Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
|- Z = ( Z/nZ ` N ) |
2 |
|
rpvmasum.l |
|- L = ( ZRHom ` Z ) |
3 |
|
rpvmasum.a |
|- ( ph -> N e. NN ) |
4 |
|
rpvmasum.g |
|- G = ( DChr ` N ) |
5 |
|
rpvmasum.d |
|- D = ( Base ` G ) |
6 |
|
rpvmasum.1 |
|- .1. = ( 0g ` G ) |
7 |
|
dchrisum.b |
|- ( ph -> X e. D ) |
8 |
|
dchrisum.n1 |
|- ( ph -> X =/= .1. ) |
9 |
|
dchrvmasum.f |
|- ( ( ph /\ m e. RR+ ) -> F e. CC ) |
10 |
|
dchrvmasum.g |
|- ( m = ( x / d ) -> F = K ) |
11 |
|
dchrvmasum.c |
|- ( ph -> C e. ( 0 [,) +oo ) ) |
12 |
|
dchrvmasum.t |
|- ( ph -> T e. CC ) |
13 |
|
dchrvmasum.1 |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( F - T ) ) <_ ( C x. ( ( log ` m ) / m ) ) ) |
14 |
|
dchrvmasum.r |
|- ( ph -> R e. RR ) |
15 |
|
dchrvmasum.2 |
|- ( ph -> A. m e. ( 1 [,) 3 ) ( abs ` ( F - T ) ) <_ R ) |
16 |
|
1red |
|- ( ph -> 1 e. RR ) |
17 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
dchrvmasumlem2 |
|- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) e. O(1) ) |
18 |
|
fzfid |
|- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin ) |
19 |
10
|
eleq1d |
|- ( m = ( x / d ) -> ( F e. CC <-> K e. CC ) ) |
20 |
9
|
ralrimiva |
|- ( ph -> A. m e. RR+ F e. CC ) |
21 |
20
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> A. m e. RR+ F e. CC ) |
22 |
|
simpr |
|- ( ( ph /\ x e. RR+ ) -> x e. RR+ ) |
23 |
|
elfznn |
|- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN ) |
24 |
23
|
nnrpd |
|- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ ) |
25 |
|
rpdivcl |
|- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ ) |
26 |
22 24 25
|
syl2an |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ ) |
27 |
19 21 26
|
rspcdva |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> K e. CC ) |
28 |
12
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> T e. CC ) |
29 |
27 28
|
subcld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( K - T ) e. CC ) |
30 |
29
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( K - T ) ) e. RR ) |
31 |
23
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN ) |
32 |
30 31
|
nndivred |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( K - T ) ) / d ) e. RR ) |
33 |
18 32
|
fsumrecl |
|- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) e. RR ) |
34 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> X e. D ) |
35 |
|
elfzelz |
|- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. ZZ ) |
36 |
35
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. ZZ ) |
37 |
4 1 5 2 34 36
|
dchrzrhcl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( X ` ( L ` d ) ) e. CC ) |
38 |
|
mucl |
|- ( d e. NN -> ( mmu ` d ) e. ZZ ) |
39 |
31 38
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. ZZ ) |
40 |
39
|
zred |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. RR ) |
41 |
40 31
|
nndivred |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. RR ) |
42 |
41
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. CC ) |
43 |
37 42
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) e. CC ) |
44 |
43 29
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) e. CC ) |
45 |
18 44
|
fsumcl |
|- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) e. CC ) |
46 |
45
|
abscld |
|- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR ) |
47 |
33
|
recnd |
|- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) e. CC ) |
48 |
47
|
abscld |
|- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) e. RR ) |
49 |
44
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR ) |
50 |
18 49
|
fsumrecl |
|- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. RR ) |
51 |
18 44
|
fsumabs |
|- ( ( 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 ) ) ) ) |
52 |
43
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) e. RR ) |
53 |
31
|
nnrecred |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 / d ) e. RR ) |
54 |
29
|
absge0d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( K - T ) ) ) |
55 |
37 42
|
absmuld |
|- ( ( ( 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 ) ) ) ) |
56 |
37
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( X ` ( L ` d ) ) ) e. RR ) |
57 |
|
