Theorem dchrvmasumlem3 25188

Description: Lemma for dchrvmasum 25214. (Contributed by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	|- Z = (ℤ/nℤ ` N )
rpvmasum.l	|- L = ( ZRHom ` Z )
rpvmasum.a	|- ( ph -> N e. NN )
rpvmasum.g	|- G = (DChr ` N )
rpvmasum.d	|- D = ( Base ` G )
rpvmasum.1	|- .1. = ( 0g ` G )
dchrisum.b	|- ( ph -> X e. D )
dchrisum.n1	|- ( ph -> X =/= .1. )
dchrvmasum.f	|- ( ( ph /\ m e. RR+ ) -> F e. CC )
dchrvmasum.g	|- ( m = ( x / d ) -> F = K )
dchrvmasum.c	|- ( ph -> C e. ( 0 [,) +oo ) )
dchrvmasum.t	|- ( ph -> T e. CC )
dchrvmasum.1	|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( F - T ) ) <_ ( C x. ( ( log ` m ) / m ) ) )
dchrvmasum.r	|- ( ph -> R e. RR )
dchrvmasum.2	|- ( ph -> A. m e. ( 1 [,) 3 ) ( abs ` ( F - T ) ) <_ R )

Assertion

Ref	Expression
dchrvmasumlem3	|- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. O(1) )

Distinct variable groups:   x, m, .1.   m, d, x, C   F, d, x   m, K   m, N, x   ph, d, m, x   T, d, m, x   R, d, m, x   m, Z, x   D, m, x   L, d, m, x   X, d, m, x

Allowed substitution hints:   D( d)   .1. ( d)   F( m)   G( x, m, d)   K( x, d)   N( d)   Z( d)

