Step |
Hyp |
Ref |
Expression |
1 |
|
stirlinglem1.1 |
|- H = ( n e. NN |-> ( ( n ^ 2 ) / ( n x. ( ( 2 x. n ) + 1 ) ) ) ) |
2 |
|
stirlinglem1.2 |
|- F = ( n e. NN |-> ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) |
3 |
|
stirlinglem1.3 |
|- G = ( n e. NN |-> ( 1 / ( ( 2 x. n ) + 1 ) ) ) |
4 |
|
stirlinglem1.4 |
|- L = ( n e. NN |-> ( 1 / n ) ) |
5 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
6 |
|
1zzd |
|- ( T. -> 1 e. ZZ ) |
7 |
|
ax-1cn |
|- 1 e. CC |
8 |
|
divcnv |
|- ( 1 e. CC -> ( n e. NN |-> ( 1 / n ) ) ~~> 0 ) |
9 |
7 8
|
ax-mp |
|- ( n e. NN |-> ( 1 / n ) ) ~~> 0 |
10 |
4 9
|
eqbrtri |
|- L ~~> 0 |
11 |
10
|
a1i |
|- ( T. -> L ~~> 0 ) |
12 |
|
nnex |
|- NN e. _V |
13 |
12
|
mptex |
|- ( n e. NN |-> ( 1 / ( ( 2 x. n ) + 1 ) ) ) e. _V |
14 |
3 13
|
eqeltri |
|- G e. _V |
15 |
14
|
a1i |
|- ( T. -> G e. _V ) |
16 |
4
|
a1i |
|- ( k e. NN -> L = ( n e. NN |-> ( 1 / n ) ) ) |
17 |
|
simpr |
|- ( ( k e. NN /\ n = k ) -> n = k ) |
18 |
17
|
oveq2d |
|- ( ( k e. NN /\ n = k ) -> ( 1 / n ) = ( 1 / k ) ) |
19 |
|
id |
|- ( k e. NN -> k e. NN ) |
20 |
|
nnrp |
|- ( k e. NN -> k e. RR+ ) |
21 |
20
|
rpreccld |
|- ( k e. NN -> ( 1 / k ) e. RR+ ) |
22 |
16 18 19 21
|
fvmptd |
|- ( k e. NN -> ( L ` k ) = ( 1 / k ) ) |
23 |
|
nnrecre |
|- ( k e. NN -> ( 1 / k ) e. RR ) |
24 |
22 23
|
eqeltrd |
|- ( k e. NN -> ( L ` k ) e. RR ) |
25 |
24
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( L ` k ) e. RR ) |
26 |
3
|
a1i |
|- ( k e. NN -> G = ( n e. NN |-> ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) |
27 |
17
|
oveq2d |
|- ( ( k e. NN /\ n = k ) -> ( 2 x. n ) = ( 2 x. k ) ) |
28 |
27
|
oveq1d |
|- ( ( k e. NN /\ n = k ) -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. k ) + 1 ) ) |
29 |
28
|
oveq2d |
|- ( ( k e. NN /\ n = k ) -> ( 1 / ( ( 2 x. n ) + 1 ) ) = ( 1 / ( ( 2 x. k ) + 1 ) ) ) |
30 |
|
2re |
|- 2 e. RR |
31 |
30
|
a1i |
|- ( k e. NN -> 2 e. RR ) |
32 |
|
nnre |
|- ( k e. NN -> k e. RR ) |
33 |
31 32
|
remulcld |
|- ( k e. NN -> ( 2 x. k ) e. RR ) |
34 |
|
0le2 |
|- 0 <_ 2 |
35 |
34
|
a1i |
|- ( k e. NN -> 0 <_ 2 ) |
36 |
20
|
rpge0d |
|- ( k e. NN -> 0 <_ k ) |
37 |
31 32 35 36
|
mulge0d |
|- ( k e. NN -> 0 <_ ( 2 x. k ) ) |
38 |
33 37
|
ge0p1rpd |
|- ( k e. NN -> ( ( 2 x. k ) + 1 ) e. RR+ ) |
39 |
38
|
rpreccld |
|- ( k e. NN -> ( 1 / ( ( 2 x. k ) + 1 ) ) e. RR+ ) |
40 |
26 29 19 39
|
fvmptd |
|- ( k e. NN -> ( G ` k ) = ( 1 / ( ( 2 x. k ) + 1 ) ) ) |
41 |
39
|
rpred |
|- ( k e. NN -> ( 1 / ( ( 2 x. k ) + 1 ) ) e. RR ) |
42 |
40 41
|
eqeltrd |
|- ( k e. NN -> ( G ` k ) e. RR ) |
43 |
42
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( G ` k ) e. RR ) |
44 |
|
1red |
|- ( k e. NN -> 1 e. RR ) |
45 |
