. . . . . . . . 9 |- 0 < 1
115	114	a1i 11	. . . . . . . 8 |- ( ph -> 0 < 1 )
116		loge 24333	. . . . . . . . 9 |- ( log ` _e ) = 1
117		mulog2sumlem1.3	. . . . . . . . . 10 |- ( ph -> _e <_ A )
118		epr 14936	. . . . . . . . . . 11 |- _e e. RR+
119		logleb 24349	. . . . . . . . . . 11 |- ( ( _e e. RR+ /\ A e. RR+ ) -> ( _e <_ A <-> ( log ` _e ) <_ ( log ` A ) ) )
120	118, 2, 119	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( _e <_ A <-> ( log ` _e ) <_ ( log ` A ) ) )
121	117, 120	mpbid 222	. . . . . . . . 9 |- ( ph -> ( log ` _e ) <_ ( log ` A ) )
122	116, 121	syl5eqbrr 4689	. . . . . . . 8 |- ( ph -> 1 <_ ( log ` A ) )
123	112, 113, 11, 115, 122	ltletrd 10197	. . . . . . 7 |- ( ph -> 0 < ( log ` A ) )
124		lemul2 10876	. . . . . . 7 |- ( ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) e. RR /\ ( 1 / A ) e. RR /\ ( ( log ` A ) e. RR /\ 0 < ( log ` A ) ) ) -> ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) <_ ( 1 / A ) <-> ( ( log ` A ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) <_ ( ( log ` A ) x. ( 1 / A ) ) ) )
125	110, 111, 11, 123, 124	syl112anc 1330	. . . . . 6 |- ( ph -> ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) <_ ( 1 / A ) <-> ( ( log ` A ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) <_ ( ( log ` A ) x. ( 1 / A ) ) ) )
126	105, 125	mpbid 222	. . . . 5 |- ( ph -> ( ( log ` A ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) <_ ( ( log ` A ) x. ( 1 / A ) ) )
127	45	rpcnd 11874	. . . . . . . . . . . 12 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> m e. CC )
128	45	rpne0d 11877	. . . . . . . . . . . 12 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> m =/= 0 )
129	63, 127, 128	divrecd 10804	. . . . . . . . . . 11 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( log ` A ) / m ) = ( ( log ` A ) x. ( 1 / m ) ) )
130	129	sumeq2dv 14433	. . . . . . . . . 10 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) x. ( 1 / m ) ) )
131	106	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 / m ) e. CC )
132	1, 38, 131	fsummulc2 14516	. . . . . . . . . 10 |- ( ph -> ( ( log ` A ) x. sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) x. ( 1 / m ) ) )
133	130, 132	eqtr4d 2659	. . . . . . . . 9 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) = ( ( log ` A ) x. sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) ) )
134	133	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) = ( ( ( log ` A ) x. sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) )
135	1, 131	fsumcl 14464	. . . . . . . . 9 |- ( ph -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) e. CC )
136	38, 135, 41	subdid 10486	. . . . . . . 8 |- ( ph -> ( ( log ` A ) x. ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) = ( ( ( log ` A ) x. sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) )
137	134, 136	eqtr4d 2659	. . . . . . 7 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) = ( ( log ` A ) x. ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) )
138	137	fveq2d 6195	. . . . . 6 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) = ( abs ` ( ( log ` A ) x. ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) )
139	135, 41	subcld 10392	. . . . . . 7 |- ( ph -> ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) e. CC )
140	38, 139	absmuld 14193	. . . . . 6 |- ( ph -> ( abs ` ( ( log ` A ) x. ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) = ( ( abs ` ( log ` A ) ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) )
141	112, 11, 123	ltled 10185	. . . . . . . 8 |- ( ph -> 0 <_ ( log ` A ) )
142	11, 141	absidd 14161	. . . . . . 7 |- ( ph -> ( abs ` ( log ` A ) ) = ( log ` A ) )
143	142	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( abs ` ( log ` A ) ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) = ( ( log ` A ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) )
144	138, 140, 143	3eqtrd 2660	. . . . 5 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) = ( ( log ` A ) x. ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( 1 / m ) - ( ( log ` A ) + gamma ) ) ) ) )
145	2	rpcnd 11874	. . . . . 6 |- ( ph -> A e. CC )
146	2	rpne0d 11877	. . . . . 6 |- ( ph -> A =/= 0 )
147	38, 145, 146	divrecd 10804	. . . . 5 |- ( ph -> ( ( log ` A ) / A ) = ( ( log ` A ) x. ( 1 / A ) ) )
