Step |
Hyp |
Ref |
Expression |
1 |
|
stirlinglem6.1 |
|- H = ( j e. NN0 |-> ( 2 x. ( ( 1 / ( ( 2 x. j ) + 1 ) ) x. ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ ( ( 2 x. j ) + 1 ) ) ) ) ) |
2 |
|
eqid |
|- ( j e. NN |-> ( ( -u 1 ^ ( j - 1 ) ) x. ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) ) = ( j e. NN |-> ( ( -u 1 ^ ( j - 1 ) ) x. ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) ) |
3 |
|
eqid |
|- ( j e. NN |-> ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) = ( j e. NN |-> ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) |
4 |
|
eqid |
|- ( j e. NN |-> ( ( ( -u 1 ^ ( j - 1 ) ) x. ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) + ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) ) = ( j e. NN |-> ( ( ( -u 1 ^ ( j - 1 ) ) x. ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) + ( ( ( 1 / ( ( 2 x. N ) + 1 ) ) ^ j ) / j ) ) ) |
5 |
|
eqid |
|- ( j e. NN0 |-> ( ( 2 x. j ) + 1 ) ) = ( j e. NN0 |-> ( ( 2 x. j ) + 1 ) ) |
6 |
|
2re |
|- 2 e. RR |
7 |
6
|
a1i |
|- ( N e. NN -> 2 e. RR ) |
8 |
|
nnre |
|- ( N e. NN -> N e. RR ) |
9 |
7 8
|
remulcld |
|- ( N e. NN -> ( 2 x. N ) e. RR ) |
10 |
|
0le2 |
|- 0 <_ 2 |
11 |
10
|
a1i |
|- ( N e. NN -> 0 <_ 2 ) |
12 |
|
0red |
|- ( N e. NN -> 0 e. RR ) |
13 |
|
nngt0 |
|- ( N e. NN -> 0 < N ) |
14 |
12 8 13
|
ltled |
|- ( N e. NN -> 0 <_ N ) |
15 |
7 8 11 14
|
mulge0d |
|- ( N e. NN -> 0 <_ ( 2 x. N ) ) |
16 |
9 15
|
ge0p1rpd |
|- ( N e. NN -> ( ( 2 x. N ) + 1 ) e. RR+ ) |
17 |
16
|
rpreccld |
|- ( N e. NN -> ( 1 / ( ( 2 x. N ) + 1 ) ) e. RR+ ) |
18 |
|
1red |
|- ( N e. NN -> 1 e. RR ) |
19 |
18
|
renegcld |
|- ( N e. NN -> -u 1 e. RR ) |
20 |
17
|
rpred |
|- ( N e. NN -> ( 1 / ( ( 2 x. N ) + 1 ) ) e. RR ) |
21 |
|
neg1lt0 |
|- -u 1 < 0 |
22 |
21
|
a1i |
|- ( N e. NN -> -u 1 < 0 ) |
23 |
17
|
rpgt0d |
|- ( N e. NN -> 0 < ( 1 / ( ( 2 x. N ) + 1 ) ) ) |
24 |
19 12 20 22 23
|
lttrd |
|- ( N e. NN -> -u 1 < ( 1 / ( ( 2 x. N ) + 1 ) ) ) |
25 |
|
1rp |
|- 1 e. RR+ |
26 |
25
|
a1i |
|- ( N e. NN -> 1 e. RR+ ) |
27 |
|
1cnd |
|- ( N e. NN -> 1 e. CC ) |
28 |
27
|
div1d |
|- ( N e. NN -> ( 1 / 1 ) = 1 ) |
29 |
|
2rp |
|- 2 e. RR+ |
30 |
29
|
a1i |
|- ( N e. NN -> 2 e. RR+ ) |
31 |
|
nnrp |
|- ( N e. NN -> N e. RR+ ) |
32 |
30 31
|
rpmulcld |
|- ( N e. NN -> ( 2 x. N ) e. RR+ ) |
33 |
18 32
|
ltaddrp2d |
|- ( N e. NN -> 1 < ( ( 2 x. N ) + 1 ) ) |
34 |
28 33
|
eqbrtrd |
|- ( N e. NN -> ( 1 / 1 ) < ( ( 2 x. N ) + 1 ) ) |
35 |
26 16 34
|
ltrec1d |
|- ( N e. NN -> ( 1 / ( ( 2 x. N ) + 1 ) ) < 1 ) |
36 |
20 18
|
absltd |
|- ( N e. NN -> ( ( abs ` ( 1 / ( ( 2 x. N ) + 1 ) ) ) < 1 <-> ( -u 1 < ( 1 / ( ( 2 x. N ) + 1 ) ) /\ ( 1 / ( ( 2 x. N ) + 1 ) ) < 1 ) ) ) |
37 |
24 35 36
|
mpbir2and |
|- ( N e. NN -> ( abs ` ( 1 / ( ( 2 x. N ) + 1 ) ) ) < 1 ) |
38 |
2 3 4 1 5 17 37
|
stirlinglem5 |
