Step |
Hyp |
Ref |
Expression |
1 |
|
leibpi.1 |
|- F = ( n e. NN0 |-> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) ) |
2 |
|
leibpilem2.2 |
|- G = ( k e. NN0 |-> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) |
3 |
|
leibpilem2.3 |
|- A e. _V |
4 |
|
2cn |
|- 2 e. CC |
5 |
|
nn0cn |
|- ( n e. NN0 -> n e. CC ) |
6 |
|
mulcl |
|- ( ( 2 e. CC /\ n e. CC ) -> ( 2 x. n ) e. CC ) |
7 |
4 5 6
|
sylancr |
|- ( n e. NN0 -> ( 2 x. n ) e. CC ) |
8 |
|
ax-1cn |
|- 1 e. CC |
9 |
|
pncan |
|- ( ( ( 2 x. n ) e. CC /\ 1 e. CC ) -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) ) |
10 |
7 8 9
|
sylancl |
|- ( n e. NN0 -> ( ( ( 2 x. n ) + 1 ) - 1 ) = ( 2 x. n ) ) |
11 |
10
|
oveq1d |
|- ( n e. NN0 -> ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) = ( ( 2 x. n ) / 2 ) ) |
12 |
|
2ne0 |
|- 2 =/= 0 |
13 |
|
divcan3 |
|- ( ( n e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( 2 x. n ) / 2 ) = n ) |
14 |
4 12 13
|
mp3an23 |
|- ( n e. CC -> ( ( 2 x. n ) / 2 ) = n ) |
15 |
5 14
|
syl |
|- ( n e. NN0 -> ( ( 2 x. n ) / 2 ) = n ) |
16 |
11 15
|
eqtrd |
|- ( n e. NN0 -> ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) = n ) |
17 |
16
|
oveq2d |
|- ( n e. NN0 -> ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) = ( -u 1 ^ n ) ) |
18 |
17
|
oveq1d |
|- ( n e. NN0 -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) = ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) ) |
19 |
18
|
mpteq2ia |
|- ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) = ( n e. NN0 |-> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) ) |
20 |
1 19
|
eqtr4i |
|- F = ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) |
21 |
|
seqeq3 |
|- ( F = ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) -> seq 0 ( + , F ) = seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ) |
22 |
20 21
|
ax-mp |
|- seq 0 ( + , F ) = seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) |
23 |
22
|
breq1i |
|- ( seq 0 ( + , F ) ~~> A <-> seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A ) |
24 |
|
neg1rr |
|- -u 1 e. RR |
25 |
|
reexpcl |
|- ( ( -u 1 e. RR /\ n e. NN0 ) -> ( -u 1 ^ n ) e. RR ) |
26 |
24 25
|
mpan |
|- ( n e. NN0 -> ( -u 1 ^ n ) e. RR ) |
27 |
|
2nn0 |
|- 2 e. NN0 |
28 |
|
nn0mulcl |
|- ( ( 2 e. NN0 /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 ) |
29 |
27 28
|
mpan |
|- ( n e. NN0 -> ( 2 x. n ) e. NN0 ) |
30 |
|
nn0p1nn |
|- ( ( 2 x. n ) e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN ) |
31 |
29 30
|
syl |
|- ( n e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN ) |
32 |
26 31
|
nndivred |
|- ( n e. NN0 -> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) e. RR ) |
33 |
32
|
recnd |
|- ( n e. NN0 -> ( ( -u 1 ^ n ) / ( ( 2 x. n ) + 1 ) ) e. CC ) |
34 |
18 33
|
eqeltrd |
|- ( n e. NN0 -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) e. CC ) |
35 |
34
|
adantl |
|- ( ( T. /\ n e. NN0 ) -> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) e. CC ) |
36 |
|
oveq1 |
|- ( k = ( ( 2 x. n ) + 1 ) -> ( k - 1 ) = ( ( ( 2 x. n ) + 1 ) - 1 ) ) |
37 |
36
|
oveq1d |
|- ( k = ( ( 2 x. n ) + 1 ) -> ( ( k - 1 ) / 2 ) = ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) |
38 |
37
|
oveq2d |
|- ( k = ( ( 2 x. n ) + 1 ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) = ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) ) |
