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 |
|
dchrvmasumif.f |
|- F = ( a e. NN |-> ( ( X ` ( L ` a ) ) / a ) ) |
10 |
|
dchrvmasumif.c |
|- ( ph -> C e. ( 0 [,) +oo ) ) |
11 |
|
dchrvmasumif.s |
|- ( ph -> seq 1 ( + , F ) ~~> S ) |
12 |
|
dchrvmasumif.1 |
|- ( ph -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) <_ ( C / y ) ) |
13 |
|
dchrvmasumif.g |
|- K = ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) ) |
14 |
|
dchrvmasumif.e |
|- ( ph -> E e. ( 0 [,) +oo ) ) |
15 |
|
dchrvmasumif.t |
|- ( ph -> seq 1 ( + , K ) ~~> T ) |
16 |
|
dchrvmasumif.2 |
|- ( ph -> A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) <_ ( E x. ( ( log ` y ) / y ) ) ) |
17 |
|
fzfid |
|- ( ( ph /\ m e. RR+ ) -> ( 1 ... ( |_ ` m ) ) e. Fin ) |
18 |
|
simpl |
|- ( ( ph /\ m e. RR+ ) -> ph ) |
19 |
|
elfznn |
|- ( k e. ( 1 ... ( |_ ` m ) ) -> k e. NN ) |
20 |
7
|
adantr |
|- ( ( ph /\ k e. NN ) -> X e. D ) |
21 |
|
nnz |
|- ( k e. NN -> k e. ZZ ) |
22 |
21
|
adantl |
|- ( ( ph /\ k e. NN ) -> k e. ZZ ) |
23 |
4 1 5 2 20 22
|
dchrzrhcl |
|- ( ( ph /\ k e. NN ) -> ( X ` ( L ` k ) ) e. CC ) |
24 |
18 19 23
|
syl2an |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( X ` ( L ` k ) ) e. CC ) |
25 |
|
simpr |
|- ( ( ph /\ m e. RR+ ) -> m e. RR+ ) |
26 |
19
|
nnrpd |
|- ( k e. ( 1 ... ( |_ ` m ) ) -> k e. RR+ ) |
27 |
|
ifcl |
|- ( ( m e. RR+ /\ k e. RR+ ) -> if ( S = 0 , m , k ) e. RR+ ) |
28 |
25 26 27
|
syl2an |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> if ( S = 0 , m , k ) e. RR+ ) |
29 |
28
|
relogcld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( log ` if ( S = 0 , m , k ) ) e. RR ) |
30 |
19
|
adantl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> k e. NN ) |
31 |
29 30
|
nndivred |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) e. RR ) |
32 |
31
|
recnd |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) e. CC ) |
33 |
24 32
|
mulcld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
34 |
17 33
|
fsumcl |
|- ( ( ph /\ m e. RR+ ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
35 |
|
fveq2 |
|- ( m = ( x / d ) -> ( |_ ` m ) = ( |_ ` ( x / d ) ) ) |
36 |
35
|
oveq2d |
|- ( m = ( x / d ) -> ( 1 ... ( |_ ` m ) ) = ( 1 ... ( |_ ` ( x / d ) ) ) ) |
37 |
|
ifeq1 |
|- ( m = ( x / d ) -> if ( S = 0 , m , k ) = if ( S = 0 , ( x / d ) , k ) ) |
38 |
37
|
fveq2d |
|- ( m = ( x / d ) -> ( log ` if ( S = 0 , m , k ) ) = ( log ` if ( S = 0 , ( x / d ) , k ) ) ) |
39 |
38
|
oveq1d |
|- ( m = ( x / d ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) = ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) |
40 |
39
|
oveq2d |
|- ( m = ( x / d ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) |
41 |
40
|
adantr |
|- ( ( m = ( x / d ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) |
42 |
36 41
|
sumeq12rdv |
|- ( m = ( x / d ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) |
43 |
10 14
|
ifcld |
|- ( ph -> if ( S = 0 , C , E ) e. ( 0 [,) +oo ) ) |
44 |
|
0cn |
|- 0 e. CC |
45 |
|
climcl |
|- ( seq 1 ( + , K ) ~~> T -> T e. CC ) |
46 |
15 45
|
syl |
|- ( ph -> T e. CC ) |
47 |
|
ifcl |
|- ( ( 0 e. CC /\ T e. CC ) -> if ( S = 0 , 0 , T ) e. CC ) |
48 |
44 46 47
|
sylancr |
|- ( ph -> if ( S = 0 , 0 , T ) e. CC ) |
49 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
50 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
51 |
|
nncn |
|- ( k e. NN -> k e. CC ) |
52 |
51
|
adantl |
|- ( ( ph /\ k e. NN ) -> k e. CC ) |
53 |
|
nnne0 |
|- ( k e. NN -> k =/= 0 ) |
54 |
53
|
adantl |
|- ( ( ph /\ k e. NN ) -> k =/= 0 ) |
55 |
23 52 54
|
divcld |
|- ( ( ph /\ k e. NN ) -> ( ( X ` ( L ` k ) ) / k ) e. CC ) |
56 |
|
2fveq3 |
|- ( a = k -> ( X ` ( L ` a ) ) = ( X ` ( L ` k ) ) ) |
57 |
|
id |
|- ( a = k -> a = k ) |
58 |
56 57
|
oveq12d |
|- ( a = k -> ( ( X ` ( L ` a ) ) / a ) = ( ( X ` ( L ` k ) ) / k ) ) |
59 |
58
|
cbvmptv |
|- ( a e. NN |-> ( ( X ` ( L ` a ) ) / a ) ) = ( k e. NN |-> ( ( X ` ( L ` k ) ) / k ) ) |
60 |
9 59
|
eqtri |
|- F = ( k e. NN |-> ( ( X ` ( L ` k ) ) / k ) ) |
61 |
55 60
|
fmptd |
|- ( ph -> F : NN --> CC ) |
62 |
|
ffvelrn |
|- ( ( F : NN --> CC /\ k e. NN ) -> ( F ` k ) e. CC ) |
63 |
61 62
|
sylan |
|- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC ) |
64 |
49 50 63
|
serf |
|- ( ph -> seq 1 ( + , F ) : NN --> CC ) |
65 |
64
|
ad2antrr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> seq 1 ( + , F ) : NN --> CC ) |
66 |
|
3re |
|- 3 e. RR |
67 |
|
elicopnf |
|- ( 3 e. RR -> ( m e. ( 3 [,) +oo ) <-> ( m e. RR /\ 3 <_ m ) ) ) |
68 |
66 67
|
mp1i |
|- ( ph -> ( m e. ( 3 [,) +oo ) <-> ( m e. RR /\ 3 <_ m ) ) ) |
69 |
68
|
simprbda |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> m e. RR ) |
70 |
|
