Step |
Hyp |
Ref |
Expression |
1 |
|
knoppndvlem6.t |
|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) ) |
2 |
|
knoppndvlem6.f |
|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) ) |
3 |
|
knoppndvlem6.w |
|- W = ( w e. RR |-> sum_ i e. NN0 ( ( F ` w ) ` i ) ) |
4 |
|
knoppndvlem6.a |
|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) |
5 |
|
knoppndvlem6.c |
|- ( ph -> C e. ( -u 1 (,) 1 ) ) |
6 |
|
knoppndvlem6.j |
|- ( ph -> J e. NN0 ) |
7 |
|
knoppndvlem6.m |
|- ( ph -> M e. ZZ ) |
8 |
|
knoppndvlem6.n |
|- ( ph -> N e. NN ) |
9 |
|
fveq2 |
|- ( w = A -> ( F ` w ) = ( F ` A ) ) |
10 |
9
|
fveq1d |
|- ( w = A -> ( ( F ` w ) ` i ) = ( ( F ` A ) ` i ) ) |
11 |
10
|
sumeq2sdv |
|- ( w = A -> sum_ i e. NN0 ( ( F ` w ) ` i ) = sum_ i e. NN0 ( ( F ` A ) ` i ) ) |
12 |
4
|
a1i |
|- ( ph -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) |
13 |
6
|
nn0zd |
|- ( ph -> J e. ZZ ) |
14 |
8 13 7
|
knoppndvlem1 |
|- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) e. RR ) |
15 |
12 14
|
eqeltrd |
|- ( ph -> A e. RR ) |
16 |
|
sumex |
|- sum_ i e. NN0 ( ( F ` A ) ` i ) e. _V |
17 |
16
|
a1i |
|- ( ph -> sum_ i e. NN0 ( ( F ` A ) ` i ) e. _V ) |
18 |
3 11 15 17
|
fvmptd3 |
|- ( ph -> ( W ` A ) = sum_ i e. NN0 ( ( F ` A ) ` i ) ) |
19 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
20 |
|
eqid |
|- ( ZZ>= ` ( J + 1 ) ) = ( ZZ>= ` ( J + 1 ) ) |
21 |
|
peano2nn0 |
|- ( J e. NN0 -> ( J + 1 ) e. NN0 ) |
22 |
6 21
|
syl |
|- ( ph -> ( J + 1 ) e. NN0 ) |
23 |
|
eqidd |
|- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) = ( ( F ` A ) ` i ) ) |
24 |
8
|
adantr |
|- ( ( ph /\ i e. NN0 ) -> N e. NN ) |
25 |
5
|
knoppndvlem3 |
|- ( ph -> ( C e. RR /\ ( abs ` C ) < 1 ) ) |
26 |
25
|
simpld |
|- ( ph -> C e. RR ) |
27 |
26
|
adantr |
|- ( ( ph /\ i e. NN0 ) -> C e. RR ) |
28 |
15
|
adantr |
|- ( ( ph /\ i e. NN0 ) -> A e. RR ) |
29 |
|
simpr |
|- ( ( ph /\ i e. NN0 ) -> i e. NN0 ) |
30 |
1 2 24 27 28 29
|
knoppcnlem3 |
|- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) e. RR ) |
31 |
30
|
recnd |
|- ( ( ph /\ i e. NN0 ) -> ( ( F ` A ) ` i ) e. CC ) |
32 |
1 2 3 15 5 8
|
knoppndvlem4 |
|- ( ph -> seq 0 ( + , ( F ` A ) ) ~~> ( W ` A ) ) |
33 |
|
seqex |
|- seq 0 ( + , ( F ` A ) ) e. _V |
34 |
|
fvex |
|- ( W ` A ) e. _V |
35 |
33 34
|
breldm |
|- ( seq 0 ( + , ( F ` A ) ) ~~> ( W ` A ) -> seq 0 ( + , ( F ` A ) ) e. dom ~~> ) |
36 |
32 35
|
syl |
|- ( ph -> seq 0 ( + , ( F ` A ) ) e. dom ~~> ) |
37 |
19 20 22 23 31 36
|
isumsplit |
|- ( ph -> sum_ i e. NN0 ( ( F ` A ) ` i ) = ( sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) ) |
38 |
6
|
nn0cnd |
|- ( ph -> J e. CC ) |
39 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
40 |
38 39
|
pncand |
|- ( ph -> ( ( J + 1 ) - 1 ) = J ) |
41 |
40
|
oveq2d |
|- ( ph -> ( 0 ... ( ( J + 1 ) - 1 ) ) = ( 0 ... J ) ) |
42 |
41
|
sumeq1d |
|- ( ph -> sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) ) |
43 |
42
|
oveq1d |
|- ( ph -> ( sum_ i e. ( 0 ... ( ( J + 1 ) - 1 ) ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) ) |
44 |
18 37 43
|
3eqtrd |
|- ( ph -> ( W ` A ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) ) |
45 |
15
|
adantr |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> A e. RR ) |
46 |
|
eluznn0 |
|- ( ( ( J + 1 ) e. NN0 /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. NN0 ) |
47 |
22 46
|
sylan |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. NN0 ) |
48 |
2 45 47
|
