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 |
|
rpvmasum2.g |
|- G = ( DChr ` N ) |
5 |
|
rpvmasum2.d |
|- D = ( Base ` G ) |
6 |
|
rpvmasum2.1 |
|- .1. = ( 0g ` G ) |
7 |
|
rpvmasum2.w |
|- W = { y e. ( D \ { .1. } ) | sum_ m e. NN ( ( y ` ( L ` m ) ) / m ) = 0 } |
8 |
|
rpvmasum2.u |
|- U = ( Unit ` Z ) |
9 |
|
rpvmasum2.b |
|- ( ph -> A e. U ) |
10 |
|
rpvmasum2.t |
|- T = ( `' L " { A } ) |
11 |
|
rpvmasum2.z1 |
|- ( ( ph /\ f e. W ) -> A = ( 1r ` Z ) ) |
12 |
3
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> N e. NN ) |
13 |
4 5
|
dchrfi |
|- ( N e. NN -> D e. Fin ) |
14 |
12 13
|
syl |
|- ( ( ph /\ x e. RR+ ) -> D e. Fin ) |
15 |
|
fzfid |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( 1 ... ( |_ ` x ) ) e. Fin ) |
16 |
|
eqid |
|- ( Base ` Z ) = ( Base ` Z ) |
17 |
|
simpr |
|- ( ( ph /\ f e. D ) -> f e. D ) |
18 |
4 1 5 16 17
|
dchrf |
|- ( ( ph /\ f e. D ) -> f : ( Base ` Z ) --> CC ) |
19 |
16 8
|
unitss |
|- U C_ ( Base ` Z ) |
20 |
19 9
|
sselid |
|- ( ph -> A e. ( Base ` Z ) ) |
21 |
20
|
adantr |
|- ( ( ph /\ f e. D ) -> A e. ( Base ` Z ) ) |
22 |
18 21
|
ffvelrnd |
|- ( ( ph /\ f e. D ) -> ( f ` A ) e. CC ) |
23 |
22
|
cjcld |
|- ( ( ph /\ f e. D ) -> ( * ` ( f ` A ) ) e. CC ) |
24 |
23
|
adantlr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( * ` ( f ` A ) ) e. CC ) |
25 |
24
|
adantrl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ f e. D ) ) -> ( * ` ( f ` A ) ) e. CC ) |
26 |
18
|
ad4ant14 |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> f : ( Base ` Z ) --> CC ) |
27 |
3
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
28 |
1 16 2
|
znzrhfo |
|- ( N e. NN0 -> L : ZZ -onto-> ( Base ` Z ) ) |
29 |
|
fof |
|- ( L : ZZ -onto-> ( Base ` Z ) -> L : ZZ --> ( Base ` Z ) ) |
30 |
27 28 29
|
3syl |
|- ( ph -> L : ZZ --> ( Base ` Z ) ) |
31 |
30
|
adantr |
|- ( ( ph /\ x e. RR+ ) -> L : ZZ --> ( Base ` Z ) ) |
32 |
|
elfzelz |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. ZZ ) |
33 |
|
ffvelrn |
|- ( ( L : ZZ --> ( Base ` Z ) /\ n e. ZZ ) -> ( L ` n ) e. ( Base ` Z ) ) |
34 |
31 32 33
|
syl2an |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( L ` n ) e. ( Base ` Z ) ) |
35 |
34
|
adantr |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( L ` n ) e. ( Base ` Z ) ) |
36 |
26 35
|
ffvelrnd |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( f ` ( L ` n ) ) e. CC ) |
37 |
36
|
anasss |
|- ( ( ( ph /\ x e. RR+ ) /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ f e. D ) ) -> ( f ` ( L ` n ) ) e. CC ) |
38 |
|
elfznn |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN ) |
39 |
38
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN ) |
40 |
|
vmacl |
|- ( n e. NN -> ( Lam ` n ) e. RR ) |
41 |
39 40
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( Lam ` n ) e. RR ) |
42 |
41 39
|
nndivred |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( Lam ` n ) / n ) e. RR ) |
43 |
42
|
recnd |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( Lam ` n ) / n ) e. CC ) |
44 |
43
|
adantrr |
|- ( ( ( ph /\ x e. RR+ ) /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ f e. D ) ) -> ( ( Lam ` n ) / n ) e. CC ) |
45 |
37 44
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ f e. D ) ) -> ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
46 |
25 45
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ ( n e. ( 1 ... ( |_ ` x ) ) /\ f e. D ) ) -> ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) e. CC ) |
47 |
46
|
anass1rs |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) e. CC ) |
48 |
15 47
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) e. CC ) |
49 |
|
relogcl |
|- ( x e. RR+ -> ( log ` x ) e. RR ) |
50 |
49
|
adantl |
|- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. RR ) |
51 |
50
|
recnd |
|- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. CC ) |
52 |
51
|
adantr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( log ` x ) e. CC ) |
53 |
|
ax-1cn |
|- 1 e. CC |
54 |
|
neg1cn |
|- -u 1 e. CC |
55 |
|
0cn |
|- 0 e. CC |
56 |
54 55
|
ifcli |
|- if ( f e. W , -u 1 , 0 ) e. CC |
57 |
53 56
|
ifcli |
|- if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) e. CC |
58 |
|
mulcl |
|- ( ( ( log ` x ) e. CC /\ if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) e. CC ) -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) e. CC ) |
59 |
52 57 58
|
sylancl |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) e. CC ) |
60 |
14 48 59
|
