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 |
|
dchrvmasumlema.f |
|- F = ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) ) |
10 |
|
fveq2 |
|- ( n = x -> ( log ` n ) = ( log ` x ) ) |
11 |
|
id |
|- ( n = x -> n = x ) |
12 |
10 11
|
oveq12d |
|- ( n = x -> ( ( log ` n ) / n ) = ( ( log ` x ) / x ) ) |
13 |
|
3nn |
|- 3 e. NN |
14 |
13
|
a1i |
|- ( ph -> 3 e. NN ) |
15 |
|
relogcl |
|- ( n e. RR+ -> ( log ` n ) e. RR ) |
16 |
|
rerpdivcl |
|- ( ( ( log ` n ) e. RR /\ n e. RR+ ) -> ( ( log ` n ) / n ) e. RR ) |
17 |
15 16
|
mpancom |
|- ( n e. RR+ -> ( ( log ` n ) / n ) e. RR ) |
18 |
17
|
adantl |
|- ( ( ph /\ n e. RR+ ) -> ( ( log ` n ) / n ) e. RR ) |
19 |
|
simp3r |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> n <_ x ) |
20 |
|
simp2l |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> n e. RR+ ) |
21 |
20
|
rpred |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> n e. RR ) |
22 |
|
ere |
|- _e e. RR |
23 |
22
|
a1i |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> _e e. RR ) |
24 |
|
3re |
|- 3 e. RR |
25 |
24
|
a1i |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> 3 e. RR ) |
26 |
|
egt2lt3 |
|- ( 2 < _e /\ _e < 3 ) |
27 |
26
|
simpri |
|- _e < 3 |
28 |
22 24 27
|
ltleii |
|- _e <_ 3 |
29 |
28
|
a1i |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> _e <_ 3 ) |
30 |
|
simp3l |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> 3 <_ n ) |
31 |
23 25 21 29 30
|
letrd |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> _e <_ n ) |
32 |
|
simp2r |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> x e. RR+ ) |
33 |
32
|
rpred |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> x e. RR ) |
34 |
23 21 33 31 19
|
letrd |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> _e <_ x ) |
35 |
|
logdivle |
|- ( ( ( n e. RR /\ _e <_ n ) /\ ( x e. RR /\ _e <_ x ) ) -> ( n <_ x <-> ( ( log ` x ) / x ) <_ ( ( log ` n ) / n ) ) ) |
36 |
21 31 33 34 35
|
syl22anc |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> ( n <_ x <-> ( ( log ` x ) / x ) <_ ( ( log ` n ) / n ) ) ) |
37 |
19 36
|
mpbid |
|- ( ( ph /\ ( n e. RR+ /\ x e. RR+ ) /\ ( 3 <_ n /\ n <_ x ) ) -> ( ( log ` x ) / x ) <_ ( ( log ` n ) / n ) ) |
38 |
|
rpcn |
|- ( n e. RR+ -> n e. CC ) |
39 |
38
|
cxp1d |
|- ( n e. RR+ -> ( n ^c 1 ) = n ) |
40 |
39
|
oveq2d |
|- ( n e. RR+ -> ( ( log ` n ) / ( n ^c 1 ) ) = ( ( log ` n ) / n ) ) |
41 |
40
|
mpteq2ia |
|- ( n e. RR+ |-> ( ( log ` n ) / ( n ^c 1 ) ) ) = ( n e. RR+ |-> ( ( log ` n ) / n ) ) |
42 |
|
1rp |
|- 1 e. RR+ |
43 |
|
cxploglim |
|- ( 1 e. RR+ -> ( n e. RR+ |-> ( ( log ` n ) / ( n ^c 1 ) ) ) ~~>r 0 ) |
44 |
42 43
|
mp1i |
|- ( ph -> ( n e. RR+ |-> ( ( log ` n ) / ( n ^c 1 ) ) ) ~~>r 0 ) |
45 |
41 44
|
eqbrtrrid |
|- ( ph -> ( n e. RR+ |-> ( ( log ` n ) / n ) ) ~~>r 0 ) |
46 |
|
2fveq3 |
|- ( a = n -> ( X ` ( L ` a ) ) = ( X ` ( L ` n ) ) ) |
47 |
|
fveq2 |
|- ( a = n -> ( log ` a ) = ( log ` n ) ) |
48 |
|
id |
|- ( a = n -> a = n ) |
49 |
47 48
|
oveq12d |
|- ( a = n -> ( ( log ` a ) / a ) = ( ( log ` n ) / n ) ) |
50 |
46 49
|
oveq12d |
|- ( a = n -> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) = ( ( X ` ( L ` n ) ) x. ( ( log ` n ) / n ) ) ) |
51 |
50
|
cbvmptv |
|- ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) ) = ( n e. NN |-> ( ( X ` ( L ` n ) ) x. ( ( log ` n ) / n ) ) ) |
52 |
9 51
|
eqtri |
|- F = ( n e. NN |-> ( ( X ` ( L ` n ) ) x. ( ( log ` n ) / n ) ) ) |
53 |
1 2 3 4 5 6 7 8 12 14 18 37 45 52
|
dchrisum |
|- ( ph -> E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. x e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) ) ) |
54 |
|
2fveq3 |
|- ( x = y -> ( seq 1 ( + , F ) ` ( |_ ` x ) ) = ( seq 1 ( + , F ) ` ( |_ ` y ) ) ) |
55 |
54
|
fvoveq1d |
|- ( x = y -> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) = ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) ) |
56 |
|
fveq2 |
|- ( x = y -> ( log ` x ) = ( log ` y ) ) |
57 |
|
id |
|- ( x = y -> x = y ) |
58 |
56 57
|
oveq12d |
|- ( x = y -> ( ( log ` x ) / x ) = ( ( log ` y ) / y ) ) |
59 |
58
|
oveq2d |
|- ( x = y -> ( c x. ( ( log ` x ) / x ) ) = ( c x. ( ( log ` y ) / y ) ) ) |
60 |
55 59
|
breq12d |
|- ( x = y -> ( ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) <-> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) ) |
61 |
60
|
cbvralvw |
|- ( A. x e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) <-> A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) |
62 |
61
|
anbi2i |
|- ( ( seq 1 ( + , F ) ~~> t /\ A. x e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) ) <-> ( seq 1 ( + , F ) ~~> t /\ A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) ) |
63 |
62
|
rexbii |
|- ( E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. x e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) ) <-> E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) ) |
64 |
63
|
exbii |
|- ( E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. x e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` x ) ) - t ) ) <_ ( c x. ( ( log ` x ) / x ) ) ) <-> E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) ) |
65 |
53 64
|
sylib |
|- ( ph -> E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , F ) ~~> t /\ A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - t ) ) <_ ( c x. ( ( log ` y ) / y ) ) ) ) |