Step |
Hyp |
Ref |
Expression |
1 |
|
cvgcmpce.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
cvgcmpce.2 |
|- ( ph -> N e. Z ) |
3 |
|
cvgcmpce.3 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) |
4 |
|
cvgcmpce.4 |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC ) |
5 |
|
cvgcmpce.5 |
|- ( ph -> seq M ( + , F ) e. dom ~~> ) |
6 |
|
cvgcmpce.6 |
|- ( ph -> C e. RR ) |
7 |
|
cvgcmpce.7 |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( abs ` ( G ` k ) ) <_ ( C x. ( F ` k ) ) ) |
8 |
2 1
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
9 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
10 |
8 9
|
syl |
|- ( ph -> M e. ZZ ) |
11 |
1 10 4
|
serf |
|- ( ph -> seq M ( + , G ) : Z --> CC ) |
12 |
11
|
ffvelrnda |
|- ( ( ph /\ n e. Z ) -> ( seq M ( + , G ) ` n ) e. CC ) |
13 |
|
fveq2 |
|- ( m = k -> ( F ` m ) = ( F ` k ) ) |
14 |
13
|
oveq2d |
|- ( m = k -> ( C x. ( F ` m ) ) = ( C x. ( F ` k ) ) ) |
15 |
|
eqid |
|- ( m e. Z |-> ( C x. ( F ` m ) ) ) = ( m e. Z |-> ( C x. ( F ` m ) ) ) |
16 |
|
ovex |
|- ( C x. ( F ` k ) ) e. _V |
17 |
14 15 16
|
fvmpt |
|- ( k e. Z -> ( ( m e. Z |-> ( C x. ( F ` m ) ) ) ` k ) = ( C x. ( F ` k ) ) ) |
18 |
17
|
adantl |
|- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( C x. ( F ` m ) ) ) ` k ) = ( C x. ( F ` k ) ) ) |
19 |
6
|
adantr |
|- ( ( ph /\ k e. Z ) -> C e. RR ) |
20 |
19 3
|
remulcld |
|- ( ( ph /\ k e. Z ) -> ( C x. ( F ` k ) ) e. RR ) |
21 |
18 20
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( C x. ( F ` m ) ) ) ` k ) e. RR ) |
22 |
|
2fveq3 |
|- ( m = k -> ( abs ` ( G ` m ) ) = ( abs ` ( G ` k ) ) ) |
23 |
|
eqid |
|- ( m e. Z |-> ( abs ` ( G ` m ) ) ) = ( m e. Z |-> ( abs ` ( G ` m ) ) ) |
24 |
|
fvex |
|- ( abs ` ( G ` k ) ) e. _V |
25 |
22 23 24
|
fvmpt |
|- ( k e. Z -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) = ( abs ` ( G ` k ) ) ) |
26 |
25
|
adantl |
|- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) = ( abs ` ( G ` k ) ) ) |
27 |
4
|
abscld |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( G ` k ) ) e. RR ) |
28 |
26 27
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) e. RR ) |
29 |
6
|
recnd |
|- ( ph -> C e. CC ) |
30 |
|
climdm |
|- ( seq M ( + , F ) e. dom ~~> <-> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) ) |
31 |
5 30
|
sylib |
|- ( ph -> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) ) |
32 |
3
|
recnd |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
33 |
1 10 29 31 32 18
|
isermulc2 |
|- ( ph -> seq M ( + , ( m e. Z |-> ( C x. ( F ` m ) ) ) ) ~~> ( C x. ( ~~> ` seq M ( + , F ) ) ) ) |
34 |
|
climrel |
|- Rel ~~> |
35 |
34
|
releldmi |
|- ( seq M ( + , ( m e. Z |-> ( C x. ( F ` m ) ) ) ) ~~> ( C x. ( ~~> ` seq M ( + , F ) ) ) -> seq M ( + , ( m e. Z |-> ( C x. ( F ` m ) ) ) ) e. dom ~~> ) |
36 |
33 35
|
syl |
|- ( ph -> seq M ( + , ( m e. Z |-> ( C x. ( F ` m ) ) ) ) e. dom ~~> ) |
37 |
1
|
uztrn2 |
|- ( ( N e. Z /\ k e. ( ZZ>= ` N ) ) -> k e. Z ) |
38 |
2 37
|
sylan |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z ) |
39 |
4
|
absge0d |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( abs ` ( G ` k ) ) ) |
40 |
39 26
|
breqtrrd |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) ) |
41 |
38 40
|
syldan |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> 0 <_ ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) ) |
42 |
38 25
|
syl |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) = ( abs ` ( G ` k ) ) ) |
43 |
38 17
|
syl |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( ( m e. Z |-> ( C x. ( F ` m ) ) ) ` k ) = ( C x. ( F ` k ) ) ) |
44 |
7 42 43
|
3brtr4d |
|- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) <_ ( ( m e. Z |-> ( C x. ( F ` m ) ) ) ` k ) ) |
45 |
1 2 21 28 36 41 44
|
cvgcmp |
|- ( ph -> seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) e. dom ~~> ) |
46 |
1
|
climcau |
|- ( ( M e. ZZ /\ seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) e. dom ~~> ) -> A. x e. RR+ E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x ) |
47 |
10 45 46
|
syl2anc |
|- ( ph -> A. x e. RR+ E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x ) |
48 |
1 10 28
|
serfre |
|- ( ph -> seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) : Z --> RR ) |
49 |
48
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) : Z --> RR ) |
50 |
1
|
uztrn2 |
|- ( ( j e. Z /\ n e. ( ZZ>= ` j ) ) -> n e. Z ) |
51 |
50
|
adantl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> n e. Z ) |
52 |
49 51
|
ffvelrnd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) e. RR ) |
53 |
|
simprl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> j e. Z ) |
54 |
49 53
|
ffvelrnd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) e. RR ) |
55 |
52 54
|
resubcld |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) e. RR ) |
56 |
|
0red |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> 0 e. RR ) |
57 |
11
|