148	126, 144, 147	3brtr4d 4685	. . . 4 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) <_ ( ( log ` A ) / A ) )
149		fveq2 6191	. . . . . . . . . . . . . 14 |- ( i = m -> ( log ` i ) = ( log ` m ) )
150		id 22	. . . . . . . . . . . . . 14 |- ( i = m -> i = m )
151	149, 150	oveq12d 6668	. . . . . . . . . . . . 13 |- ( i = m -> ( ( log ` i ) / i ) = ( ( log ` m ) / m ) )
152	151	cbvsumv 14426	. . . . . . . . . . . 12 |- sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) = sum_ m e. ( 1 ... ( |_ ` y ) ) ( ( log ` m ) / m )
153		fveq2 6191	. . . . . . . . . . . . . 14 |- ( y = A -> ( |_ ` y ) = ( |_ ` A ) )
154	153	oveq2d 6666	. . . . . . . . . . . . 13 |- ( y = A -> ( 1 ... ( |_ ` y ) ) = ( 1 ... ( |_ ` A ) ) )
155	154	sumeq1d 14431	. . . . . . . . . . . 12 |- ( y = A -> sum_ m e. ( 1 ... ( |_ ` y ) ) ( ( log ` m ) / m ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) )
156	152, 155	syl5eq 2668	. . . . . . . . . . 11 |- ( y = A -> sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) )
157		fveq2 6191	. . . . . . . . . . . . 13 |- ( y = A -> ( log ` y ) = ( log ` A ) )
158	157	oveq1d 6665	. . . . . . . . . . . 12 |- ( y = A -> ( ( log ` y ) ^ 2 ) = ( ( log ` A ) ^ 2 ) )
159	158	oveq1d 6665	. . . . . . . . . . 11 |- ( y = A -> ( ( ( log ` y ) ^ 2 ) / 2 ) = ( ( ( log ` A ) ^ 2 ) / 2 ) )
160	156, 159	oveq12d 6668	. . . . . . . . . 10 |- ( y = A -> ( sum_ i e. ( 1 ... ( |_ ` y ) ) ( ( log ` i ) / i ) - ( ( ( log ` y ) ^ 2 ) / 2 ) ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
161		ovex 6678	. . . . . . . . . 10 |- ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) e. _V
162	160, 19, 161	fvmpt 6282	. . . . . . . . 9 |- ( A e. RR+ -> ( F ` A ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
163	2, 162	syl 17	. . . . . . . 8 |- ( ph -> ( F ` A ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) )
164	163	oveq1d 6665	. . . . . . 7 |- ( ph -> ( ( F ` A ) - L ) = ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) - L ) )
165	49, 50, 51	subsub4d 10423	. . . . . . 7 |- ( ph -> ( ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( log ` A ) ^ 2 ) / 2 ) ) - L ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) )
166	164, 165	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( F ` A ) - L ) = ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) )
167	166	fveq2d 6195	. . . . 5 |- ( ph -> ( abs ` ( ( F ` A ) - L ) ) = ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) )
168	20	simp3i 1072	. . . . . 6 |- ( ( F ~~> r L /\ A e. RR+ /\ _e <_ A ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( ( log ` A ) / A ) )
169	24, 2, 117, 168	syl3anc 1326	. . . . 5 |- ( ph -> ( abs ` ( ( F ` A ) - L ) ) <_ ( ( log ` A ) / A ) )
170	167, 169	eqbrtrrd 4677	. . . 4 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) <_ ( ( log ` A ) / A ) )
171	44, 54, 57, 57, 148, 170	le2addd 10646	. . 3 |- ( ph -> ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) + ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) <_ ( ( ( log ` A ) / A ) + ( ( log ` A ) / A ) ) )
172	57	recnd 10068	. . . 4 |- ( ph -> ( ( log ` A ) / A ) e. CC )
173	172	2timesd 11275	. . 3 |- ( ph -> ( 2 x. ( ( log ` A ) / A ) ) = ( ( ( log ` A ) / A ) + ( ( log ` A ) / A ) ) )
174	171, 173	breqtrrd 4681	. 2 |- ( ph -> ( ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` A ) / m ) - ( ( log ` A ) x. ( ( log ` A ) + gamma ) ) ) ) + ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` m ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + L ) ) ) ) <_ ( 2 x. ( ( log ` A ) / A ) ) )
175	33, 55, 59, 103, 174	letrd 10194	1 |- ( ph -> ( abs ` ( sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( log ` ( A / m ) ) / m ) - ( ( ( ( log ` A ) ^ 2 ) / 2 ) + ( ( gamma x. ( log ` A ) ) - L ) ) ) ) <_ ( 2 x. ( ( log ` A ) / A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   ...cfz 12326   |_cfl 12591   ^cexp 12860   abscabs 13974   ~~> r crli 14216   sum_csu 14416   _eceu 14793   logclog 24301   gammacem 24718

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-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-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-pi 14803  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

This theorem is referenced by:  mulog2sumlem2  25224