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 |
|
dchrmusum.g |
|- G = ( DChr ` N ) |
5 |
|
dchrmusum.d |
|- D = ( Base ` G ) |
6 |
|
dchrmusum.1 |
|- .1. = ( 0g ` G ) |
7 |
|
dchrmusum.b |
|- ( ph -> X e. D ) |
8 |
|
dchrmusum.n1 |
|- ( ph -> X =/= .1. ) |
9 |
|
dchrmusum.f |
|- F = ( a e. NN |-> ( ( X ` ( L ` a ) ) / a ) ) |
10 |
|
dchrmusum.c |
|- ( ph -> C e. ( 0 [,) +oo ) ) |
11 |
|
dchrmusum.t |
|- ( ph -> seq 1 ( + , F ) ~~> T ) |
12 |
|
dchrmusum.2 |
|- ( ph -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - T ) ) <_ ( C / y ) ) |
13 |
|
fzfid |
|- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin ) |
14 |
7
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> X e. D ) |
15 |
|
elfzelz |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. ZZ ) |
16 |
15
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. ZZ ) |
17 |
4 1 5 2 14 16
|
dchrzrhcl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( X ` ( L ` n ) ) e. CC ) |
18 |
|
elfznn |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN ) |
19 |
18
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN ) |
20 |
|
mucl |
|- ( n e. NN -> ( mmu ` n ) e. ZZ ) |
21 |
19 20
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` n ) e. ZZ ) |
22 |
21
|
zred |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` n ) e. RR ) |
23 |
22 19
|
nndivred |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` n ) / n ) e. RR ) |
24 |
23
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` n ) / n ) e. CC ) |
25 |
17 24
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) e. CC ) |
26 |
13 25
|
fsumcl |
|- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) e. CC ) |
27 |
|
climcl |
|- ( seq 1 ( + , F ) ~~> T -> T e. CC ) |
28 |
11 27
|
syl |
|- ( ph -> T e. CC ) |
29 |
28
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> T e. CC ) |
30 |
26 29
|
mulcld |
|- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) e. CC ) |
31 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dchrisumn0 |
|- ( ph -> T =/= 0 ) |
32 |
31
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> T =/= 0 ) |
33 |
30 29 32
|
divrecd |
|- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) / T ) = ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) x. ( 1 / T ) ) ) |
34 |
26 29 32
|
divcan4d |
|- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) / T ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) ) |
35 |
33 34
|
eqtr3d |
|- ( ( ph /\ x e. RR+ ) -> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) x. ( 1 / T ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) ) |
36 |
35
|
mpteq2dva |
|- ( ph -> ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) x. ( 1 / T ) ) ) = ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) ) ) |
37 |
28 31
|
reccld |
|- ( ph -> ( 1 / T ) e. CC ) |
38 |
37
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> ( 1 / T ) e. CC ) |
39 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dchrmusum2 |
|- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) ) e. O(1) ) |
40 |
|
rpssre |
|- RR+ C_ RR |
41 |
|
o1const |
|- ( ( RR+ C_ RR /\ ( 1 / T ) e. CC ) -> ( x e. RR+ |-> ( 1 / T ) ) e. O(1) ) |
42 |
40 37 41
|
sylancr |
|- ( ph -> ( x e. RR+ |-> ( 1 / T ) ) e. O(1) ) |
43 |
30 38 39 42
|
o1mul2 |
|- ( ph -> ( x e. RR+ |-> ( ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) x. T ) x. ( 1 / T ) ) ) e. O(1) ) |
44 |
36 43
|
eqeltrrd |
|- ( ph -> ( x e. RR+ |-> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( ( mmu ` n ) / n ) ) ) e. O(1) ) |