ad2antrr |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> seq M ( + , G ) : Z --> CC ) |
58 |
57 51
|
ffvelrnd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( seq M ( + , G ) ` n ) e. CC ) |
59 |
57 53
|
ffvelrnd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( seq M ( + , G ) ` j ) e. CC ) |
60 |
58 59
|
subcld |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) e. CC ) |
61 |
60
|
abscld |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) e. RR ) |
62 |
60
|
absge0d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> 0 <_ ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) ) |
63 |
|
fzfid |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( M ... n ) e. Fin ) |
64 |
|
difss |
|- ( ( M ... n ) \ ( M ... j ) ) C_ ( M ... n ) |
65 |
|
ssfi |
|- ( ( ( M ... n ) e. Fin /\ ( ( M ... n ) \ ( M ... j ) ) C_ ( M ... n ) ) -> ( ( M ... n ) \ ( M ... j ) ) e. Fin ) |
66 |
63 64 65
|
sylancl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( M ... n ) \ ( M ... j ) ) e. Fin ) |
67 |
|
eldifi |
|- ( k e. ( ( M ... n ) \ ( M ... j ) ) -> k e. ( M ... n ) ) |
68 |
|
simpll |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ph ) |
69 |
|
elfzuz |
|- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) ) |
70 |
69 1
|
eleqtrrdi |
|- ( k e. ( M ... n ) -> k e. Z ) |
71 |
68 70 4
|
syl2an |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... n ) ) -> ( G ` k ) e. CC ) |
72 |
67 71
|
sylan2 |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( ( M ... n ) \ ( M ... j ) ) ) -> ( G ` k ) e. CC ) |
73 |
66 72
|
fsumabs |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( abs ` sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) ) <_ sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( abs ` ( G ` k ) ) ) |
74 |
|
eqidd |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... n ) ) -> ( G ` k ) = ( G ` k ) ) |
75 |
51 1
|
eleqtrdi |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> n e. ( ZZ>= ` M ) ) |
76 |
74 75 71
|
fsumser |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... n ) ( G ` k ) = ( seq M ( + , G ) ` n ) ) |
77 |
|
eqidd |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... j ) ) -> ( G ` k ) = ( G ` k ) ) |
78 |
53 1
|
eleqtrdi |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> j e. ( ZZ>= ` M ) ) |
79 |
|
elfzuz |
|- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) ) |
80 |
79 1
|
eleqtrrdi |
|- ( k e. ( M ... j ) -> k e. Z ) |
81 |
68 80 4
|
syl2an |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... j ) ) -> ( G ` k ) e. CC ) |
82 |
77 78 81
|
fsumser |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... j ) ( G ` k ) = ( seq M ( + , G ) ` j ) ) |
83 |
76 82
|
oveq12d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( sum_ k e. ( M ... n ) ( G ` k ) - sum_ k e. ( M ... j ) ( G ` k ) ) = ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) |
84 |
|
fzfid |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( M ... j ) e. Fin ) |
85 |
84 81
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... j ) ( G ` k ) e. CC ) |
86 |
66 72
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) e. CC ) |
87 |
|
disjdif |
|- ( ( M ... j ) i^i ( ( M ... n ) \ ( M ... j ) ) ) = (/) |
88 |
87
|
a1i |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( M ... j ) i^i ( ( M ... n ) \ ( M ... j ) ) ) = (/) ) |
89 |
|
undif2 |
|- ( ( M ... j ) u. ( ( M ... n ) \ ( M ... j ) ) ) = ( ( M ... j ) u. ( M ... n ) ) |
90 |
|
fzss2 |
|- ( n e. ( ZZ>= ` j ) -> ( M ... j ) C_ ( M ... n ) ) |
91 |
90
|
ad2antll |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( M ... j ) C_ ( M ... n ) ) |
92 |
|
ssequn1 |
|- ( ( M ... j ) C_ ( M ... n ) <-> ( ( M ... j ) u. ( M ... n ) ) = ( M ... n ) ) |
93 |
91 92
|
sylib |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( M ... j ) u. ( M ... n ) ) = ( M ... n ) ) |
94 |
89 93
|
eqtr2id |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( M ... n ) = ( ( M ... j ) u. ( ( M ... n ) \ ( M ... j ) ) ) ) |
95 |
88 94 63 71
|
fsumsplit |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... n ) ( G ` k ) = ( sum_ k e. ( M ... j ) ( G ` k ) + sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) ) ) |
96 |
85 86 95
|
mvrladdd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( sum_ k e. ( M ... n ) ( G ` k ) - sum_ k e. ( M ... j ) ( G ` k ) ) = sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) ) |
97 |
83 96
|
eqtr3d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) = sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) ) |
98 |
97
|
fveq2d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) = ( abs ` sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( G ` k ) ) ) |
99 |
70
|
adantl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... n ) ) -> k e. Z ) |
100 |
99 25
|
syl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... n ) ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) = ( abs ` ( G ` k ) ) ) |
101 |
|
abscl |