39 |
|
id |
|- ( k = ( ( 2 x. n ) + 1 ) -> k = ( ( 2 x. n ) + 1 ) ) |
40 |
38 39
|
oveq12d |
|- ( k = ( ( 2 x. n ) + 1 ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) = ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) |
41 |
35 40
|
iserodd |
|- ( T. -> ( seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A ) ) |
42 |
41
|
mptru |
|- ( seq 0 ( + , ( n e. NN0 |-> ( ( -u 1 ^ ( ( ( ( 2 x. n ) + 1 ) - 1 ) / 2 ) ) / ( ( 2 x. n ) + 1 ) ) ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A ) |
43 |
|
addid2 |
|- ( n e. CC -> ( 0 + n ) = n ) |
44 |
43
|
adantl |
|- ( ( T. /\ n e. CC ) -> ( 0 + n ) = n ) |
45 |
|
0cnd |
|- ( T. -> 0 e. CC ) |
46 |
|
1eluzge0 |
|- 1 e. ( ZZ>= ` 0 ) |
47 |
46
|
a1i |
|- ( T. -> 1 e. ( ZZ>= ` 0 ) ) |
48 |
|
1nn0 |
|- 1 e. NN0 |
49 |
|
0cnd |
|- ( ( k e. NN0 /\ ( k = 0 \/ 2 || k ) ) -> 0 e. CC ) |
50 |
|
ioran |
|- ( -. ( k = 0 \/ 2 || k ) <-> ( -. k = 0 /\ -. 2 || k ) ) |
51 |
|
leibpilem1 |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( k e. NN /\ ( ( k - 1 ) / 2 ) e. NN0 ) ) |
52 |
51
|
simprd |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( k - 1 ) / 2 ) e. NN0 ) |
53 |
|
reexpcl |
|- ( ( -u 1 e. RR /\ ( ( k - 1 ) / 2 ) e. NN0 ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) e. RR ) |
54 |
24 52 53
|
sylancr |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) e. RR ) |
55 |
51
|
simpld |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> k e. NN ) |
56 |
54 55
|
nndivred |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. RR ) |
57 |
56
|
recnd |
|- ( ( k e. NN0 /\ ( -. k = 0 /\ -. 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. CC ) |
58 |
50 57
|
sylan2b |
|- ( ( k e. NN0 /\ -. ( k = 0 \/ 2 || k ) ) -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) e. CC ) |
59 |
49 58
|
ifclda |
|- ( k e. NN0 -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) e. CC ) |
60 |
2 59
|
fmpti |
|- G : NN0 --> CC |
61 |
60
|
ffvelrni |
|- ( 1 e. NN0 -> ( G ` 1 ) e. CC ) |
62 |
48 61
|
mp1i |
|- ( T. -> ( G ` 1 ) e. CC ) |
63 |
|
simpr |
|- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n e. ( 0 ... ( 1 - 1 ) ) ) |
64 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
65 |
64
|
oveq2i |
|- ( 0 ... ( 1 - 1 ) ) = ( 0 ... 0 ) |
66 |
63 65
|
eleqtrdi |
|- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n e. ( 0 ... 0 ) ) |
67 |
|
elfz1eq |
|- ( n e. ( 0 ... 0 ) -> n = 0 ) |
68 |
66 67
|
syl |
|- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> n = 0 ) |
69 |
68
|
fveq2d |
|- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> ( G ` n ) = ( G ` 0 ) ) |
70 |
|
0nn0 |
|- 0 e. NN0 |
71 |
|
iftrue |
|- ( ( k = 0 \/ 2 || k ) -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = 0 ) |
72 |
71
|
orcs |
|- ( k = 0 -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = 0 ) |
73 |
|
c0ex |
|- 0 e. _V |
74 |
72 2 73
|
fvmpt |
|- ( 0 e. NN0 -> ( G ` 0 ) = 0 ) |
75 |
70 74
|
ax-mp |
|- ( G ` 0 ) = 0 |
76 |
69 75
|
eqtrdi |
|- ( ( T. /\ n e. ( 0 ... ( 1 - 1 ) ) ) -> ( G ` n ) = 0 ) |
77 |
44 45 47 62 76
|
seqid |
|- ( T. -> ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , G ) ) |
78 |
|
1zzd |
|- ( T. -> 1 e. ZZ ) |
79 |
|
simpr |
|- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> n e. ( ZZ>= ` 1 ) ) |
80 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