fsumsub |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. D ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ f e. D sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - sum_ f e. D ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
61 |
45
|
anass1rs |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
62 |
15 61
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
63 |
24 62 59
|
subdid |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( ( ( * ` ( f ` A ) ) x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( * ` ( f ` A ) ) x. ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) ) |
64 |
15 24 61
|
fsummulc2 |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
65 |
57
|
a1i |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) e. CC ) |
66 |
24 52 65
|
mul12d |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( ( log ` x ) x. ( ( * ` ( f ` A ) ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
67 |
|
ovif2 |
|- ( ( * ` ( f ` A ) ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = if ( f = .1. , ( ( * ` ( f ` A ) ) x. 1 ) , ( ( * ` ( f ` A ) ) x. if ( f e. W , -u 1 , 0 ) ) ) |
68 |
|
fveq1 |
|- ( f = .1. -> ( f ` A ) = ( .1. ` A ) ) |
69 |
3
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> N e. NN ) |
70 |
9
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> A e. U ) |
71 |
4 1 6 8 69 70
|
dchr1 |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( .1. ` A ) = 1 ) |
72 |
68 71
|
sylan9eqr |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f = .1. ) -> ( f ` A ) = 1 ) |
73 |
72
|
fveq2d |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f = .1. ) -> ( * ` ( f ` A ) ) = ( * ` 1 ) ) |
74 |
|
1re |
|- 1 e. RR |
75 |
|
cjre |
|- ( 1 e. RR -> ( * ` 1 ) = 1 ) |
76 |
74 75
|
ax-mp |
|- ( * ` 1 ) = 1 |
77 |
73 76
|
eqtrdi |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f = .1. ) -> ( * ` ( f ` A ) ) = 1 ) |
78 |
77
|
oveq1d |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f = .1. ) -> ( ( * ` ( f ` A ) ) x. 1 ) = ( 1 x. 1 ) ) |
79 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
80 |
78 79
|
eqtrdi |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f = .1. ) -> ( ( * ` ( f ` A ) ) x. 1 ) = 1 ) |
81 |
|
df-ne |
|- ( f =/= .1. <-> -. f = .1. ) |
82 |
|
ovif2 |
|- ( ( * ` ( f ` A ) ) x. if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , ( ( * ` ( f ` A ) ) x. -u 1 ) , ( ( * ` ( f ` A ) ) x. 0 ) ) |
83 |
11
|
fveq2d |
|- ( ( ph /\ f e. W ) -> ( f ` A ) = ( f ` ( 1r ` Z ) ) ) |
84 |
83
|
ad5ant15 |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( f ` A ) = ( f ` ( 1r ` Z ) ) ) |
85 |
4 1 5
|
dchrmhm |
|- D C_ ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) |
86 |
|
simpr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> f e. D ) |
87 |
85 86
|
sselid |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> f e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) ) |
88 |
|
eqid |
|- ( mulGrp ` Z ) = ( mulGrp ` Z ) |
89 |
|
eqid |
|- ( 1r ` Z ) = ( 1r ` Z ) |
90 |
88 89
|
ringidval |
|- ( 1r ` Z ) = ( 0g ` ( mulGrp ` Z ) ) |
91 |
|
eqid |
|- ( mulGrp ` CCfld ) = ( mulGrp ` CCfld ) |
92 |
|
cnfld1 |
|- 1 = ( 1r ` CCfld ) |
93 |
91 92
|
ringidval |
|- 1 = ( 0g ` ( mulGrp ` CCfld ) ) |
94 |
90 93
|
mhm0 |
|- ( f e. ( ( mulGrp ` Z ) MndHom ( mulGrp ` CCfld ) ) -> ( f ` ( 1r ` Z ) ) = 1 ) |
95 |
87 94
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( f ` ( 1r ` Z ) ) = 1 ) |
96 |
95
|
ad2antrr |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( f ` ( 1r ` Z ) ) = 1 ) |
97 |
84 96
|
eqtrd |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( f ` A ) = 1 ) |
98 |
97
|
fveq2d |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( * ` ( f ` A ) ) = ( * ` 1 ) ) |
99 |
98 76
|
eqtrdi |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( * ` ( f ` A ) ) = 1 ) |
100 |
99
|
oveq1d |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( ( * ` ( f ` A ) ) x. -u 1 ) = ( 1 x. -u 1 ) ) |
101 |
54
|
mulid2i |
|- ( 1 x. -u 1 ) = -u 1 |
102 |
100 101
|
eqtrdi |
|- ( ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) /\ f e. W ) -> ( ( * ` ( f ` A ) ) x. -u 1 ) = -u 1 ) |
103 |
102
|
ifeq1da |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> if ( f e. W , ( ( * ` ( f ` A ) ) x. -u 1 ) , ( ( * ` ( f ` A ) ) x. 0 ) ) = if ( f e. W , -u 1 , ( ( * ` ( f ` A ) ) x. 0 ) ) ) |
104 |
24
|
adantr |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> ( * ` ( f ` A ) ) e. CC ) |
105 |
104