|
0le1 |
|- 0 <_ 1 |
46 |
45
|
a1i |
|- ( k e. NN -> 0 <_ 1 ) |
47 |
33 44
|
readdcld |
|- ( k e. NN -> ( ( 2 x. k ) + 1 ) e. RR ) |
48 |
|
nncn |
|- ( k e. NN -> k e. CC ) |
49 |
48
|
mulid2d |
|- ( k e. NN -> ( 1 x. k ) = k ) |
50 |
|
1lt2 |
|- 1 < 2 |
51 |
50
|
a1i |
|- ( k e. NN -> 1 < 2 ) |
52 |
44 31 20 51
|
ltmul1dd |
|- ( k e. NN -> ( 1 x. k ) < ( 2 x. k ) ) |
53 |
49 52
|
eqbrtrrd |
|- ( k e. NN -> k < ( 2 x. k ) ) |
54 |
33
|
ltp1d |
|- ( k e. NN -> ( 2 x. k ) < ( ( 2 x. k ) + 1 ) ) |
55 |
32 33 47 53 54
|
lttrd |
|- ( k e. NN -> k < ( ( 2 x. k ) + 1 ) ) |
56 |
32 47 55
|
ltled |
|- ( k e. NN -> k <_ ( ( 2 x. k ) + 1 ) ) |
57 |
20 38 44 46 56
|
lediv2ad |
|- ( k e. NN -> ( 1 / ( ( 2 x. k ) + 1 ) ) <_ ( 1 / k ) ) |
58 |
57 40 22
|
3brtr4d |
|- ( k e. NN -> ( G ` k ) <_ ( L ` k ) ) |
59 |
58
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( G ` k ) <_ ( L ` k ) ) |
60 |
39
|
rpge0d |
|- ( k e. NN -> 0 <_ ( 1 / ( ( 2 x. k ) + 1 ) ) ) |
61 |
60 40
|
breqtrrd |
|- ( k e. NN -> 0 <_ ( G ` k ) ) |
62 |
61
|
adantl |
|- ( ( T. /\ k e. NN ) -> 0 <_ ( G ` k ) ) |
63 |
5 6 11 15 25 43 59 62
|
climsqz2 |
|- ( T. -> G ~~> 0 ) |
64 |
|
1cnd |
|- ( T. -> 1 e. CC ) |
65 |
12
|
mptex |
|- ( n e. NN |-> ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) e. _V |
66 |
2 65
|
eqeltri |
|- F e. _V |
67 |
66
|
a1i |
|- ( T. -> F e. _V ) |
68 |
43
|
recnd |
|- ( ( T. /\ k e. NN ) -> ( G ` k ) e. CC ) |
69 |
2
|
a1i |
|- ( k e. NN -> F = ( n e. NN |-> ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) |
70 |
29
|
oveq2d |
|- ( ( k e. NN /\ n = k ) -> ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) = ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) |
71 |
|
1cnd |
|- ( k e. NN -> 1 e. CC ) |
72 |
|
2cnd |
|- ( k e. NN -> 2 e. CC ) |
73 |
72 48
|
mulcld |
|- ( k e. NN -> ( 2 x. k ) e. CC ) |
74 |
73 71
|
addcld |
|- ( k e. NN -> ( ( 2 x. k ) + 1 ) e. CC ) |
75 |
38
|
rpne0d |
|- ( k e. NN -> ( ( 2 x. k ) + 1 ) =/= 0 ) |
76 |
74 75
|
reccld |
|- ( k e. NN -> ( 1 / ( ( 2 x. k ) + 1 ) ) e. CC ) |
77 |
71 76
|
subcld |
|- ( k e. NN -> ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) e. CC ) |
78 |
69 70 19 77
|
fvmptd |
|- ( k e. NN -> ( F ` k ) = ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) |
79 |
40
|
eqcomd |
|- ( k e. NN -> ( 1 / ( ( 2 x. k ) + 1 ) ) = ( G ` k ) ) |
80 |
79
|
oveq2d |
|- ( k e. NN -> ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) = ( 1 - ( G ` k ) ) ) |
81 |
78 80
|
eqtrd |
|- ( k e. NN -> ( F ` k ) = ( 1 - ( G ` k ) ) ) |
82 |
81
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( F ` k ) = ( 1 - ( G ` k ) ) ) |
83 |
5 6 63 64 67 68 82
|
climsubc2 |
|- ( T. -> F ~~> ( 1 - 0 ) ) |