|- ( N e. NN -> seq 0 ( + , H ) ~~> ( log ` ( ( 1 + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( 1 - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) ) ) |
39 |
|
2cnd |
|- ( N e. NN -> 2 e. CC ) |
40 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
41 |
39 40
|
mulcld |
|- ( N e. NN -> ( 2 x. N ) e. CC ) |
42 |
41 27
|
addcld |
|- ( N e. NN -> ( ( 2 x. N ) + 1 ) e. CC ) |
43 |
9 18
|
readdcld |
|- ( N e. NN -> ( ( 2 x. N ) + 1 ) e. RR ) |
44 |
|
2pos |
|- 0 < 2 |
45 |
44
|
a1i |
|- ( N e. NN -> 0 < 2 ) |
46 |
7 8 45 13
|
mulgt0d |
|- ( N e. NN -> 0 < ( 2 x. N ) ) |
47 |
9
|
ltp1d |
|- ( N e. NN -> ( 2 x. N ) < ( ( 2 x. N ) + 1 ) ) |
48 |
12 9 43 46 47
|
lttrd |
|- ( N e. NN -> 0 < ( ( 2 x. N ) + 1 ) ) |
49 |
48
|
gt0ne0d |
|- ( N e. NN -> ( ( 2 x. N ) + 1 ) =/= 0 ) |
50 |
42 49
|
dividd |
|- ( N e. NN -> ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) = 1 ) |
51 |
50
|
eqcomd |
|- ( N e. NN -> 1 = ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) ) |
52 |
51
|
oveq1d |
|- ( N e. NN -> ( 1 + ( 1 / ( ( 2 x. N ) + 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) |
53 |
51
|
oveq1d |
|- ( N e. NN -> ( 1 - ( 1 / ( ( 2 x. N ) + 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) |
54 |
52 53
|
oveq12d |
|- ( N e. NN -> ( ( 1 + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( 1 - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) = ( ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) ) |
55 |
42 27 42 49
|
divdird |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) + 1 ) / ( ( 2 x. N ) + 1 ) ) = ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) |
56 |
55
|
eqcomd |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) + 1 ) / ( ( 2 x. N ) + 1 ) ) ) |
57 |
42 27 42 49
|
divsubdird |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) - 1 ) / ( ( 2 x. N ) + 1 ) ) = ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) |
58 |
57
|
eqcomd |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) = ( ( ( ( 2 x. N ) + 1 ) - 1 ) / ( ( 2 x. N ) + 1 ) ) ) |
59 |
56 58
|
oveq12d |
|- ( N e. NN -> ( ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) = ( ( ( ( ( 2 x. N ) + 1 ) + 1 ) / ( ( 2 x. N ) + 1 ) ) / ( ( ( ( 2 x. N ) + 1 ) - 1 ) / ( ( 2 x. N ) + 1 ) ) ) ) |
60 |
41 27 27
|
addassd |
|- ( N e. NN -> ( ( ( 2 x. N ) + 1 ) + 1 ) = ( ( 2 x. N ) + ( 1 + 1 ) ) ) |
61 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
62 |
61
|
a1i |
|- ( N e. NN -> ( 1 + 1 ) = 2 ) |
63 |
62
|
oveq2d |
|- ( N e. NN -> ( ( 2 x. N ) + ( 1 + 1 ) ) = ( ( 2 x. N ) + 2 ) ) |
64 |
39
|
mulid1d |
|- ( N e. NN -> ( 2 x. 1 ) = 2 ) |
65 |
64
|
eqcomd |
|- ( N e. NN -> 2 = ( 2 x. 1 ) ) |
66 |
65
|
oveq2d |
|- ( N e. NN -> ( ( 2 x. N ) + 2 ) = ( ( 2 x. N ) + ( 2 x. 1 ) ) ) |
67 |
39 40 27
|
adddid |