1red |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 1 e. RR ) |
71 |
66
|
a1i |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 3 e. RR ) |
72 |
|
1le3 |
|- 1 <_ 3 |
73 |
72
|
a1i |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 1 <_ 3 ) |
74 |
68
|
simplbda |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 3 <_ m ) |
75 |
70 71 69 73 74
|
letrd |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 1 <_ m ) |
76 |
|
flge1nn |
|- ( ( m e. RR /\ 1 <_ m ) -> ( |_ ` m ) e. NN ) |
77 |
69 75 76
|
syl2anc |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( |_ ` m ) e. NN ) |
78 |
77
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( |_ ` m ) e. NN ) |
79 |
65 78
|
ffvelrnd |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( seq 1 ( + , F ) ` ( |_ ` m ) ) e. CC ) |
80 |
79
|
abscld |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) e. RR ) |
81 |
|
simpl |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ph ) |
82 |
|
0red |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 0 e. RR ) |
83 |
|
3pos |
|- 0 < 3 |
84 |
83
|
a1i |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 0 < 3 ) |
85 |
82 71 69 84 74
|
ltletrd |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 0 < m ) |
86 |
69 85
|
elrpd |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> m e. RR+ ) |
87 |
81 86
|
jca |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( ph /\ m e. RR+ ) ) |
88 |
|
elrege0 |
|- ( C e. ( 0 [,) +oo ) <-> ( C e. RR /\ 0 <_ C ) ) |
89 |
88
|
simplbi |
|- ( C e. ( 0 [,) +oo ) -> C e. RR ) |
90 |
10 89
|
syl |
|- ( ph -> C e. RR ) |
91 |
|
rerpdivcl |
|- ( ( C e. RR /\ m e. RR+ ) -> ( C / m ) e. RR ) |
92 |
90 91
|
sylan |
|- ( ( ph /\ m e. RR+ ) -> ( C / m ) e. RR ) |
93 |
87 92
|
syl |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( C / m ) e. RR ) |
94 |
93
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( C / m ) e. RR ) |
95 |
86
|
relogcld |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( log ` m ) e. RR ) |
96 |
69 75
|
logge0d |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> 0 <_ ( log ` m ) ) |
97 |
95 96
|
jca |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( ( log ` m ) e. RR /\ 0 <_ ( log ` m ) ) ) |
98 |
97
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( ( log ` m ) e. RR /\ 0 <_ ( log ` m ) ) ) |
99 |
|
oveq2 |
|- ( S = 0 -> ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) = ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - 0 ) ) |
100 |
64
|
adantr |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> seq 1 ( + , F ) : NN --> CC ) |
101 |
100 77
|
ffvelrnd |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( seq 1 ( + , F ) ` ( |_ ` m ) ) e. CC ) |
102 |
101
|
subid1d |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - 0 ) = ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) |
103 |
99 102
|
sylan9eqr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) = ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) |
104 |
103
|
fveq2d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) ) = ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) |
105 |
|
2fveq3 |
|- ( y = m -> ( seq 1 ( + , F ) ` ( |_ ` y ) ) = ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) |
106 |
105
|
fvoveq1d |
|- ( y = m -> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) = ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) ) ) |
107 |
|
oveq2 |
|- ( y = m -> ( C / y ) = ( C / m ) ) |
108 |
106 107
|
breq12d |
|- ( y = m -> ( ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) <_ ( C / y ) <-> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) ) <_ ( C / m ) ) ) |
109 |
12
|
adantr |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) <_ ( C / y ) ) |
110 |
|
1re |
|- 1 e. RR |
111 |
|
elicopnf |
|- ( 1 e. RR -> ( m e. ( 1 [,) +oo ) <-> ( m e. RR /\ 1 <_ m ) ) ) |
112 |
110 111
|
ax-mp |
|- ( m e. ( 1 [,) +oo ) <-> ( m e. RR /\ 1 <_ m ) ) |
113 |
69 75 112
|
sylanbrc |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> m e. ( 1 [,) +oo ) ) |
114 |
108 109 113
|
rspcdva |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) ) <_ ( C / m ) ) |
115 |
114
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` m ) ) - S ) ) <_ ( C / m ) ) |
116 |
104 115
|
eqbrtrrd |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) <_ ( C / m ) ) |
117 |
|
lemul2a |
|- ( ( ( ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) e. RR /\ ( C / m ) e. RR /\ ( ( log ` m ) e. RR /\ 0 <_ ( log ` m ) ) ) /\ ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) <_ ( C / m ) ) -> ( ( log ` m ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) <_ ( ( log ` m ) x. ( C / m ) ) ) |
118 |
80 94 98 116 117
|