81 |
79 80
|
eleqtrrdi |
|- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> n e. NN ) |
82 |
|
nnne0 |
|- ( n e. NN -> n =/= 0 ) |
83 |
82
|
neneqd |
|- ( n e. NN -> -. n = 0 ) |
84 |
|
biorf |
|- ( -. n = 0 -> ( 2 || n <-> ( n = 0 \/ 2 || n ) ) ) |
85 |
83 84
|
syl |
|- ( n e. NN -> ( 2 || n <-> ( n = 0 \/ 2 || n ) ) ) |
86 |
85
|
ifbid |
|- ( n e. NN -> if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
87 |
|
breq2 |
|- ( k = n -> ( 2 || k <-> 2 || n ) ) |
88 |
|
oveq1 |
|- ( k = n -> ( k - 1 ) = ( n - 1 ) ) |
89 |
88
|
oveq1d |
|- ( k = n -> ( ( k - 1 ) / 2 ) = ( ( n - 1 ) / 2 ) ) |
90 |
89
|
oveq2d |
|- ( k = n -> ( -u 1 ^ ( ( k - 1 ) / 2 ) ) = ( -u 1 ^ ( ( n - 1 ) / 2 ) ) ) |
91 |
|
id |
|- ( k = n -> k = n ) |
92 |
90 91
|
oveq12d |
|- ( k = n -> ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) = ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) |
93 |
87 92
|
ifbieq2d |
|- ( k = n -> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
94 |
|
eqid |
|- ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) = ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) |
95 |
|
ovex |
|- ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) e. _V |
96 |
73 95
|
ifex |
|- if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) e. _V |
97 |
93 94 96
|
fvmpt |
|- ( n e. NN -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = if ( 2 || n , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
98 |
|
nnnn0 |
|- ( n e. NN -> n e. NN0 ) |
99 |
|
eqeq1 |
|- ( k = n -> ( k = 0 <-> n = 0 ) ) |
100 |
99 87
|
orbi12d |
|- ( k = n -> ( ( k = 0 \/ 2 || k ) <-> ( n = 0 \/ 2 || n ) ) ) |
101 |
100 92
|
ifbieq2d |
|- ( k = n -> if ( ( k = 0 \/ 2 || k ) , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
102 |
73 95
|
ifex |
|- if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) e. _V |
103 |
101 2 102
|
fvmpt |
|- ( n e. NN0 -> ( G ` n ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
104 |
98 103
|
syl |
|- ( n e. NN -> ( G ` n ) = if ( ( n = 0 \/ 2 || n ) , 0 , ( ( -u 1 ^ ( ( n - 1 ) / 2 ) ) / n ) ) ) |
105 |
86 97 104
|
3eqtr4d |
|- ( n e. NN -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = ( G ` n ) ) |
106 |
81 105
|
syl |
|- ( ( T. /\ n e. ( ZZ>= ` 1 ) ) -> ( ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ` n ) = ( G ` n ) ) |
107 |
78 106
|
seqfeq |
|- ( T. -> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) = seq 1 ( + , G ) ) |
108 |
77 107
|
eqtr4d |
|- ( T. -> ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ) |
109 |
108
|
mptru |
|- ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) = seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) |
110 |
109
|
breq1i |
|- ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A ) |
111 |
|
1z |
|- 1 e. ZZ |
112 |
|
seqex |
|- seq 0 ( + , G ) e. _V |
113 |
|
climres |
|- ( ( 1 e. ZZ /\ seq 0 ( + , G ) e. _V ) -> ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 0 ( + , G ) ~~> A ) ) |
114 |
111 112 113
|
mp2an |
|- ( ( seq 0 ( + , G ) |` ( ZZ>= ` 1 ) ) ~~> A <-> seq 0 ( + , G ) ~~> A ) |
115 |
110 114
|
bitr3i |
|- ( seq 1 ( + , ( k e. NN |-> if ( 2 || k , 0 , ( ( -u 1 ^ ( ( k - 1 ) / 2 ) ) / k ) ) ) ) ~~> A <-> seq 0 ( + , G ) ~~> A ) |
116 |
23 42 115
|
3bitri |
|- ( seq 0 ( + , F ) ~~> A <-> seq 0 ( + , G ) ~~> A ) |