|
mul01d |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> ( ( * ` ( f ` A ) ) x. 0 ) = 0 ) |
106 |
105
|
ifeq2d |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> if ( f e. W , -u 1 , ( ( * ` ( f ` A ) ) x. 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
107 |
103 106
|
eqtrd |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> if ( f e. W , ( ( * ` ( f ` A ) ) x. -u 1 ) , ( ( * ` ( f ` A ) ) x. 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
108 |
82 107
|
syl5eq |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ f =/= .1. ) -> ( ( * ` ( f ` A ) ) x. if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
109 |
81 108
|
sylan2br |
|- ( ( ( ( ph /\ x e. RR+ ) /\ f e. D ) /\ -. f = .1. ) -> ( ( * ` ( f ` A ) ) x. if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
110 |
80 109
|
ifeq12da |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> if ( f = .1. , ( ( * ` ( f ` A ) ) x. 1 ) , ( ( * ` ( f ` A ) ) x. if ( f e. W , -u 1 , 0 ) ) ) = if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) |
111 |
67 110
|
syl5eq |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) |
112 |
111
|
oveq2d |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( log ` x ) x. ( ( * ` ( f ` A ) ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) |
113 |
66 112
|
eqtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) |
114 |
64 113
|
oveq12d |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( ( * ` ( f ` A ) ) x. sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( * ` ( f ` A ) ) x. ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
115 |
63 114
|
eqtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
116 |
115
|
sumeq2dv |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. D ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = sum_ f e. D ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
117 |
|
fzfid |
|- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin ) |
118 |
|
inss1 |
|- ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) |
119 |
|
ssfi |
|- ( ( ( 1 ... ( |_ ` x ) ) e. Fin /\ ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) e. Fin ) |
120 |
117 118 119
|
sylancl |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) e. Fin ) |
121 |
12
|
phicld |
|- ( ( ph /\ x e. RR+ ) -> ( phi ` N ) e. NN ) |
122 |
121
|
nncnd |
|- ( ( ph /\ x e. RR+ ) -> ( phi ` N ) e. CC ) |
123 |
118
|
a1i |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) ) |
124 |
123
|
sselda |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> n e. ( 1 ... ( |_ ` x ) ) ) |
125 |
124 43
|
syldan |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( ( Lam ` n ) / n ) e. CC ) |
126 |
120 122 125
|
fsummulc2 |
|- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) = sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) ) |
127 |
122
|
adantr |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( phi ` N ) e. CC ) |
128 |
127 43
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
129 |
124 128
|
syldan |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
130 |
129
|
ralrimiva |
|- ( ( ph /\ x e. RR+ ) -> A. n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
131 |
117
|
olcd |
|- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... ( |_ ` x ) ) e. Fin ) ) |
132 |
|
sumss2 |
|- ( ( ( ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) /\ A. n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) e. CC ) /\ ( ( 1 ... ( |_ ` x ) ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... ( |_ ` x ) ) e. Fin ) ) -> sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) if ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , 0 ) ) |
133 |
123 130 131 132
|
syl21anc |
|- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) if ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , 0 ) ) |
134 |
|
elin |
|- ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) <-> ( n e. ( 1 ... ( |_ ` x ) ) /\ n e. T ) ) |
135 |
134
|
baib |
|- ( n e. ( 1 ... ( |_ ` x ) ) -> ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) <-> n e. T ) ) |
136 |
135
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) <-> n e. T ) ) |
137 |
10
|
eleq2i |
|- ( n e. T <-> n e. ( `' L " { A } ) ) |
138 |
31
|
ffnd |
|- ( ( ph /\ x e. RR+ ) -> L Fn ZZ ) |
139 |
|
fniniseg |
|- ( L Fn ZZ -> ( n e. ( `' L " { A } ) <-> ( n e. ZZ /\ ( L ` n ) = A ) ) ) |
140 |
139
|
baibd |
|- ( ( L Fn ZZ /\ n e. ZZ ) -> ( n e. ( `' L " { A } ) <-> ( L ` n ) = A ) ) |