syl31anc |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( ( log ` m ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) <_ ( ( log ` m ) x. ( C / m ) ) ) |
119 |
|
iftrue |
|- ( S = 0 -> if ( S = 0 , m , k ) = m ) |
120 |
119
|
fveq2d |
|- ( S = 0 -> ( log ` if ( S = 0 , m , k ) ) = ( log ` m ) ) |
121 |
120
|
oveq1d |
|- ( S = 0 -> ( ( log ` if ( S = 0 , m , k ) ) / k ) = ( ( log ` m ) / k ) ) |
122 |
121
|
ad2antlr |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) = ( ( log ` m ) / k ) ) |
123 |
122
|
oveq2d |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` m ) / k ) ) ) |
124 |
24
|
adantlr |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( X ` ( L ` k ) ) e. CC ) |
125 |
|
relogcl |
|- ( m e. RR+ -> ( log ` m ) e. RR ) |
126 |
125
|
adantl |
|- ( ( ph /\ m e. RR+ ) -> ( log ` m ) e. RR ) |
127 |
126
|
recnd |
|- ( ( ph /\ m e. RR+ ) -> ( log ` m ) e. CC ) |
128 |
127
|
ad2antrr |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( log ` m ) e. CC ) |
129 |
19
|
adantl |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> k e. NN ) |
130 |
129
|
nncnd |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> k e. CC ) |
131 |
129
|
nnne0d |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> k =/= 0 ) |
132 |
124 128 130 131
|
div12d |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` m ) / k ) ) = ( ( log ` m ) x. ( ( X ` ( L ` k ) ) / k ) ) ) |
133 |
123 132
|
eqtrd |
|- ( ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( log ` m ) x. ( ( X ` ( L ` k ) ) / k ) ) ) |
134 |
133
|
sumeq2dv |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( log ` m ) x. ( ( X ` ( L ` k ) ) / k ) ) ) |
135 |
|
iftrue |
|- ( S = 0 -> if ( S = 0 , 0 , T ) = 0 ) |
136 |
135
|
oveq2d |
|- ( S = 0 -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - 0 ) ) |
137 |
34
|
subid1d |
|- ( ( ph /\ m e. RR+ ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - 0 ) = sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) |
138 |
136 137
|
sylan9eqr |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) |
139 |
|
ovex |
|- ( ( X ` ( L ` k ) ) / k ) e. _V |
140 |
58 9 139
|
fvmpt |
|- ( k e. NN -> ( F ` k ) = ( ( X ` ( L ` k ) ) / k ) ) |
141 |
30 140
|
syl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( F ` k ) = ( ( X ` ( L ` k ) ) / k ) ) |
142 |
61
|
adantr |
|- ( ( ph /\ m e. RR+ ) -> F : NN --> CC ) |
143 |
142 19 62
|
syl2an |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( F ` k ) e. CC ) |
144 |
141 143
|
eqeltrrd |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) / k ) e. CC ) |
145 |
17 127 144
|
fsummulc2 |
|- ( ( ph /\ m e. RR+ ) -> ( ( log ` m ) x. sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) ) = sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( log ` m ) x. ( ( X ` ( L ` k ) ) / k ) ) ) |
146 |
145
|
adantr |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( ( log ` m ) x. sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) ) = sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( log ` m ) x. ( ( X ` ( L ` k ) ) / k ) ) ) |
147 |
134 138 146
|
3eqtr4d |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = ( ( log ` m ) x. sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) ) ) |
148 |
87 147
|
sylan |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = ( ( log ` m ) x. sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) ) ) |
149 |
87 141
|
sylan |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( F ` k ) = ( ( X ` ( L ` k ) ) / k ) ) |
150 |
77 49
|
eleqtrdi |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( |_ ` m ) e. ( ZZ>= ` 1 ) ) |
151 |
81 19 55
|
syl2an |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) / k ) e. CC ) |
152 |
149 150 151
|
fsumser |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) = ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) |
153 |
152
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) = ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) |
154 |
153
|
oveq2d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( ( log ` m ) x. sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) / k ) ) = ( ( log ` m ) x. ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) |
155 |
148 154
|
eqtrd |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = ( ( log ` m ) x. ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) |
156 |
155
|
fveq2d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = ( abs ` ( ( log ` m ) x. ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) ) |
157 |
125
|
ad2antlr |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( log ` m ) e. RR ) |
158 |
157
|
recnd |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( log ` m ) e. CC ) |
159 |
87 158
|