knoppcnlem1 |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( F ` A ) ` i ) = ( ( C ^ i ) x. ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) ) ) |
49 |
4
|
a1i |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) |
50 |
49
|
oveq2d |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. A ) = ( ( ( 2 x. N ) ^ i ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) ) |
51 |
8
|
adantr |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> N e. NN ) |
52 |
47
|
nn0zd |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> i e. ZZ ) |
53 |
13
|
adantr |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> J e. ZZ ) |
54 |
7
|
adantr |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> M e. ZZ ) |
55 |
|
eluzle |
|- ( i e. ( ZZ>= ` ( J + 1 ) ) -> ( J + 1 ) <_ i ) |
56 |
55
|
adantl |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J + 1 ) <_ i ) |
57 |
53 52
|
jca |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J e. ZZ /\ i e. ZZ ) ) |
58 |
|
zltp1le |
|- ( ( J e. ZZ /\ i e. ZZ ) -> ( J < i <-> ( J + 1 ) <_ i ) ) |
59 |
57 58
|
syl |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( J < i <-> ( J + 1 ) <_ i ) ) |
60 |
56 59
|
mpbird |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> J < i ) |
61 |
51 52 53 54 60
|
knoppndvlem2 |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) e. ZZ ) |
62 |
50 61
|
eqeltrd |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( ( 2 x. N ) ^ i ) x. A ) e. ZZ ) |
63 |
1 62
|
dnizeq0 |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) = 0 ) |
64 |
63
|
oveq2d |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( C ^ i ) x. ( T ` ( ( ( 2 x. N ) ^ i ) x. A ) ) ) = ( ( C ^ i ) x. 0 ) ) |
65 |
26
|
recnd |
|- ( ph -> C e. CC ) |
66 |
65
|
adantr |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> C e. CC ) |
67 |
66 47
|
expcld |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( C ^ i ) e. CC ) |
68 |
67
|
mul01d |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( C ^ i ) x. 0 ) = 0 ) |
69 |
48 64 68
|
3eqtrd |
|- ( ( ph /\ i e. ( ZZ>= ` ( J + 1 ) ) ) -> ( ( F ` A ) ` i ) = 0 ) |
70 |
69
|
sumeq2dv |
|- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) = sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 ) |
71 |
|
ssidd |
|- ( ph -> ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) ) |
72 |
71
|
orcd |
|- ( ph -> ( ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) \/ ( ZZ>= ` ( J + 1 ) ) e. Fin ) ) |
73 |
|
sumz |
|- ( ( ( ZZ>= ` ( J + 1 ) ) C_ ( ZZ>= ` ( J + 1 ) ) \/ ( ZZ>= ` ( J + 1 ) ) e. Fin ) -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 = 0 ) |
74 |
72 73
|
syl |
|- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) 0 = 0 ) |
75 |
70 74
|
eqtrd |
|- ( ph -> sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) = 0 ) |
76 |
75
|
oveq2d |
|- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + 0 ) ) |
77 |
1 2 15 26 8
|
knoppndvlem5 |
|- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) e. RR ) |
78 |
77
|
recnd |
|- ( ph -> sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) e. CC ) |
79 |
78
|
addid1d |
|- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + 0 ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) ) |
80 |
76 79
|
eqtrd |
|- ( ph -> ( sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) + sum_ i e. ( ZZ>= ` ( J + 1 ) ) ( ( F ` A ) ` i ) ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) ) |
81 |
44 80
|
eqtrd |
|- ( ph -> ( W ` A ) = sum_ i e. ( 0 ... J ) ( ( F ` A ) ` i ) ) |