141 |
138 32 140
|
syl2an |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( n e. ( `' L " { A } ) <-> ( L ` n ) = A ) ) |
142 |
137 141
|
syl5bb |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( n e. T <-> ( L ` n ) = A ) ) |
143 |
136 142
|
bitr2d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( L ` n ) = A <-> n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) ) |
144 |
43
|
mul02d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( 0 x. ( ( Lam ` n ) / n ) ) = 0 ) |
145 |
143 144
|
ifbieq2d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> if ( ( L ` n ) = A , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , ( 0 x. ( ( Lam ` n ) / n ) ) ) = if ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , 0 ) ) |
146 |
|
ovif |
|- ( if ( ( L ` n ) = A , ( phi ` N ) , 0 ) x. ( ( Lam ` n ) / n ) ) = if ( ( L ` n ) = A , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , ( 0 x. ( ( Lam ` n ) / n ) ) ) |
147 |
3
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> N e. NN ) |
148 |
147 13
|
syl |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> D e. Fin ) |
149 |
23
|
ad4ant14 |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( * ` ( f ` A ) ) e. CC ) |
150 |
36 149
|
mulcld |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) e. CC ) |
151 |
148 43 150
|
fsummulc1 |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( sum_ f e. D ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = sum_ f e. D ( ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) ) |
152 |
9
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> A e. U ) |
153 |
4 5 1 16 8 147 34 152
|
sum2dchr |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ f e. D ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) = if ( ( L ` n ) = A , ( phi ` N ) , 0 ) ) |
154 |
153
|
oveq1d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( sum_ f e. D ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = ( if ( ( L ` n ) = A , ( phi ` N ) , 0 ) x. ( ( Lam ` n ) / n ) ) ) |
155 |
43
|
adantr |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( ( Lam ` n ) / n ) e. CC ) |
156 |
|
mulass |
|- ( ( ( f ` ( L ` n ) ) e. CC /\ ( * ` ( f ` A ) ) e. CC /\ ( ( Lam ` n ) / n ) e. CC ) -> ( ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = ( ( f ` ( L ` n ) ) x. ( ( * ` ( f ` A ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
157 |
|
mul12 |
|- ( ( ( f ` ( L ` n ) ) e. CC /\ ( * ` ( f ` A ) ) e. CC /\ ( ( Lam ` n ) / n ) e. CC ) -> ( ( f ` ( L ` n ) ) x. ( ( * ` ( f ` A ) ) x. ( ( Lam ` n ) / n ) ) ) = ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
158 |
156 157
|
eqtrd |
|- ( ( ( f ` ( L ` n ) ) e. CC /\ ( * ` ( f ` A ) ) e. CC /\ ( ( Lam ` n ) / n ) e. CC ) -> ( ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
159 |
36 149 155 158
|
syl3anc |
|- ( ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) /\ f e. D ) -> ( ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
160 |
159
|
sumeq2dv |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> sum_ f e. D ( ( ( f ` ( L ` n ) ) x. ( * ` ( f ` A ) ) ) x. ( ( Lam ` n ) / n ) ) = sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
161 |
151 154 160
|
3eqtr3d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( if ( ( L ` n ) = A , ( phi ` N ) , 0 ) x. ( ( Lam ` n ) / n ) ) = sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
162 |
146 161
|
eqtr3id |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> if ( ( L ` n ) = A , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , ( 0 x. ( ( Lam ` n ) / n ) ) ) = sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
163 |
145 162
|
eqtr3d |
|- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> if ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , 0 ) = sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
164 |
163
|
sumeq2dv |
|- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) if ( n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) , ( ( phi ` N ) x. ( ( Lam ` n ) / n ) ) , 0 ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
165 |
126 133 164
|
3eqtrd |
|- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
166 |
117 14 46
|
fsumcom |
|- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) sum_ f e. D ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) = sum_ f e. D sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
167 |
165 166
|
eqtrd |
|- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) = sum_ f e. D sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) ) |