sylan |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( log ` m ) e. CC ) |
160 |
159 79
|
absmuld |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( ( log ` m ) x. ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) = ( ( abs ` ( log ` m ) ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) ) |
161 |
95 96
|
absidd |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( log ` m ) ) = ( log ` m ) ) |
162 |
161
|
oveq1d |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( ( abs ` ( log ` m ) ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) = ( ( log ` m ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) ) |
163 |
162
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( ( abs ` ( log ` m ) ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) = ( ( log ` m ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) ) |
164 |
156 160 163
|
3eqtrd |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = ( ( log ` m ) x. ( abs ` ( seq 1 ( + , F ) ` ( |_ ` m ) ) ) ) ) |
165 |
|
iftrue |
|- ( S = 0 -> if ( S = 0 , C , E ) = C ) |
166 |
165
|
adantl |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> if ( S = 0 , C , E ) = C ) |
167 |
166
|
oveq1d |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) = ( C x. ( ( log ` m ) / m ) ) ) |
168 |
90
|
recnd |
|- ( ph -> C e. CC ) |
169 |
168
|
ad2antrr |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> C e. CC ) |
170 |
|
rpcnne0 |
|- ( m e. RR+ -> ( m e. CC /\ m =/= 0 ) ) |
171 |
170
|
ad2antlr |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( m e. CC /\ m =/= 0 ) ) |
172 |
|
div12 |
|- ( ( C e. CC /\ ( log ` m ) e. CC /\ ( m e. CC /\ m =/= 0 ) ) -> ( C x. ( ( log ` m ) / m ) ) = ( ( log ` m ) x. ( C / m ) ) ) |
173 |
169 158 171 172
|
syl3anc |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( C x. ( ( log ` m ) / m ) ) = ( ( log ` m ) x. ( C / m ) ) ) |
174 |
167 173
|
eqtrd |
|- ( ( ( ph /\ m e. RR+ ) /\ S = 0 ) -> ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) = ( ( log ` m ) x. ( C / m ) ) ) |
175 |
87 174
|
sylan |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) = ( ( log ` m ) x. ( C / m ) ) ) |
176 |
118 164 175
|
3brtr4d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S = 0 ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) ) |
177 |
|
2fveq3 |
|- ( y = m -> ( seq 1 ( + , K ) ` ( |_ ` y ) ) = ( seq 1 ( + , K ) ` ( |_ ` m ) ) ) |
178 |
177
|
fvoveq1d |
|- ( y = m -> ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) = ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) ) |
179 |
|
fveq2 |
|- ( y = m -> ( log ` y ) = ( log ` m ) ) |
180 |
|
id |
|- ( y = m -> y = m ) |
181 |
179 180
|
oveq12d |
|- ( y = m -> ( ( log ` y ) / y ) = ( ( log ` m ) / m ) ) |
182 |
181
|
oveq2d |
|- ( y = m -> ( E x. ( ( log ` y ) / y ) ) = ( E x. ( ( log ` m ) / m ) ) ) |
183 |
178 182
|
breq12d |
|- ( y = m -> ( ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) <_ ( E x. ( ( log ` y ) / y ) ) <-> ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) <_ ( E x. ( ( log ` m ) / m ) ) ) ) |
184 |
183
|
rspccva |
|- ( ( A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) <_ ( E x. ( ( log ` y ) / y ) ) /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) <_ ( E x. ( ( log ` m ) / m ) ) ) |
185 |
16 184
|
sylan |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) <_ ( E x. ( ( log ` m ) / m ) ) ) |
186 |
185
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) <_ ( E x. ( ( log ` m ) / m ) ) ) |
187 |
|
fveq2 |
|- ( a = k -> ( log ` a ) = ( log ` k ) ) |
188 |
187 57
|
oveq12d |
|- ( a = k -> ( ( log ` a ) / a ) = ( ( log ` k ) / k ) ) |
189 |
56 188
|
oveq12d |
|- ( a = k -> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
190 |
|
ovex |
|- ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) e. _V |
191 |
189 13 190
|
fvmpt |
|- ( k e. NN -> ( K ` k ) = ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
192 |
19 191
|
syl |
|- ( k e. ( 1 ... ( |_ ` m ) ) -> ( K ` k ) = ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
193 |
|
ifnefalse |
|- ( S =/= 0 -> if ( S = 0 , m , k ) = k ) |
194 |
193
|
fveq2d |
|- ( S =/= 0 -> ( log ` if ( S = 0 , m , k ) ) = ( log ` k ) ) |
195 |
194
|
oveq1d |
|- ( S =/= 0 -> ( ( log ` if ( S = 0 , m , k ) ) / k ) = ( ( log ` k ) / k ) ) |
196 |
195
|
oveq2d |
|- ( S =/= 0 -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
197 |
196
|
adantl |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
198 |
197
|
eqcomd |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) = ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) |
199 |
192 198
|
sylan9eqr |
|- ( ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( K ` k ) = ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) |