|- ( N e. NN -> ( 2 x. ( N + 1 ) ) = ( ( 2 x. N ) + ( 2 x. 1 ) ) ) |
68 |
66 67
|
eqtr4d |
|- ( N e. NN -> ( ( 2 x. N ) + 2 ) = ( 2 x. ( N + 1 ) ) ) |
69 |
60 63 68
|
3eqtrd |
|- ( N e. NN -> ( ( ( 2 x. N ) + 1 ) + 1 ) = ( 2 x. ( N + 1 ) ) ) |
70 |
69
|
oveq1d |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) + 1 ) / ( ( 2 x. N ) + 1 ) ) = ( ( 2 x. ( N + 1 ) ) / ( ( 2 x. N ) + 1 ) ) ) |
71 |
41 27
|
pncand |
|- ( N e. NN -> ( ( ( 2 x. N ) + 1 ) - 1 ) = ( 2 x. N ) ) |
72 |
71
|
oveq1d |
|- ( N e. NN -> ( ( ( ( 2 x. N ) + 1 ) - 1 ) / ( ( 2 x. N ) + 1 ) ) = ( ( 2 x. N ) / ( ( 2 x. N ) + 1 ) ) ) |
73 |
70 72
|
oveq12d |
|- ( N e. NN -> ( ( ( ( ( 2 x. N ) + 1 ) + 1 ) / ( ( 2 x. N ) + 1 ) ) / ( ( ( ( 2 x. N ) + 1 ) - 1 ) / ( ( 2 x. N ) + 1 ) ) ) = ( ( ( 2 x. ( N + 1 ) ) / ( ( 2 x. N ) + 1 ) ) / ( ( 2 x. N ) / ( ( 2 x. N ) + 1 ) ) ) ) |
74 |
59 73
|
eqtrd |
|- ( N e. NN -> ( ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( ( ( ( 2 x. N ) + 1 ) / ( ( 2 x. N ) + 1 ) ) - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) = ( ( ( 2 x. ( N + 1 ) ) / ( ( 2 x. N ) + 1 ) ) / ( ( 2 x. N ) / ( ( 2 x. N ) + 1 ) ) ) ) |
75 |
40 27
|
addcld |
|- ( N e. NN -> ( N + 1 ) e. CC ) |
76 |
39 75
|
mulcld |
|- ( N e. NN -> ( 2 x. ( N + 1 ) ) e. CC ) |
77 |
46
|
gt0ne0d |
|- ( N e. NN -> ( 2 x. N ) =/= 0 ) |
78 |
76 41 42 77 49
|
divcan7d |
|- ( N e. NN -> ( ( ( 2 x. ( N + 1 ) ) / ( ( 2 x. N ) + 1 ) ) / ( ( 2 x. N ) / ( ( 2 x. N ) + 1 ) ) ) = ( ( 2 x. ( N + 1 ) ) / ( 2 x. N ) ) ) |
79 |
45
|
gt0ne0d |
|- ( N e. NN -> 2 =/= 0 ) |
80 |
13
|
gt0ne0d |
|- ( N e. NN -> N =/= 0 ) |
81 |
39 39 75 40 79 80
|
divmuldivd |
|- ( N e. NN -> ( ( 2 / 2 ) x. ( ( N + 1 ) / N ) ) = ( ( 2 x. ( N + 1 ) ) / ( 2 x. N ) ) ) |
82 |
81
|
eqcomd |
|- ( N e. NN -> ( ( 2 x. ( N + 1 ) ) / ( 2 x. N ) ) = ( ( 2 / 2 ) x. ( ( N + 1 ) / N ) ) ) |
83 |
39 79
|
dividd |
|- ( N e. NN -> ( 2 / 2 ) = 1 ) |
84 |
83
|
oveq1d |
|- ( N e. NN -> ( ( 2 / 2 ) x. ( ( N + 1 ) / N ) ) = ( 1 x. ( ( N + 1 ) / N ) ) ) |
85 |
75 40 80
|
divcld |
|- ( N e. NN -> ( ( N + 1 ) / N ) e. CC ) |
86 |
85
|
mulid2d |
|- ( N e. NN -> ( 1 x. ( ( N + 1 ) / N ) ) = ( ( N + 1 ) / N ) ) |
87 |
84 86
|
eqtrd |
|- ( N e. NN -> ( ( 2 / 2 ) x. ( ( N + 1 ) / N ) ) = ( ( N + 1 ) / N ) ) |
88 |
78 82 87
|
3eqtrd |
|- ( N e. NN -> ( ( ( 2 x. ( N + 1 ) ) / ( ( 2 x. N ) + 1 ) ) / ( ( 2 x. N ) / ( ( 2 x. N ) + 1 ) ) ) = ( ( N + 1 ) / N ) ) |
89 |
54 74 88
|
3eqtrd |
|- ( N e. NN -> ( ( 1 + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( 1 - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) = ( ( N + 1 ) / N ) ) |
90 |
89
|
fveq2d |
|- ( N e. NN -> ( log ` ( ( 1 + ( 1 / ( ( 2 x. N ) + 1 ) ) ) / ( 1 - ( 1 / ( ( 2 x. N ) + 1 ) ) ) ) ) = ( log ` ( ( N + 1 ) / N ) ) ) |
91 |
38 90
|
breqtrd |
|- ( N e. NN -> seq 0 ( + , H ) ~~> ( log ` ( ( N + 1 ) / N ) ) ) |