168 |
4
|
dchrabl |
|- ( N e. NN -> G e. Abel ) |
169 |
|
ablgrp |
|- ( G e. Abel -> G e. Grp ) |
170 |
5 6
|
grpidcl |
|- ( G e. Grp -> .1. e. D ) |
171 |
12 168 169 170
|
4syl |
|- ( ( ph /\ x e. RR+ ) -> .1. e. D ) |
172 |
51
|
mulid1d |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. 1 ) = ( log ` x ) ) |
173 |
172 51
|
eqeltrd |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. 1 ) e. CC ) |
174 |
|
iftrue |
|- ( f = .1. -> if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) = 1 ) |
175 |
174
|
oveq2d |
|- ( f = .1. -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( ( log ` x ) x. 1 ) ) |
176 |
175
|
sumsn |
|- ( ( .1. e. D /\ ( ( log ` x ) x. 1 ) e. CC ) -> sum_ f e. { .1. } ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( ( log ` x ) x. 1 ) ) |
177 |
171 173 176
|
syl2anc |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. { .1. } ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( ( log ` x ) x. 1 ) ) |
178 |
|
eldifsn |
|- ( f e. ( D \ { .1. } ) <-> ( f e. D /\ f =/= .1. ) ) |
179 |
|
ifnefalse |
|- ( f =/= .1. -> if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
180 |
179
|
ad2antll |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , -u 1 , 0 ) ) |
181 |
|
negeq |
|- ( if ( f e. W , 1 , 0 ) = 1 -> -u if ( f e. W , 1 , 0 ) = -u 1 ) |
182 |
|
negeq |
|- ( if ( f e. W , 1 , 0 ) = 0 -> -u if ( f e. W , 1 , 0 ) = -u 0 ) |
183 |
|
neg0 |
|- -u 0 = 0 |
184 |
182 183
|
eqtrdi |
|- ( if ( f e. W , 1 , 0 ) = 0 -> -u if ( f e. W , 1 , 0 ) = 0 ) |
185 |
181 184
|
ifsb |
|- -u if ( f e. W , 1 , 0 ) = if ( f e. W , -u 1 , 0 ) |
186 |
180 185
|
eqtr4di |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) = -u if ( f e. W , 1 , 0 ) ) |
187 |
186
|
oveq2d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( ( log ` x ) x. -u if ( f e. W , 1 , 0 ) ) ) |
188 |
51
|
adantr |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> ( log ` x ) e. CC ) |
189 |
53 55
|
ifcli |
|- if ( f e. W , 1 , 0 ) e. CC |
190 |
|
mulneg2 |
|- ( ( ( log ` x ) e. CC /\ if ( f e. W , 1 , 0 ) e. CC ) -> ( ( log ` x ) x. -u if ( f e. W , 1 , 0 ) ) = -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
191 |
188 189 190
|
sylancl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> ( ( log ` x ) x. -u if ( f e. W , 1 , 0 ) ) = -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
192 |
187 191
|
eqtrd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( f e. D /\ f =/= .1. ) ) -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
193 |
178 192
|
sylan2b |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. ( D \ { .1. } ) ) -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
194 |
193
|
sumeq2dv |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = sum_ f e. ( D \ { .1. } ) -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
195 |
|
diffi |
|- ( D e. Fin -> ( D \ { .1. } ) e. Fin ) |
196 |
14 195
|
syl |
|- ( ( ph /\ x e. RR+ ) -> ( D \ { .1. } ) e. Fin ) |
197 |
51
|
adantr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. ( D \ { .1. } ) ) -> ( log ` x ) e. CC ) |
198 |
|
mulcl |
|- ( ( ( log ` x ) e. CC /\ if ( f e. W , 1 , 0 ) e. CC ) -> ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) e. CC ) |
199 |
197 189 198
|
sylancl |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. ( D \ { .1. } ) ) -> ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) e. CC ) |
200 |
196 199
|
fsumneg |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. ( D \ { .1. } ) -u ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) = -u sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
201 |
189
|
a1i |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. ( D \ { .1. } ) ) -> if ( f e. W , 1 , 0 ) e. CC ) |
202 |
196 51 201
|
fsummulc2 |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. sum_ f e. ( D \ { .1. } ) if ( f e. W , 1 , 0 ) ) = sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) ) |
203 |
7
|
ssrab3 |
|- W C_ ( D \ { .1. } ) |
204 |
|
difss |
|- ( D \ { .1. } ) C_ D |
205 |
203 204
|
sstri |
|- W C_ D |
206 |
|
ssfi |
|- ( ( D e. Fin /\ W C_ D ) -> W e. Fin ) |
207 |
14 205 206
|
sylancl |
|- ( ( ph /\ x e. RR+ ) -> W e. Fin ) |
208 |
|
fsumconst |
|- ( ( W e. Fin /\ 1 e. CC ) -> sum_ f e. W 1 = ( ( # ` W ) x. 1 ) ) |
209 |
207 53 208
|
sylancl |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. W 1 = ( ( # ` W ) x. 1 ) ) |
210 |
203
|
a1i |