200 |
150
|
adantr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( |_ ` m ) e. ( ZZ>= ` 1 ) ) |
201 |
|
nnrp |
|- ( k e. NN -> k e. RR+ ) |
202 |
201
|
adantl |
|- ( ( ph /\ k e. NN ) -> k e. RR+ ) |
203 |
202
|
relogcld |
|- ( ( ph /\ k e. NN ) -> ( log ` k ) e. RR ) |
204 |
203
|
recnd |
|- ( ( ph /\ k e. NN ) -> ( log ` k ) e. CC ) |
205 |
204 52 54
|
divcld |
|- ( ( ph /\ k e. NN ) -> ( ( log ` k ) / k ) e. CC ) |
206 |
23 205
|
mulcld |
|- ( ( ph /\ k e. NN ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) e. CC ) |
207 |
189
|
cbvmptv |
|- ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) ) = ( k e. NN |-> ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
208 |
13 207
|
eqtri |
|- K = ( k e. NN |-> ( ( X ` ( L ` k ) ) x. ( ( log ` k ) / k ) ) ) |
209 |
206 208
|
fmptd |
|- ( ph -> K : NN --> CC ) |
210 |
209
|
ad2antrr |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> K : NN --> CC ) |
211 |
|
ffvelrn |
|- ( ( K : NN --> CC /\ k e. NN ) -> ( K ` k ) e. CC ) |
212 |
210 19 211
|
syl2an |
|- ( ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( K ` k ) e. CC ) |
213 |
199 212
|
eqeltrrd |
|- ( ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
214 |
199 200 213
|
fsumser |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( seq 1 ( + , K ) ` ( |_ ` m ) ) ) |
215 |
|
ifnefalse |
|- ( S =/= 0 -> if ( S = 0 , 0 , T ) = T ) |
216 |
215
|
adantl |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> if ( S = 0 , 0 , T ) = T ) |
217 |
214 216
|
oveq12d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) = ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) |
218 |
217
|
fveq2d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` m ) ) - T ) ) ) |
219 |
|
ifnefalse |
|- ( S =/= 0 -> if ( S = 0 , C , E ) = E ) |
220 |
219
|
adantl |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> if ( S = 0 , C , E ) = E ) |
221 |
220
|
oveq1d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) = ( E x. ( ( log ` m ) / m ) ) ) |
222 |
186 218 221
|
3brtr4d |
|- ( ( ( ph /\ m e. ( 3 [,) +oo ) ) /\ S =/= 0 ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) ) |
223 |
176 222
|
pm2.61dane |
|- ( ( ph /\ m e. ( 3 [,) +oo ) ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( if ( S = 0 , C , E ) x. ( ( log ` m ) / m ) ) ) |
224 |
|
fzfid |
|- ( ph -> ( 1 ... 2 ) e. Fin ) |
225 |
7
|
adantr |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> X e. D ) |
226 |
|
elfzelz |
|- ( k e. ( 1 ... 2 ) -> k e. ZZ ) |
227 |
226
|
adantl |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> k e. ZZ ) |
228 |
4 1 5 2 225 227
|
dchrzrhcl |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> ( X ` ( L ` k ) ) e. CC ) |
229 |
228
|
abscld |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( X ` ( L ` k ) ) ) e. RR ) |
230 |
|
3rp |
|- 3 e. RR+ |
231 |
|
relogcl |
|- ( 3 e. RR+ -> ( log ` 3 ) e. RR ) |
232 |
230 231
|
ax-mp |
|- ( log ` 3 ) e. RR |
233 |
|
elfznn |
|- ( k e. ( 1 ... 2 ) -> k e. NN ) |
234 |
233
|
adantl |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> k e. NN ) |
235 |
|
nndivre |
|- ( ( ( log ` 3 ) e. RR /\ k e. NN ) -> ( ( log ` 3 ) / k ) e. RR ) |
236 |
232 234 235
|
sylancr |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> ( ( log ` 3 ) / k ) e. RR ) |
237 |
229 236
|
remulcld |
|- ( ( ph /\ k e. ( 1 ... 2 ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) e. RR ) |
238 |
224 237
|
fsumrecl |
|- ( ph -> sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) e. RR ) |
239 |
48
|
abscld |
|- ( ph -> ( abs ` if ( S = 0 , 0 , T ) ) e. RR ) |
240 |
238 239
|
readdcld |
|- ( ph -> ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) e. RR ) |
241 |
|
simpl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ph ) |
242 |
66
|
rexri |
|- 3 e. RR* |
243 |
|
elico2 |
|- ( ( 1 e. RR /\ 3 e. RR* ) -> ( m e. ( 1 [,) 3 ) <-> ( m e. RR /\ 1 <_ m /\ m < 3 ) ) ) |
244 |
110 242 243
|
mp2an |
|- ( m e. ( 1 [,) 3 ) <-> ( m e. RR /\ 1 <_ m /\ m < 3 ) ) |
245 |
244
|
simp1bi |
|- ( m e. ( 1 [,) 3 ) -> m e. RR ) |
246 |
245
|
adantl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> m e. RR ) |
247 |
|
0red |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 0 e. RR ) |
248 |
|
1red |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 1 e. RR ) |
249 |
|
0lt1 |
|- 0 < 1 |
250 |
249
|
a1i |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 0 < 1 ) |
251 |
244
|
simp2bi |
|- ( m e. ( 1 [,) 3 ) -> 1 <_ m ) |
252 |
251
|
adantl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 1 <_ m ) |
253 |
247 248 246 250 252
|
ltletrd |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 0 < m ) |
254 |
246 253
|