1red |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 1 e. RR ) |
58 |
42
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) e. RR ) |
59 |
37
|
absge0d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( X ` ( L ` d ) ) ) ) |
60 |
42
|
absge0d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( abs ` ( ( mmu ` d ) / d ) ) ) |
61 |
|
eqid |
|- ( Base ` Z ) = ( Base ` Z ) |
62 |
3
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
63 |
1 61 2
|
znzrhfo |
|- ( N e. NN0 -> L : ZZ -onto-> ( Base ` Z ) ) |
64 |
62 63
|
syl |
|- ( ph -> L : ZZ -onto-> ( Base ` Z ) ) |
65 |
|
fof |
|- ( L : ZZ -onto-> ( Base ` Z ) -> L : ZZ --> ( Base ` Z ) ) |
66 |
64 65
|
syl |
|- ( ph -> L : ZZ --> ( Base ` Z ) ) |
67 |
66
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> L : ZZ --> ( Base ` Z ) ) |
68 |
67 36
|
ffvelrnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( L ` d ) e. ( Base ` Z ) ) |
69 |
4 5 1 61 34 68
|
dchrabs2 |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( X ` ( L ` d ) ) ) <_ 1 ) |
70 |
40
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. CC ) |
71 |
31
|
nncnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. CC ) |
72 |
31
|
nnne0d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d =/= 0 ) |
73 |
70 71 72
|
absdivd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) = ( ( abs ` ( mmu ` d ) ) / ( abs ` d ) ) ) |
74 |
31
|
nnrpd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. RR+ ) |
75 |
74
|
rprege0d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( d e. RR /\ 0 <_ d ) ) |
76 |
|
absid |
|- ( ( d e. RR /\ 0 <_ d ) -> ( abs ` d ) = d ) |
77 |
75 76
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` d ) = d ) |
78 |
77
|
oveq2d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( mmu ` d ) ) / ( abs ` d ) ) = ( ( abs ` ( mmu ` d ) ) / d ) ) |
79 |
73 78
|
eqtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) = ( ( abs ` ( mmu ` d ) ) / d ) ) |
80 |
70
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( mmu ` d ) ) e. RR ) |
81 |
|
mule1 |
|- ( d e. NN -> ( abs ` ( mmu ` d ) ) <_ 1 ) |
82 |
31 81
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( mmu ` d ) ) <_ 1 ) |
83 |
80 57 74 82
|
lediv1dd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( mmu ` d ) ) / d ) <_ ( 1 / d ) ) |
84 |
79 83
|
eqbrtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( mmu ` d ) / d ) ) <_ ( 1 / d ) ) |
85 |
56 57 58 53 59 60 69 84
|
lemul12ad |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( X ` ( L ` d ) ) ) x. ( abs ` ( ( mmu ` d ) / d ) ) ) <_ ( 1 x. ( 1 / d ) ) ) |
86 |
53
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 / d ) e. CC ) |
87 |
86
|
mulid2d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 x. ( 1 / d ) ) = ( 1 / d ) ) |
88 |
85 87
|
breqtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( X ` ( L ` d ) ) ) x. ( abs ` ( ( mmu ` d ) / d ) ) ) <_ ( 1 / d ) ) |
89 |
55 88
|
eqbrtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) ) <_ ( 1 / d ) ) |
90 |
52 53 30 54 89
|
lemul1ad |
|- ( ( ( 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 ) ) ) ) |
91 |
43 29
|
absmuld |
|- ( ( ( 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 ) ) ) ) |
92 |
30
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( K - T ) ) e. CC ) |
93 |
92 71 72
|
divrec2d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( abs ` ( K - T ) ) / d ) = ( ( 1 / d ) x. ( abs ` ( K - T ) ) ) ) |
94 |
90 91 93
|
3brtr4d |
|- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( abs ` ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) <_ ( ( abs ` ( K - T ) ) / d ) ) |
95 |
18 49 32 94
|
fsumle |
|- ( ( 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 ) ) |
96 |
46 50 33 51 95
|
letrd |
|- ( ( 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 ) ) |
97 |
33
|
leabsd |
|- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( abs ` ( K - T ) ) / d ) ) ) |
98 |
46 33 48 96 97
|
letrd |
|- ( ( 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 ) ) ) |
99 |
98
|
adantrr |
|- ( ( 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 ) ) ) |
100 |
16 17 33 45 99
|
o1le |
|- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( K - T ) ) ) e. O(1) ) |