|- ( ( ph /\ x e. RR+ ) -> W C_ ( D \ { .1. } ) ) |
211 |
53
|
a1i |
|- ( ( ph /\ x e. RR+ ) -> 1 e. CC ) |
212 |
211
|
ralrimivw |
|- ( ( ph /\ x e. RR+ ) -> A. f e. W 1 e. CC ) |
213 |
196
|
olcd |
|- ( ( ph /\ x e. RR+ ) -> ( ( D \ { .1. } ) C_ ( ZZ>= ` 1 ) \/ ( D \ { .1. } ) e. Fin ) ) |
214 |
|
sumss2 |
|- ( ( ( W C_ ( D \ { .1. } ) /\ A. f e. W 1 e. CC ) /\ ( ( D \ { .1. } ) C_ ( ZZ>= ` 1 ) \/ ( D \ { .1. } ) e. Fin ) ) -> sum_ f e. W 1 = sum_ f e. ( D \ { .1. } ) if ( f e. W , 1 , 0 ) ) |
215 |
210 212 213 214
|
syl21anc |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. W 1 = sum_ f e. ( D \ { .1. } ) if ( f e. W , 1 , 0 ) ) |
216 |
|
hashcl |
|- ( W e. Fin -> ( # ` W ) e. NN0 ) |
217 |
207 216
|
syl |
|- ( ( ph /\ x e. RR+ ) -> ( # ` W ) e. NN0 ) |
218 |
217
|
nn0cnd |
|- ( ( ph /\ x e. RR+ ) -> ( # ` W ) e. CC ) |
219 |
218
|
mulid1d |
|- ( ( ph /\ x e. RR+ ) -> ( ( # ` W ) x. 1 ) = ( # ` W ) ) |
220 |
209 215 219
|
3eqtr3d |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. ( D \ { .1. } ) if ( f e. W , 1 , 0 ) = ( # ` W ) ) |
221 |
220
|
oveq2d |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. sum_ f e. ( D \ { .1. } ) if ( f e. W , 1 , 0 ) ) = ( ( log ` x ) x. ( # ` W ) ) ) |
222 |
202 221
|
eqtr3d |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) = ( ( log ` x ) x. ( # ` W ) ) ) |
223 |
222
|
negeqd |
|- ( ( ph /\ x e. RR+ ) -> -u sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f e. W , 1 , 0 ) ) = -u ( ( log ` x ) x. ( # ` W ) ) ) |
224 |
194 200 223
|
3eqtrd |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = -u ( ( log ` x ) x. ( # ` W ) ) ) |
225 |
177 224
|
oveq12d |
|- ( ( ph /\ x e. RR+ ) -> ( sum_ f e. { .1. } ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) + sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( ( ( log ` x ) x. 1 ) + -u ( ( log ` x ) x. ( # ` W ) ) ) ) |
226 |
51 218
|
mulcld |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. ( # ` W ) ) e. CC ) |
227 |
173 226
|
negsubd |
|- ( ( ph /\ x e. RR+ ) -> ( ( ( log ` x ) x. 1 ) + -u ( ( log ` x ) x. ( # ` W ) ) ) = ( ( ( log ` x ) x. 1 ) - ( ( log ` x ) x. ( # ` W ) ) ) ) |
228 |
225 227
|
eqtrd |
|- ( ( ph /\ x e. RR+ ) -> ( sum_ f e. { .1. } ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) + sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( ( ( log ` x ) x. 1 ) - ( ( log ` x ) x. ( # ` W ) ) ) ) |
229 |
|
disjdif |
|- ( { .1. } i^i ( D \ { .1. } ) ) = (/) |
230 |
229
|
a1i |
|- ( ( ph /\ x e. RR+ ) -> ( { .1. } i^i ( D \ { .1. } ) ) = (/) ) |
231 |
|
undif2 |
|- ( { .1. } u. ( D \ { .1. } ) ) = ( { .1. } u. D ) |
232 |
171
|
snssd |
|- ( ( ph /\ x e. RR+ ) -> { .1. } C_ D ) |
233 |
|
ssequn1 |
|- ( { .1. } C_ D <-> ( { .1. } u. D ) = D ) |
234 |
232 233
|
sylib |
|- ( ( ph /\ x e. RR+ ) -> ( { .1. } u. D ) = D ) |
235 |
231 234
|
eqtr2id |
|- ( ( ph /\ x e. RR+ ) -> D = ( { .1. } u. ( D \ { .1. } ) ) ) |
236 |
230 235 14 59
|
fsumsplit |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. D ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( sum_ f e. { .1. } ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) + sum_ f e. ( D \ { .1. } ) ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
237 |
51 211 218
|
subdid |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) = ( ( ( log ` x ) x. 1 ) - ( ( log ` x ) x. ( # ` W ) ) ) ) |
238 |
228 236 237
|
3eqtr4rd |
|- ( ( ph /\ x e. RR+ ) -> ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) = sum_ f e. D ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) |
239 |
167 238
|
oveq12d |
|- ( ( ph /\ x e. RR+ ) -> ( ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) ) = ( sum_ f e. D sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( * ` ( f ` A ) ) x. ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) - sum_ f e. D ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) |
240 |
60 116 239
|
3eqtr4d |
|- ( ( ph /\ x e. RR+ ) -> sum_ f e. D ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) ) ) |
241 |
240
|
mpteq2dva |
|- ( ph -> ( x e. RR+ |-> sum_ f e. D ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) ) = ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) ) ) ) |
242 |
|
rpssre |
|- RR+ C_ RR |
243 |
242
|
a1i |
|- ( ph -> RR+ C_ RR ) |
244 |
3 13
|
syl |
|- ( ph -> D e. Fin ) |
245 |
22
|
adantlr |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( f ` A ) e. CC ) |
246 |
245
|
cjcld |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( * ` ( f ` A ) ) e. CC ) |