elrpd |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> m e. RR+ ) |
255 |
241 254
|
jca |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( ph /\ m e. RR+ ) ) |
256 |
48
|
adantr |
|- ( ( ph /\ m e. RR+ ) -> if ( S = 0 , 0 , T ) e. CC ) |
257 |
34 256
|
subcld |
|- ( ( ph /\ m e. RR+ ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) e. CC ) |
258 |
255 257
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) e. CC ) |
259 |
258
|
abscld |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) e. RR ) |
260 |
255 34
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
261 |
260
|
abscld |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
262 |
239
|
adantr |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` if ( S = 0 , 0 , T ) ) e. RR ) |
263 |
261 262
|
readdcld |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) e. RR ) |
264 |
238
|
adantr |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) e. RR ) |
265 |
264 262
|
readdcld |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) e. RR ) |
266 |
34 256
|
abs2dif2d |
|- ( ( ph /\ m e. RR+ ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) |
267 |
255 266
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) |
268 |
33
|
abscld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... ( |_ ` m ) ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
269 |
17 268
|
fsumrecl |
|- ( ( ph /\ m e. RR+ ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
270 |
255 269
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
271 |
17 33
|
fsumabs |
|- ( ( ph /\ m e. RR+ ) -> ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
272 |
255 271
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
273 |
|
fzfid |
|- ( ( ph /\ m e. RR+ ) -> ( 1 ... 2 ) e. Fin ) |
274 |
228
|
adantlr |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( X ` ( L ` k ) ) e. CC ) |
275 |
25
|
adantr |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> m e. RR+ ) |
276 |
233
|
adantl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> k e. NN ) |
277 |
276
|
nnrpd |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> k e. RR+ ) |
278 |
275 277
|
ifcld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> if ( S = 0 , m , k ) e. RR+ ) |
279 |
278
|
relogcld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( log ` if ( S = 0 , m , k ) ) e. RR ) |
280 |
279 276
|
nndivred |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) e. RR ) |
281 |
280
|
recnd |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) e. CC ) |
282 |
274 281
|
mulcld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
283 |
282
|
abscld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
284 |
273 283
|
fsumrecl |
|- ( ( ph /\ m e. RR+ ) -> sum_ k e. ( 1 ... 2 ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
285 |
255 284
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... 2 ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
286 |
|
fzfid |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( 1 ... 2 ) e. Fin ) |
287 |
255 282
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. CC ) |
288 |
287
|
abscld |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
289 |
287
|
absge0d |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> 0 <_ ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
290 |
246
|
flcld |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( |_ ` m ) e. ZZ ) |
291 |
|
2z |
|- 2 e. ZZ |
292 |
291
|
a1i |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 2 e. ZZ ) |
293 |
244
|
simp3bi |
|- ( m e. ( 1 [,) 3 ) -> m < 3 ) |
294 |
293
|
adantl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> m < 3 ) |
295 |
|
3z |
|- 3 e. ZZ |
296 |
|
fllt |
|- ( ( m e. RR /\ 3 e. ZZ ) -> ( m < 3 <-> ( |_ ` m ) < 3 ) ) |
297 |
246 295 296
|
sylancl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( m < 3 <-> ( |_ ` m ) < 3 ) ) |
298 |
294 297
|
mpbid |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( |_ ` m ) < 3 ) |
299 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
300 |
298 299
|
breqtrdi |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( |_ ` m ) < ( 2 + 1 ) ) |
301 |
|
rpre |
|- ( m e. RR+ -> m e. RR ) |
302 |
301
|
adantl |
|- ( ( ph /\ m e. RR+ ) -> m e. RR ) |
303 |
302
|
flcld |
|- ( ( ph /\ m e. RR+ ) -> ( |_ ` m ) e. ZZ ) |
304 |
|
zleltp1 |
|- ( ( ( |_ ` m ) e. ZZ /\ 2 e. ZZ ) -> ( ( |_ ` m ) <_ 2 <-> ( |_ ` m ) < ( 2 + 1 ) ) ) |
305 |
303 291 304
|
sylancl |
|- ( ( ph /\ m e. RR+ ) -> ( ( |_ ` m ) <_ 2 <-> ( |_ ` m ) < ( 2 + 1 ) ) ) |
306 |
255 305
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( ( |_ ` m ) <_ 2 <-> ( |_ ` m ) < ( 2 + 1 ) ) ) |
307 |
300 306
|
mpbird |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( |_ ` m ) <_ 2 ) |
