Step |
Hyp |
Ref |
Expression |
1 |
|
knoppndvlem15.t |
|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) ) |
2 |
|
knoppndvlem15.f |
|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) ) |
3 |
|
knoppndvlem15.w |
|- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) |
4 |
|
knoppndvlem15.a |
|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) |
5 |
|
knoppndvlem15.b |
|- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) |
6 |
|
knoppndvlem15.c |
|- ( ph -> C e. ( -u 1 (,) 1 ) ) |
7 |
|
knoppndvlem15.j |
|- ( ph -> J e. NN0 ) |
8 |
|
knoppndvlem15.m |
|- ( ph -> M e. ZZ ) |
9 |
|
knoppndvlem15.n |
|- ( ph -> N e. NN ) |
10 |
|
knoppndvlem15.1 |
|- ( ph -> 1 < ( N x. ( abs ` C ) ) ) |
11 |
6
|
knoppndvlem3 |
|- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) ) |
12 |
11
|
simpld |
|- ( ph -> C e. RR ) |
13 |
12
|
recnd |
|- ( ph -> C e. CC ) |
14 |
13
|
abscld |
|- ( ph -> ( abs ` C ) e. RR ) |
15 |
14 7
|
reexpcld |
|- ( ph -> ( ( abs ` C ) ^ J ) e. RR ) |
16 |
|
2re |
|- 2 e. RR |
17 |
16
|
a1i |
|- ( ph -> 2 e. RR ) |
18 |
|
2ne0 |
|- 2 =/= 0 |
19 |
18
|
a1i |
|- ( ph -> 2 =/= 0 ) |
20 |
15 17 19
|
redivcld |
|- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) e. RR ) |
21 |
|
1red |
|- ( ph -> 1 e. RR ) |
22 |
9
|
nnred |
|- ( ph -> N e. RR ) |
23 |
17 22
|
remulcld |
|- ( ph -> ( 2 x. N ) e. RR ) |
24 |
23 14
|
remulcld |
|- ( ph -> ( ( 2 x. N ) x. ( abs ` C ) ) e. RR ) |
25 |
24 21
|
resubcld |
|- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR ) |
26 |
|
0red |
|- ( ph -> 0 e. RR ) |
27 |
|
0lt1 |
|- 0 < 1 |
28 |
27
|
a1i |
|- ( ph -> 0 < 1 ) |
29 |
6 9 10
|
knoppndvlem12 |
|- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) =/= 1 /\ 1 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) |
30 |
29
|
simprd |
|- ( ph -> 1 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) |
31 |
26 21 25 28 30
|
lttrd |
|- ( ph -> 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) |
32 |
25 31
|
jca |
|- ( ph -> ( ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR /\ 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) |
33 |
|
gt0ne0 |
|- ( ( ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) e. RR /\ 0 < ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) =/= 0 ) |
34 |
32 33
|
syl |
|- ( ph -> ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) =/= 0 ) |
35 |
21 25 34
|
redivcld |
|- ( ph -> ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) e. RR ) |
36 |
21 35
|
resubcld |
|- ( ph -> ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) |
37 |
20 36
|
remulcld |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) e. RR ) |
38 |
4
|
a1i |
|- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) |
39 |
7
|
nn0zd |
|- ( ph -> J e. ZZ ) |
40 |
9 39 8
|
knoppndvlem1 |
|- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR ) |
41 |
38 40
|
eqeltrd |
|- ( ph -> A e. RR ) |
42 |
1 2 9 12 41 7
|
knoppcnlem3 |
|- ( ph -> ( ( F ` A ) ` J ) e. RR ) |
43 |
42
|
recnd |
|- ( ph -> ( ( F ` A ) ` J ) e. CC ) |
44 |
5
|
a1i |
|- ( ph -> B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) ) |
45 |
8
|
peano2zd |
|- ( ph -> ( M + 1 ) e. ZZ ) |
46 |
9 39 45
|
knoppndvlem1 |
|- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( M + 1 ) ) e. RR ) |
47 |
44 46
|
eqeltrd |
|- ( ph -> B e. RR ) |
48 |
1 2 9 12 47 7
|
knoppcnlem3 |
|- ( ph -> ( ( F ` B ) ` J ) e. RR ) |
49 |
48
|
recnd |
|- ( ph -> ( ( F ` B ) ` J ) e. CC ) |
50 |
43 49
|
subcld |
|- ( ph -> ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) e. CC ) |
51 |
50
|
abscld |
|- ( ph -> ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) e. RR ) |
52 |
1 2 47 12 9
|
knoppndvlem5 |
|- ( ph -> sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) e. RR ) |
53 |
52
|
recnd |
|- ( ph -> sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) e. CC ) |
54 |
1 2 41 12 9
|
knoppndvlem5 |
|- ( ph -> sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) e. RR ) |
55 |
54
|