84 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
85 |
83 84
|
breqtrdi |
|- ( T. -> F ~~> 1 ) |
86 |
64
|
halfcld |
|- ( T. -> ( 1 / 2 ) e. CC ) |
87 |
12
|
mptex |
|- ( n e. NN |-> ( ( n ^ 2 ) / ( n x. ( ( 2 x. n ) + 1 ) ) ) ) e. _V |
88 |
1 87
|
eqeltri |
|- H e. _V |
89 |
88
|
a1i |
|- ( T. -> H e. _V ) |
90 |
78 77
|
eqeltrd |
|- ( k e. NN -> ( F ` k ) e. CC ) |
91 |
90
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( F ` k ) e. CC ) |
92 |
|
nncn |
|- ( n e. NN -> n e. CC ) |
93 |
92
|
sqcld |
|- ( n e. NN -> ( n ^ 2 ) e. CC ) |
94 |
93
|
mulid2d |
|- ( n e. NN -> ( 1 x. ( n ^ 2 ) ) = ( n ^ 2 ) ) |
95 |
94
|
eqcomd |
|- ( n e. NN -> ( n ^ 2 ) = ( 1 x. ( n ^ 2 ) ) ) |
96 |
|
2cnd |
|- ( n e. NN -> 2 e. CC ) |
97 |
96 92
|
mulcld |
|- ( n e. NN -> ( 2 x. n ) e. CC ) |
98 |
|
1cnd |
|- ( n e. NN -> 1 e. CC ) |
99 |
92 97 98
|
adddid |
|- ( n e. NN -> ( n x. ( ( 2 x. n ) + 1 ) ) = ( ( n x. ( 2 x. n ) ) + ( n x. 1 ) ) ) |
100 |
92 96 92
|
mul12d |
|- ( n e. NN -> ( n x. ( 2 x. n ) ) = ( 2 x. ( n x. n ) ) ) |
101 |
92
|
sqvald |
|- ( n e. NN -> ( n ^ 2 ) = ( n x. n ) ) |
102 |
101
|
eqcomd |
|- ( n e. NN -> ( n x. n ) = ( n ^ 2 ) ) |
103 |
102
|
oveq2d |
|- ( n e. NN -> ( 2 x. ( n x. n ) ) = ( 2 x. ( n ^ 2 ) ) ) |
104 |
100 103
|
eqtrd |
|- ( n e. NN -> ( n x. ( 2 x. n ) ) = ( 2 x. ( n ^ 2 ) ) ) |
105 |
92
|
mulid1d |
|- ( n e. NN -> ( n x. 1 ) = n ) |
106 |
104 105
|
oveq12d |
|- ( n e. NN -> ( ( n x. ( 2 x. n ) ) + ( n x. 1 ) ) = ( ( 2 x. ( n ^ 2 ) ) + n ) ) |
107 |
|
2ne0 |
|- 2 =/= 0 |
108 |
107
|
a1i |
|- ( n e. NN -> 2 =/= 0 ) |
109 |
92 96 108
|
divcan2d |
|- ( n e. NN -> ( 2 x. ( n / 2 ) ) = n ) |
110 |
109
|
eqcomd |
|- ( n e. NN -> n = ( 2 x. ( n / 2 ) ) ) |
111 |
110
|
oveq2d |
|- ( n e. NN -> ( ( 2 x. ( n ^ 2 ) ) + n ) = ( ( 2 x. ( n ^ 2 ) ) + ( 2 x. ( n / 2 ) ) ) ) |
112 |
92
|
halfcld |
|- ( n e. NN -> ( n / 2 ) e. CC ) |
113 |
96 93 112
|
adddid |
|- ( n e. NN -> ( 2 x. ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( ( 2 x. ( n ^ 2 ) ) + ( 2 x. ( n / 2 ) ) ) ) |
114 |
111 113
|
eqtr4d |
|- ( n e. NN -> ( ( 2 x. ( n ^ 2 ) ) + n ) = ( 2 x. ( ( n ^ 2 ) + ( n / 2 ) ) ) ) |
115 |
99 106 114
|
3eqtrd |
|- ( n e. NN -> ( n x. ( ( 2 x. n ) + 1 ) ) = ( 2 x. ( ( n ^ 2 ) + ( n / 2 ) ) ) ) |
116 |
95 115
|
oveq12d |
|- ( n e. NN -> ( ( n ^ 2 ) / ( n x. ( ( 2 x. n ) + 1 ) ) ) = ( ( 1 x. ( n ^ 2 ) ) / ( 2 x. ( ( n ^ 2 ) + ( n / 2 ) ) ) ) ) |
117 |
93 112
|
addcld |
|- ( n e. NN -> ( ( n ^ 2 ) + ( n / 2 ) ) e. CC ) |
118 |
|
nnrp |
|- ( n e. NN -> n e. RR+ ) |
119 |
|
2z |
|- 2 e. ZZ |
120 |
119
|
a1i |
|- ( n e. NN -> 2 e. ZZ ) |
121 |
118 120
|
rpexpcld |
|- ( n e. NN -> ( n ^ 2 ) e. RR+ ) |
122 |
118
|
rphalfcld |