247 |
62 59
|
subcld |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) e. CC ) |
248 |
246 247
|
mulcld |
|- ( ( ( ph /\ x e. RR+ ) /\ f e. D ) -> ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) e. CC ) |
249 |
248
|
anasss |
|- ( ( ph /\ ( x e. RR+ /\ f e. D ) ) -> ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) e. CC ) |
250 |
23
|
adantr |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( * ` ( f ` A ) ) e. CC ) |
251 |
247
|
an32s |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) e. CC ) |
252 |
|
o1const |
|- ( ( RR+ C_ RR /\ ( * ` ( f ` A ) ) e. CC ) -> ( x e. RR+ |-> ( * ` ( f ` A ) ) ) e. O(1) ) |
253 |
242 23 252
|
sylancr |
|- ( ( ph /\ f e. D ) -> ( x e. RR+ |-> ( * ` ( f ` A ) ) ) e. O(1) ) |
254 |
|
fveq1 |
|- ( f = .1. -> ( f ` ( L ` n ) ) = ( .1. ` ( L ` n ) ) ) |
255 |
254
|
oveq1d |
|- ( f = .1. -> ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) = ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) |
256 |
255
|
sumeq2sdv |
|- ( f = .1. -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) = sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) ) |
257 |
256 175
|
oveq12d |
|- ( f = .1. -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. 1 ) ) ) |
258 |
257
|
adantl |
|- ( ( ( ph /\ f e. D ) /\ f = .1. ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. 1 ) ) ) |
259 |
49
|
recnd |
|- ( x e. RR+ -> ( log ` x ) e. CC ) |
260 |
259
|
mulid1d |
|- ( x e. RR+ -> ( ( log ` x ) x. 1 ) = ( log ` x ) ) |
261 |
260
|
oveq2d |
|- ( x e. RR+ -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. 1 ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( log ` x ) ) ) |
262 |
258 261
|
sylan9eq |
|- ( ( ( ( ph /\ f e. D ) /\ f = .1. ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( log ` x ) ) ) |
263 |
262
|
mpteq2dva |
|- ( ( ( ph /\ f e. D ) /\ f = .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( log ` x ) ) ) ) |
264 |
1 2 3 4 5 6
|
rpvmasumlem |
|- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( log ` x ) ) ) e. O(1) ) |
265 |
264
|
ad2antrr |
|- ( ( ( ph /\ f e. D ) /\ f = .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( .1. ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( log ` x ) ) ) e. O(1) ) |
266 |
263 265
|
eqeltrd |
|- ( ( ( ph /\ f e. D ) /\ f = .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) e. O(1) ) |
267 |
179
|
oveq2d |
|- ( f =/= .1. -> ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) = ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) ) |
268 |
267
|
oveq2d |
|- ( f =/= .1. -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) ) ) |
269 |
51
|
adantlr |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( log ` x ) e. CC ) |
270 |
|
mulcom |
|- ( ( ( log ` x ) e. CC /\ -u 1 e. CC ) -> ( ( log ` x ) x. -u 1 ) = ( -u 1 x. ( log ` x ) ) ) |
271 |
269 54 270
|
sylancl |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( ( log ` x ) x. -u 1 ) = ( -u 1 x. ( log ` x ) ) ) |
272 |
269
|
mulm1d |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( -u 1 x. ( log ` x ) ) = -u ( log ` x ) ) |
273 |
271 272
|
eqtrd |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( ( log ` x ) x. -u 1 ) = -u ( log ` x ) ) |
274 |
269
|
mul01d |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( ( log ` x ) x. 0 ) = 0 ) |
275 |
273 274
|
ifeq12d |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> if ( f e. W , ( ( log ` x ) x. -u 1 ) , ( ( log ` x ) x. 0 ) ) = if ( f e. W , -u ( log ` x ) , 0 ) ) |
276 |
|
ovif2 |
|- ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) = if ( f e. W , ( ( log ` x ) x. -u 1 ) , ( ( log ` x ) x. 0 ) ) |
277 |
|
negeq |
|- ( if ( f e. W , ( log ` x ) , 0 ) = ( log ` x ) -> -u if ( f e. W , ( log ` x ) , 0 ) = -u ( log ` x ) ) |
278 |
|
negeq |
|- ( if ( f e. W , ( log ` x ) , 0 ) = 0 -> -u if ( f e. W , ( log ` x ) , 0 ) = -u 0 ) |
279 |
278 183
|
eqtrdi |
|- ( if ( f e. W , ( log ` x ) , 0 ) = 0 -> -u if ( f e. W , ( log ` x ) , 0 ) = 0 ) |
280 |
277 279
|
ifsb |
|- -u if ( f e. W , ( log ` x ) , 0 ) = if ( f e. W , -u ( log ` x ) , 0 ) |
281 |
275 276 280
|
3eqtr4g |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) = -u if ( f e. W , ( log ` x ) , 0 ) ) |
282 |
281
|
oveq2d |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - -u if ( f e. W , ( log ` x ) , 0 ) ) ) |
283 |
62
|
an32s |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) e. CC ) |
284 |
|
ifcl |