308 |
|
eluz2 |
|- ( 2 e. ( ZZ>= ` ( |_ ` m ) ) <-> ( ( |_ ` m ) e. ZZ /\ 2 e. ZZ /\ ( |_ ` m ) <_ 2 ) ) |
309 |
290 292 307 308
|
syl3anbrc |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> 2 e. ( ZZ>= ` ( |_ ` m ) ) ) |
310 |
|
fzss2 |
|- ( 2 e. ( ZZ>= ` ( |_ ` m ) ) -> ( 1 ... ( |_ ` m ) ) C_ ( 1 ... 2 ) ) |
311 |
309 310
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( 1 ... ( |_ ` m ) ) C_ ( 1 ... 2 ) ) |
312 |
286 288 289 311
|
fsumless |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
313 |
237
|
adantlr |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) e. RR ) |
314 |
274 281
|
absmuld |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) = ( ( abs ` ( X ` ( L ` k ) ) ) x. ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
315 |
255 314
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) = ( ( abs ` ( X ` ( L ` k ) ) ) x. ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) ) |
316 |
255 280
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) e. RR ) |
317 |
255 279
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( log ` if ( S = 0 , m , k ) ) e. RR ) |
318 |
|
log1 |
|- ( log ` 1 ) = 0 |
319 |
|
elfzle1 |
|- ( k e. ( 1 ... 2 ) -> 1 <_ k ) |
320 |
|
breq2 |
|- ( m = if ( S = 0 , m , k ) -> ( 1 <_ m <-> 1 <_ if ( S = 0 , m , k ) ) ) |
321 |
|
breq2 |
|- ( k = if ( S = 0 , m , k ) -> ( 1 <_ k <-> 1 <_ if ( S = 0 , m , k ) ) ) |
322 |
320 321
|
ifboth |
|- ( ( 1 <_ m /\ 1 <_ k ) -> 1 <_ if ( S = 0 , m , k ) ) |
323 |
252 319 322
|
syl2an |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> 1 <_ if ( S = 0 , m , k ) ) |
324 |
|
1rp |
|- 1 e. RR+ |
325 |
|
logleb |
|- ( ( 1 e. RR+ /\ if ( S = 0 , m , k ) e. RR+ ) -> ( 1 <_ if ( S = 0 , m , k ) <-> ( log ` 1 ) <_ ( log ` if ( S = 0 , m , k ) ) ) ) |
326 |
324 278 325
|
sylancr |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( 1 <_ if ( S = 0 , m , k ) <-> ( log ` 1 ) <_ ( log ` if ( S = 0 , m , k ) ) ) ) |
327 |
255 326
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( 1 <_ if ( S = 0 , m , k ) <-> ( log ` 1 ) <_ ( log ` if ( S = 0 , m , k ) ) ) ) |
328 |
323 327
|
mpbid |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( log ` 1 ) <_ ( log ` if ( S = 0 , m , k ) ) ) |
329 |
318 328
|
eqbrtrrid |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> 0 <_ ( log ` if ( S = 0 , m , k ) ) ) |
330 |
277
|
rpregt0d |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( k e. RR /\ 0 < k ) ) |
331 |
255 330
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( k e. RR /\ 0 < k ) ) |
332 |
|
divge0 |
|- ( ( ( ( log ` if ( S = 0 , m , k ) ) e. RR /\ 0 <_ ( log ` if ( S = 0 , m , k ) ) ) /\ ( k e. RR /\ 0 < k ) ) -> 0 <_ ( ( log ` if ( S = 0 , m , k ) ) / k ) ) |
333 |
317 329 331 332
|
syl21anc |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> 0 <_ ( ( log ` if ( S = 0 , m , k ) ) / k ) ) |
334 |
316 333
|
absidd |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) = ( ( log ` if ( S = 0 , m , k ) ) / k ) ) |
335 |
334 316
|
eqeltrd |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. RR ) |
336 |
236
|
adantlr |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` 3 ) / k ) e. RR ) |
337 |
229
|
adantlr |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( X ` ( L ` k ) ) ) e. RR ) |
338 |
274
|
absge0d |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> 0 <_ ( abs ` ( X ` ( L ` k ) ) ) ) |
339 |
337 338
|
jca |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) e. RR /\ 0 <_ ( abs ` ( X ` ( L ` k ) ) ) ) ) |
340 |
255 339
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) e. RR /\ 0 <_ ( abs ` ( X ` ( L ` k ) ) ) ) ) |
341 |
293
|
ad2antlr |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> m < 3 ) |
342 |
276
|
nnred |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> k e. RR ) |
343 |
|
2re |
|- 2 e. RR |
344 |
343
|
a1i |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> 2 e. RR ) |
345 |
66
|
a1i |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> 3 e. RR ) |
346 |
|
elfzle2 |
|- ( k e. ( 1 ... 2 ) -> k <_ 2 ) |
347 |
346
|
adantl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> k <_ 2 ) |
348 |
|
2lt3 |
|- 2 < 3 |
349 |
348
|
a1i |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> 2 < 3 ) |
350 |
342 344 345 347 349
|
lelttrd |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> k < 3 ) |
351 |
255 350
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> k < 3 ) |
352 |
|
breq1 |
|- ( m = if ( S = 0 , m , k ) -> ( m < 3 <-> if ( S = 0 , m , k ) < 3 ) ) |
353 |
|
breq1 |
|- ( k = if ( S = 0 , m , k ) -> ( k < 3 <-> if ( S = 0 , m , k ) < 3 ) ) |
354 |
352 353
|
ifboth |
|- ( ( m < 3 /\ k < 3 ) -> if ( S = 0 , m , k ) < 3 ) |
355 |
341 351 354
|