|- ( n e. NN -> ( n / 2 ) e. RR+ ) |
123 |
121 122
|
rpaddcld |
|- ( n e. NN -> ( ( n ^ 2 ) + ( n / 2 ) ) e. RR+ ) |
124 |
123
|
rpne0d |
|- ( n e. NN -> ( ( n ^ 2 ) + ( n / 2 ) ) =/= 0 ) |
125 |
98 96 93 117 108 124
|
divmuldivd |
|- ( n e. NN -> ( ( 1 / 2 ) x. ( ( n ^ 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) = ( ( 1 x. ( n ^ 2 ) ) / ( 2 x. ( ( n ^ 2 ) + ( n / 2 ) ) ) ) ) |
126 |
93 112
|
pncand |
|- ( n e. NN -> ( ( ( n ^ 2 ) + ( n / 2 ) ) - ( n / 2 ) ) = ( n ^ 2 ) ) |
127 |
126
|
eqcomd |
|- ( n e. NN -> ( n ^ 2 ) = ( ( ( n ^ 2 ) + ( n / 2 ) ) - ( n / 2 ) ) ) |
128 |
127
|
oveq1d |
|- ( n e. NN -> ( ( n ^ 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( ( ( ( n ^ 2 ) + ( n / 2 ) ) - ( n / 2 ) ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) |
129 |
117 112 117 124
|
divsubdird |
|- ( n e. NN -> ( ( ( ( n ^ 2 ) + ( n / 2 ) ) - ( n / 2 ) ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( ( ( ( n ^ 2 ) + ( n / 2 ) ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) - ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) ) |
130 |
117 124
|
dividd |
|- ( n e. NN -> ( ( ( n ^ 2 ) + ( n / 2 ) ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = 1 ) |
131 |
130
|
oveq1d |
|- ( n e. NN -> ( ( ( ( n ^ 2 ) + ( n / 2 ) ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) - ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) = ( 1 - ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) ) |
132 |
128 129 131
|
3eqtrd |
|- ( n e. NN -> ( ( n ^ 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( 1 - ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) ) |
133 |
|
nnne0 |
|- ( n e. NN -> n =/= 0 ) |
134 |
96 92 133
|
divcld |
|- ( n e. NN -> ( 2 / n ) e. CC ) |
135 |
96 92 108 133
|
divne0d |
|- ( n e. NN -> ( 2 / n ) =/= 0 ) |
136 |
112 117 134 124 135
|
divcan5rd |
|- ( n e. NN -> ( ( ( n / 2 ) x. ( 2 / n ) ) / ( ( ( n ^ 2 ) + ( n / 2 ) ) x. ( 2 / n ) ) ) = ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) |
137 |
92 96 133 108
|
divcan6d |
|- ( n e. NN -> ( ( n / 2 ) x. ( 2 / n ) ) = 1 ) |
138 |
93 112 134
|
adddird |
|- ( n e. NN -> ( ( ( n ^ 2 ) + ( n / 2 ) ) x. ( 2 / n ) ) = ( ( ( n ^ 2 ) x. ( 2 / n ) ) + ( ( n / 2 ) x. ( 2 / n ) ) ) ) |
139 |
93 96 92 133
|
div12d |
|- ( n e. NN -> ( ( n ^ 2 ) x. ( 2 / n ) ) = ( 2 x. ( ( n ^ 2 ) / n ) ) ) |
140 |
|
1e2m1 |
|- 1 = ( 2 - 1 ) |
141 |
140
|
oveq2i |
|- ( n ^ 1 ) = ( n ^ ( 2 - 1 ) ) |
142 |
92
|
exp1d |
|- ( n e. NN -> ( n ^ 1 ) = n ) |
143 |
92 133 120
|
expm1d |
|- ( n e. NN -> ( n ^ ( 2 - 1 ) ) = ( ( n ^ 2 ) / n ) ) |
144 |
141 142 143
|
3eqtr3a |
|- ( n e. NN -> n = ( ( n ^ 2 ) / n ) ) |
145 |
144
|
eqcomd |
|- ( n e. NN -> ( ( n ^ 2 ) / n ) = n ) |
146 |
145
|