recnd |
|- ( ph -> sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) e. CC ) |
56 |
53 55
|
subcld |
|- ( ph -> ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) e. CC ) |
57 |
56
|
abscld |
|- ( ph -> ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) e. RR ) |
58 |
51 57
|
resubcld |
|- ( ph -> ( ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) - ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) e. RR ) |
59 |
50 56
|
subcld |
|- ( ph -> ( ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) e. CC ) |
60 |
59
|
abscld |
|- ( ph -> ( abs ` ( ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) e. RR ) |
61 |
20 35
|
jca |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) e. RR /\ ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) e. RR ) ) |
62 |
|
remulcl |
|- ( ( ( ( ( abs ` C ) ^ J ) / 2 ) e. RR /\ ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) e. RR ) -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) |
63 |
61 62
|
syl |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) |
64 |
20 63
|
jca |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) e. RR /\ ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) ) |
65 |
|
resubcl |
|- ( ( ( ( ( abs ` C ) ^ J ) / 2 ) e. RR /\ ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) -> ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) e. RR ) |
66 |
64 65
|
syl |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) e. RR ) |
67 |
20
|
recnd |
|- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) e. CC ) |
68 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
69 |
35
|
recnd |
|- ( ph -> ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) e. CC ) |
70 |
67 68 69
|
subdid |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) = ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. 1 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) ) |
71 |
67
|
mulid1d |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. 1 ) = ( ( ( abs ` C ) ^ J ) / 2 ) ) |
72 |
71
|
oveq1d |
|- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. 1 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) = ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) ) |
73 |
66
|
leidd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) ) |
74 |
72 73
|
eqbrtrd |
|- ( ph -> ( ( ( ( ( abs ` C ) ^ J ) / 2 ) x. 1 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) ) |
75 |
70 74
|
eqbrtrd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) ) |
76 |
20 35
|
remulcld |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) e. RR ) |
77 |
20
|
leidd |
|- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) <_ ( ( ( abs ` C ) ^ J ) / 2 ) ) |
78 |
43 49
|
abssubd |
|- ( ph -> ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) = ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) ) |
79 |
1 2 4 5 6 7 8 9
|
knoppndvlem10 |
|- ( ph -> ( abs ` ( ( ( F ` B ) ` J ) - ( ( F ` A ) ` J ) ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) ) |
80 |
78 79
|
eqtrd |
|- ( ph -> ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) = ( ( ( abs ` C ) ^ J ) / 2 ) ) |
81 |
80
|
eqcomd |
|- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) = ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) |
82 |
77 81
|
breqtrd |
|- ( ph -> ( ( ( abs ` C ) ^ J ) / 2 ) <_ ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) |
83 |
1 2 4 5 6 7 8 9 10
|
knoppndvlem14 |
|- ( ph -> ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) <_ ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) |
84 |
20 57 51 76 82 83
|
le2subd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) - ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) - ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) ) |
85 |
37 66 58 75 84
|
letrd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) - ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) ) |
86 |
50 56
|
abs2difd |