syl2anc |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> if ( S = 0 , m , k ) < 3 ) |
356 |
278
|
rpred |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> if ( S = 0 , m , k ) e. RR ) |
357 |
|
ltle |
|- ( ( if ( S = 0 , m , k ) e. RR /\ 3 e. RR ) -> ( if ( S = 0 , m , k ) < 3 -> if ( S = 0 , m , k ) <_ 3 ) ) |
358 |
356 66 357
|
sylancl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( if ( S = 0 , m , k ) < 3 -> if ( S = 0 , m , k ) <_ 3 ) ) |
359 |
255 358
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( if ( S = 0 , m , k ) < 3 -> if ( S = 0 , m , k ) <_ 3 ) ) |
360 |
355 359
|
mpd |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> if ( S = 0 , m , k ) <_ 3 ) |
361 |
|
logleb |
|- ( ( if ( S = 0 , m , k ) e. RR+ /\ 3 e. RR+ ) -> ( if ( S = 0 , m , k ) <_ 3 <-> ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) ) ) |
362 |
278 230 361
|
sylancl |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( if ( S = 0 , m , k ) <_ 3 <-> ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) ) ) |
363 |
255 362
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( if ( S = 0 , m , k ) <_ 3 <-> ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) ) ) |
364 |
360 363
|
mpbid |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) ) |
365 |
232
|
a1i |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( log ` 3 ) e. RR ) |
366 |
279 365 277
|
lediv1d |
|- ( ( ( ph /\ m e. RR+ ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) <-> ( ( log ` if ( S = 0 , m , k ) ) / k ) <_ ( ( log ` 3 ) / k ) ) ) |
367 |
255 366
|
sylan |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) <_ ( log ` 3 ) <-> ( ( log ` if ( S = 0 , m , k ) ) / k ) <_ ( ( log ` 3 ) / k ) ) ) |
368 |
364 367
|
mpbid |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( log ` if ( S = 0 , m , k ) ) / k ) <_ ( ( log ` 3 ) / k ) ) |
369 |
334 368
|
eqbrtrd |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) <_ ( ( log ` 3 ) / k ) ) |
370 |
|
lemul2a |
|- ( ( ( ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) e. RR /\ ( ( log ` 3 ) / k ) e. RR /\ ( ( abs ` ( X ` ( L ` k ) ) ) e. RR /\ 0 <_ ( abs ` ( X ` ( L ` k ) ) ) ) ) /\ ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) <_ ( ( log ` 3 ) / k ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) x. ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
371 |
335 336 340 369 370
|
syl31anc |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( ( abs ` ( X ` ( L ` k ) ) ) x. ( abs ` ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
372 |
315 371
|
eqbrtrd |
|- ( ( ( ph /\ m e. ( 1 [,) 3 ) ) /\ k e. ( 1 ... 2 ) ) -> ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
373 |
286 288 313 372
|
fsumle |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... 2 ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
374 |
270 285 264 312 373
|
letrd |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> sum_ k e. ( 1 ... ( |_ ` m ) ) ( abs ` ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
375 |
261 270 264 272 374
|
letrd |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) ) |
376 |
34
|
abscld |
|- ( ( ph /\ m e. RR+ ) -> ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) e. RR ) |
377 |
238
|
adantr |
|- ( ( ph /\ m e. RR+ ) -> sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) e. RR ) |
378 |
256
|
abscld |
|- ( ( ph /\ m e. RR+ ) -> ( abs ` if ( S = 0 , 0 , T ) ) e. RR ) |
379 |
376 377 378
|
leadd1d |
|- ( ( ph /\ m e. RR+ ) -> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) <-> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) <_ ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) ) |
380 |
255 379
|
syl |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) <_ sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) <-> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) <_ ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) ) |
381 |
375 380
|
mpbid |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( ( abs ` sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) <_ ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) |
382 |
259 263 265 267 381
|
letrd |
|- ( ( ph /\ m e. ( 1 [,) 3 ) ) -> ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) |
383 |
382
|
ralrimiva |
|- ( ph -> A. m e. ( 1 [,) 3 ) ( abs ` ( sum_ k e. ( 1 ... ( |_ ` m ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , m , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) <_ ( sum_ k e. ( 1 ... 2 ) ( ( abs ` ( X ` ( L ` k ) ) ) x. ( ( log ` 3 ) / k ) ) + ( abs ` if ( S = 0 , 0 , T ) ) ) ) |
384 |
1 2 3 4 5 6 7 8 34 42 43 48 223 240 383
|
dchrvmasumlem3 |
|- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) ) e. O(1) ) |