|- ( ( G ` k ) e. CC -> ( abs ` ( G ` k ) ) e. RR ) |
102 |
101
|
recnd |
|- ( ( G ` k ) e. CC -> ( abs ` ( G ` k ) ) e. CC ) |
103 |
71 102
|
syl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... n ) ) -> ( abs ` ( G ` k ) ) e. CC ) |
104 |
100 75 103
|
fsumser |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... n ) ( abs ` ( G ` k ) ) = ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) ) |
105 |
80
|
adantl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... j ) ) -> k e. Z ) |
106 |
105 25
|
syl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... j ) ) -> ( ( m e. Z |-> ( abs ` ( G ` m ) ) ) ` k ) = ( abs ` ( G ` k ) ) ) |
107 |
81 102
|
syl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( M ... j ) ) -> ( abs ` ( G ` k ) ) e. CC ) |
108 |
106 78 107
|
fsumser |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... j ) ( abs ` ( G ` k ) ) = ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) |
109 |
104 108
|
oveq12d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( sum_ k e. ( M ... n ) ( abs ` ( G ` k ) ) - sum_ k e. ( M ... j ) ( abs ` ( G ` k ) ) ) = ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) |
110 |
84 107
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... j ) ( abs ` ( G ` k ) ) e. CC ) |
111 |
72 102
|
syl |
|- ( ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) /\ k e. ( ( M ... n ) \ ( M ... j ) ) ) -> ( abs ` ( G ` k ) ) e. CC ) |
112 |
66 111
|
fsumcl |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( abs ` ( G ` k ) ) e. CC ) |
113 |
88 94 63 103
|
fsumsplit |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> sum_ k e. ( M ... n ) ( abs ` ( G ` k ) ) = ( sum_ k e. ( M ... j ) ( abs ` ( G ` k ) ) + sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( abs ` ( G ` k ) ) ) ) |
114 |
110 112 113
|
mvrladdd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( sum_ k e. ( M ... n ) ( abs ` ( G ` k ) ) - sum_ k e. ( M ... j ) ( abs ` ( G ` k ) ) ) = sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( abs ` ( G ` k ) ) ) |
115 |
109 114
|
eqtr3d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) = sum_ k e. ( ( M ... n ) \ ( M ... j ) ) ( abs ` ( G ` k ) ) ) |
116 |
73 98 115
|
3brtr4d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) <_ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) |
117 |
56 61 55 62 116
|
letrd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> 0 <_ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) |
118 |
55 117
|
absidd |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) = ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) |
119 |
118
|
breq1d |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x <-> ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) < x ) ) |
120 |
|
rpre |
|- ( x e. RR+ -> x e. RR ) |
121 |
120
|
ad2antlr |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> x e. RR ) |
122 |
|
lelttr |
|- ( ( ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) e. RR /\ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) e. RR /\ x e. RR ) -> ( ( ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) <_ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) /\ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) < x ) -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
123 |
61 55 121 122
|
syl3anc |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) <_ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) /\ ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) < x ) -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
124 |
116 123
|
mpand |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) < x -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
125 |
119 124
|
sylbid |
|- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ n e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
126 |
125
|
anassrs |
|- ( ( ( ( ph /\ x e. RR+ ) /\ j e. Z ) /\ n e. ( ZZ>= ` j ) ) -> ( ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x -> ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
127 |
126
|
ralimdva |
|- ( ( ( ph /\ x e. RR+ ) /\ j e. Z ) -> ( A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x -> A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
128 |
127
|
reximdva |
|- ( ( ph /\ x e. RR+ ) -> ( E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x -> E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
129 |
128
|
ralimdva |
|- ( ph -> ( A. x e. RR+ E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` n ) - ( seq M ( + , ( m e. Z |-> ( abs ` ( G ` m ) ) ) ) ` j ) ) ) < x -> A. x e. RR+ E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) ) |
130 |
47 129
|
mpd |
|- ( ph -> A. x e. RR+ E. j e. Z A. n e. ( ZZ>= ` j ) ( abs ` ( ( seq M ( + , G ) ` n ) - ( seq M ( + , G ) ` j ) ) ) < x ) |
131 |
|
seqex |
|- seq M ( + , G ) e. _V |
132 |
131
|
a1i |
|- ( ph -> seq M ( + , G ) e. _V ) |
133 |
1 12 130 132
|
caucvg |
|- ( ph -> seq M ( + , G ) e. dom ~~> ) |