|- ( ph -> ( ( abs ` ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) - ( abs ` ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) <_ ( abs ` ( ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) ) |
87 |
37 58 60 85 86
|
letrd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( abs ` ( ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) ) |
88 |
50 56
|
abssubd |
|- ( ph -> ( abs ` ( ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) ) ) = ( abs ` ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) ) |
89 |
87 88
|
breqtrd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( abs ` ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) ) |
90 |
1 2 3 5 6 7 45 9
|
knoppndvlem6 |
|- ( ph -> ( W ` B ) = sum_ i e. ( 0 ... J ) ( ( F ` B ) ` i ) ) |
91 |
|
elnn0uz |
|- ( J e. NN0 <-> J e. ( ZZ>= ` 0 ) ) |
92 |
7 91
|
sylib |
|- ( ph -> J e. ( ZZ>= ` 0 ) ) |
93 |
9
|
adantr |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> N e. NN ) |
94 |
12
|
adantr |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> C e. RR ) |
95 |
47
|
adantr |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> B e. RR ) |
96 |
|
elfznn0 |
|- ( i e. ( 0 ... J ) -> i e. NN0 ) |
97 |
96
|
adantl |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> i e. NN0 ) |
98 |
1 2 93 94 95 97
|
knoppcnlem3 |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> ( ( F ` B ) ` i ) e. RR ) |
99 |
98
|
recnd |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> ( ( F ` B ) ` i ) e. CC ) |
100 |
|
fveq2 |
|- ( i = J -> ( ( F ` B ) ` i ) = ( ( F ` B ) ` J ) ) |
101 |
92 99 100
|
fsumm1 |
|- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` B ) ` i ) = ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) + ( ( F ` B ) ` J ) ) ) |
102 |
90 101
|
eqtrd |
|- ( ph -> ( W ` B ) = ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) + ( ( F ` B ) ` J ) ) ) |
103 |
1 2 3 4 6 7 8 9
|
knoppndvlem6 |
|- ( ph -> ( W ` A ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) ) |
104 |
41
|
adantr |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> A e. RR ) |
105 |
1 2 93 94 104 97
|
knoppcnlem3 |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> ( ( F ` A ) ` i ) e. RR ) |
106 |
105
|
recnd |
|- ( ( ph /\ i e. ( 0 ... J ) ) -> ( ( F ` A ) ` i ) e. CC ) |
107 |
|
fveq2 |
|- ( i = J -> ( ( F ` A ) ` i ) = ( ( F ` A ) ` J ) ) |
108 |
92 106 107
|
fsumm1 |
|- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) = ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) + ( ( F ` A ) ` J ) ) ) |
109 |
103 108
|
eqtrd |
|- ( ph -> ( W ` A ) = ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) + ( ( F ` A ) ` J ) ) ) |
110 |
102 109
|
oveq12d |
|- ( ph -> ( ( W ` B ) - ( W ` A ) ) = ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) + ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) + ( ( F ` A ) ` J ) ) ) ) |
111 |
53 55 43 49
|
subadd4d |
|- ( ph -> ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) = ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) + ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) + ( ( F ` A ) ` J ) ) ) ) |
112 |
111
|
eqcomd |
|- ( ph -> ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) + ( ( F ` B ) ` J ) ) - ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) + ( ( F ` A ) ` J ) ) ) = ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) |
113 |
110 112
|
eqtrd |
|- ( ph -> ( ( W ` B ) - ( W ` A ) ) = ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) |
114 |
113
|
fveq2d |
|- ( ph -> ( abs ` ( ( W ` B ) - ( W ` A ) ) ) = ( abs ` ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) ) |
115 |
114
|
eqcomd |
|- ( ph -> ( abs ` ( ( sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` B ) ` i ) - sum_ i e. ( 0 ... ( J - 1 ) ) ( ( F ` A ) ` i ) ) - ( ( ( F ` A ) ` J ) - ( ( F ` B ) ` J ) ) ) ) = ( abs ` ( ( W ` B ) - ( W ` A ) ) ) ) |
116 |
89 115
|
breqtrd |
|- ( ph -> ( ( ( ( abs ` C ) ^ J ) / 2 ) x. ( 1 - ( 1 / ( ( ( 2 x. N ) x. ( abs ` C ) ) - 1 ) ) ) ) <_ ( abs ` ( ( W ` B ) - ( W ` A ) ) ) ) |