Proof of Theorem dchrvmasumlem3

Step	Hyp	Ref	Expression
1		1red 10055	. 2 |- ( ph -> 1 e. RR )
2		rpvmasum.z	. . 3 |- Z = (ℤ/nℤ ` N )
3		rpvmasum.l	. . 3 |- L = ( ZRHom ` Z )
4		rpvmasum.a	. . 3 |- ( ph -> N e. NN )
5		rpvmasum.g	. . 3 |- G = (DChr ` N )
6		rpvmasum.d	. . 3 |- D = ( Base ` G )
7		rpvmasum.1	. . 3 |- .1. = ( 0g ` G )
8		dchrisum.b	. . 3 |- ( ph -> X e. D )
9		dchrisum.n1	. . 3 |- ( ph -> X =/= .1. )
10		dchrvmasum.f	. . 3 |- ( ( ph /\ m e. RR+ ) -> F e. CC )
11		dchrvmasum.g	. . 3 |- ( m = ( x / d ) -> F = K )
12		dchrvmasum.c	. . 3 |- ( ph -> C e. ( 0 [,) +oo ) )
13		dchrvmasum.t	. . 3 |- ( ph -> T e. CC )
14		dchrvmasum.1	. . 3 |- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( F - T ) ) <_ ( C x. ( ( log ` m ) / m ) ) )
15		dchrvmasum.r	. . 3 |- ( ph -> R e. RR )
16		dchrvmasum.2	. . 3 |- ( ph -> A. m e. ( 1 [,) 3 ) ( abs ` ( F - T ) ) <_ R )
17	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	dchrvmasumlem2 25187	. 2 |- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) e. O(1) )
18		fzfid 12772	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin )
19		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
20		elfznn 12370	. . . . . . . . 9 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
21	20	nnrpd 11870	. . . . . . . 8 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
22		rpdivcl 11856	. . . . . . . 8 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
23	19, 21, 22	syl2an 494	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
24	10	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. m e. RR+ F e. CC )
25	24	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> A. m e. RR+ F e. CC )
26	11	eleq1d 2686	. . . . . . . 8 |- ( m = ( x / d ) -> ( F e. CC <-> K e. CC ) )
27	26	rspcv 3305	. . . . . . 7 |- ( ( x / d ) e. RR+ -> ( A. m e. RR+ F e. CC -> K e. CC ) )
28	23, 25, 27	sylc 65	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> K e. CC )
29	13	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> T e. CC )
30	28, 29	subcld 10392	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( K - T ) e. CC )
31	30	abscld 14175	. . . 4 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( K - T ) ) e. RR )
32	20	adantl 482	. . . 4 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
33	31, 32	nndivred 11069	. . 3 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( K - T ) ) / d ) e. RR )
34	18, 33	fsumrecl 14465	. 2 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) e. RR )
35	8	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> X e. D )
36		elfzelz 12342	. . . . . . 7 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. ZZ )
37	36	adantl 482	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. ZZ )
38	5, 2, 6, 3, 35, 37	dchrzrhcl 24970	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( X ` ( L ` d ) ) e. CC )
39		mucl 24867	. . . . . . . . 9 |- ( d e. NN -> ( mmu ` d ) e. ZZ )
40	32, 39	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. ZZ )
41	40	zred 11482	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. RR )
42	41, 32	nndivred 11069	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. RR )
43	42	recnd 10068	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. CC )
44	38, 43	mulcld 10060	. . . 4 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) e. CC )
45	44, 30	mulcld 10060	. . 3 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) e. CC )
46	18, 45	fsumcl 14464	. 2 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) e. CC )
47	46	abscld 14175	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR )
48	34	recnd 10068	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) e. CC )
49	48	abscld 14175	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) e. RR )
50	45	abscld 14175	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR )
51	18, 50	fsumrecl 14465	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR )
52	18, 45	fsumabs 14533	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ sum_ d e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) )
53	44	abscld 14175	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) e. RR )
54	32	nnrecred 11066	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 / d ) e. RR )
55	30	absge0d 14183	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( K - T ) ) )
56	38, 43	absmuld 14193	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) = ( ( abs ` ( X ` ( L ` d ) ) ) x. ( abs ` ( ( mmu ` d ) / d ) ) ) )
57	38	abscld 14175	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( X ` ( L ` d ) ) ) e. RR )
58		1red 10055	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 1 e. RR )
59	43	abscld 14175	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) e. RR )
60	38	absge0d 14183	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( X ` ( L ` d ) ) ) )
61	43	absge0d 14183	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( ( mmu ` d ) / d ) ) )
62		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` Z ) = ( Base ` Z )
63	4	nnnn0d 11351	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. NN0 )
64	2, 62, 3	znzrhfo 19896	. . . . . . . . . . . . . . . 16 |- ( N e. NN0 -> L : ZZ -onto-> ( Base ` Z ) )
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> L : ZZ -onto-> ( Base ` Z ) )
66		fof 6115	. . . . . . . . . . . . . . 15 |- ( L : ZZ -onto-> ( Base ` Z ) -> L : ZZ --> ( Base ` Z ) )
67	65, 66	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> L : ZZ --> ( Base ` Z ) )
68	67	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> L : ZZ --> ( Base ` Z ) )
69	68, 37	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( L ` d ) e. ( Base ` Z ) )
70	5, 6, 2, 62, 35, 69	dchrabs2 24987	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( X ` ( L ` d ) ) ) <_ 1 )
71	41	recnd 10068	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. CC )
72	32	nncnd 11036	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. CC )
73	32	nnne0d 11065	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d =/= 0 )
74	71, 72, 73	absdivd 14194	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) = ( ( abs ` ( mmu ` d ) ) / ( abs ` d ) ) )
75	32	nnrpd 11870	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. RR+ )
76	75	rprege0d 11879	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( d e. RR /\ 0 <_ d ) )
77		absid 14036	. . . . . . . . . . . . . . 15 |- ( ( d e. RR /\ 0 <_ d ) -> ( abs ` d ) = d )
78	76, 77	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` d ) = d )
79	78	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( mmu ` d ) ) / ( abs ` d ) ) = ( ( abs ` ( mmu ` d ) ) / d ) )
80	74, 79	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) = ( ( abs ` ( mmu ` d ) ) / d ) )
81	71	abscld 14175	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( mmu ` d ) ) e. RR )
82		mule1 24874	. . . . . . . . . . . . . 14 |- ( d e. NN -> ( abs ` ( mmu ` d ) ) <_ 1 )
83	32, 82	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( mmu ` d ) ) <_ 1 )
84	81, 58, 75, 83	lediv1dd 11930	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( mmu ` d ) ) / d ) <_ ( 1 / d ) )
85	80, 84	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) <_ ( 1 / d ) )
86	57, 58, 59, 54, 60, 61, 70, 85	lemul12ad 10966	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( X ` ( L ` d ) ) ) x. ( abs ` ( ( mmu ` d ) / d ) ) ) <_ ( 1 x. ( 1 / d ) ) )
87	54	recnd 10068	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 / d ) e. CC )
88	87	mulid2d 10058	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 x. ( 1 / d ) ) = ( 1 / d ) )
89	86, 88	breqtrd 4679	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( X ` ( L ` d ) ) ) x. ( abs ` ( ( mmu ` d ) / d ) ) ) <_ ( 1 / d ) )
90	56, 89	eqbrtrd 4675	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) <_ ( 1 / d ) )
91	53, 54, 31, 55, 90	lemul1ad 10963	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) x. ( abs ` ( K - T ) ) ) <_ ( ( 1 / d ) x. ( abs ` ( K - T ) ) ) )
92	44, 30	absmuld 14193	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) = ( ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) x. ( abs ` ( K - T ) ) ) )
93	31	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( K - T ) ) e. CC )
94	93, 72, 73	divrec2d 10805	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( K - T ) ) / d ) = ( ( 1 / d ) x. ( abs ` ( K - T ) ) ) )
95	91, 92, 94	3brtr4d 4685	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ ( ( abs ` ( K - T ) ) / d ) )
96	18, 50, 33, 95	fsumle 14531	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) )
97	47, 51, 34, 52, 96	letrd 10194	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) )
98	34	leabsd 14153	. . . 4 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) )
99	47, 34, 49, 97, 98	letrd 10194	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) )
100	99	adantrr 753	. 2 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) )
101	1, 17, 34, 46, 100	o1le 14383	1 |- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. O(1) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   3c3 11071   NN0cn0 11292   ZZcz 11377   RR+crp 11832   [,)cico 12177   ...cfz 12326   |_cfl 12591   abscabs 13974   O(1)co1 14217   sum_csu 14416   Basecbs 15857   0gc0g 16100   ZRHomczrh 19848  ℤ/nℤczn 19851   logclog 24301   mmucmu 24821  DChrcdchr 24957

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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  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-acn 8768  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-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-prm 15386  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-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  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-zring 19819  df-zrh 19852  df-zn 19855  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  df-dchr 24958

This theorem is referenced by:  dchrvmasumiflem1  25190