|- ( ( ( log ` x ) e. CC /\ 0 e. CC ) -> if ( f e. W , ( log ` x ) , 0 ) e. CC ) |
285 |
269 55 284
|
sylancl |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> if ( f e. W , ( log ` x ) , 0 ) e. CC ) |
286 |
283 285
|
subnegd |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - -u if ( f e. W , ( log ` x ) , 0 ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) |
287 |
282 286
|
eqtrd |
|- ( ( ( ph /\ f e. D ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f e. W , -u 1 , 0 ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) |
288 |
268 287
|
sylan9eqr |
|- ( ( ( ( ph /\ f e. D ) /\ x e. RR+ ) /\ f =/= .1. ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) |
289 |
288
|
an32s |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) |
290 |
289
|
mpteq2dva |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) ) |
291 |
3
|
ad2antrr |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> N e. NN ) |
292 |
|
simplr |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> f e. D ) |
293 |
|
simpr |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> f =/= .1. ) |
294 |
|
eqid |
|- ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) = ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) |
295 |
1 2 291 4 5 6 292 293 294
|
dchrmusumlema |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) |
296 |
3
|
adantr |
|- ( ( ph /\ f e. D ) -> N e. NN ) |
297 |
296
|
ad2antrr |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> N e. NN ) |
298 |
292
|
adantr |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> f e. D ) |
299 |
|
simplr |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> f =/= .1. ) |
300 |
|
simprl |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> c e. ( 0 [,) +oo ) ) |
301 |
|
simprrl |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t ) |
302 |
|
simprrr |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) |
303 |
1 2 297 4 5 6 298 299 294 300 301 302 7
|
dchrvmaeq0 |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> ( f e. W <-> t = 0 ) ) |
304 |
|
ifbi |
|- ( ( f e. W <-> t = 0 ) -> if ( f e. W , ( log ` x ) , 0 ) = if ( t = 0 , ( log ` x ) , 0 ) ) |
305 |
304
|
oveq2d |
|- ( ( f e. W <-> t = 0 ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) = ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( t = 0 , ( log ` x ) , 0 ) ) ) |
306 |
305
|
mpteq2dv |
|- ( ( f e. W <-> t = 0 ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( t = 0 , ( log ` x ) , 0 ) ) ) ) |
307 |
303 306
|
syl |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( t = 0 , ( log ` x ) , 0 ) ) ) ) |
308 |
1 2 297 4 5 6 298 299 294 300 301 302
|
dchrvmasumif |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( t = 0 , ( log ` x ) , 0 ) ) ) e. O(1) ) |
309 |
307 308
|
eqeltrd |
|- ( ( ( ( ph /\ f e. D ) /\ f =/= .1. ) /\ ( c e. ( 0 [,) +oo ) /\ ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) ) ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) e. O(1) ) |
310 |
309
|
rexlimdvaa |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> ( E. c e. ( 0 [,) +oo ) ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) e. O(1) ) ) |
311 |
310
|
exlimdv |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> ( E. t E. c e. ( 0 [,) +oo ) ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ~~> t /\ A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , ( a e. NN |-> ( ( f ` ( L ` a ) ) / a ) ) ) ` ( |_ ` y ) ) - t ) ) <_ ( c / y ) ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) e. O(1) ) ) |
312 |
295 311
|
mpd |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) + if ( f e. W , ( log ` x ) , 0 ) ) ) e. O(1) ) |
313 |
290 312
|
eqeltrd |
|- ( ( ( ph /\ f e. D ) /\ f =/= .1. ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) e. O(1) ) |
314 |
266 313
|
pm2.61dane |
|- ( ( ph /\ f e. D ) -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) e. O(1) ) |
315 |
250 251 253 314
|
o1mul2 |
|- ( ( ph /\ f e. D ) -> ( x e. RR+ |-> ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) ) e. O(1) ) |
316 |
243 244 249 315
|
fsumo1 |
|- ( ph -> ( x e. RR+ |-> sum_ f e. D ( ( * ` ( f ` A ) ) x. ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( f ` ( L ` n ) ) x. ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. if ( f = .1. , 1 , if ( f e. W , -u 1 , 0 ) ) ) ) ) ) e. O(1) ) |
317 |
241 316
|
eqeltrrd |
|- ( ph -> ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( Lam ` n ) / n ) ) - ( ( log ` x ) x. ( 1 - ( # ` W ) ) ) ) ) e. O(1) ) |