oveq2d |
|- ( n e. NN -> ( 2 x. ( ( n ^ 2 ) / n ) ) = ( 2 x. n ) ) |
147 |
139 146
|
eqtrd |
|- ( n e. NN -> ( ( n ^ 2 ) x. ( 2 / n ) ) = ( 2 x. n ) ) |
148 |
147 137
|
oveq12d |
|- ( n e. NN -> ( ( ( n ^ 2 ) x. ( 2 / n ) ) + ( ( n / 2 ) x. ( 2 / n ) ) ) = ( ( 2 x. n ) + 1 ) ) |
149 |
138 148
|
eqtrd |
|- ( n e. NN -> ( ( ( n ^ 2 ) + ( n / 2 ) ) x. ( 2 / n ) ) = ( ( 2 x. n ) + 1 ) ) |
150 |
137 149
|
oveq12d |
|- ( n e. NN -> ( ( ( n / 2 ) x. ( 2 / n ) ) / ( ( ( n ^ 2 ) + ( n / 2 ) ) x. ( 2 / n ) ) ) = ( 1 / ( ( 2 x. n ) + 1 ) ) ) |
151 |
136 150
|
eqtr3d |
|- ( n e. NN -> ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( 1 / ( ( 2 x. n ) + 1 ) ) ) |
152 |
151
|
oveq2d |
|- ( n e. NN -> ( 1 - ( ( n / 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) = ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) |
153 |
132 152
|
eqtrd |
|- ( n e. NN -> ( ( n ^ 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) = ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) |
154 |
153
|
oveq2d |
|- ( n e. NN -> ( ( 1 / 2 ) x. ( ( n ^ 2 ) / ( ( n ^ 2 ) + ( n / 2 ) ) ) ) = ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) |
155 |
116 125 154
|
3eqtr2d |
|- ( n e. NN -> ( ( n ^ 2 ) / ( n x. ( ( 2 x. n ) + 1 ) ) ) = ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) |
156 |
155
|
mpteq2ia |
|- ( n e. NN |-> ( ( n ^ 2 ) / ( n x. ( ( 2 x. n ) + 1 ) ) ) ) = ( n e. NN |-> ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) |
157 |
1 156
|
eqtri |
|- H = ( n e. NN |-> ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) |
158 |
157
|
a1i |
|- ( k e. NN -> H = ( n e. NN |-> ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) ) ) |
159 |
70
|
oveq2d |
|- ( ( k e. NN /\ n = k ) -> ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. n ) + 1 ) ) ) ) = ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) ) |
160 |
71
|
halfcld |
|- ( k e. NN -> ( 1 / 2 ) e. CC ) |
161 |
160 77
|
mulcld |
|- ( k e. NN -> ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) e. CC ) |
162 |
158 159 19 161
|
fvmptd |
|- ( k e. NN -> ( H ` k ) = ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) ) |
163 |
78
|
oveq2d |
|- ( k e. NN -> ( ( 1 / 2 ) x. ( F ` k ) ) = ( ( 1 / 2 ) x. ( 1 - ( 1 / ( ( 2 x. k ) + 1 ) ) ) ) ) |
164 |
162 163
|
eqtr4d |
|- ( k e. NN -> ( H ` k ) = ( ( 1 / 2 ) x. ( F ` k ) ) ) |
165 |
164
|
adantl |
|- ( ( T. /\ k e. NN ) -> ( H ` k ) = ( ( 1 / 2 ) x. ( F ` k ) ) ) |
166 |
5 6 85 86 89 91 165
|
climmulc2 |
|- ( T. -> H ~~> ( ( 1 / 2 ) x. 1 ) ) |
167 |
166
|
mptru |
|- H ~~> ( ( 1 / 2 ) x. 1 ) |
168 |
|
halfcn |
|- ( 1 / 2 ) e. CC |
169 |
168
|
mulid1i |
|- ( ( 1 / 2 ) x. 1 ) = ( 1 / 2 ) |
170 |
167 169
|
breqtri |
|- H ~~> ( 1 / 2 ) |