Step |
Hyp |
Ref |
Expression |
1 |
|
abelth.1 |
|
2 |
|
abelth.2 |
|
3 |
|
abelth.3 |
|
4 |
|
abelth.4 |
|
5 |
|
abelth.5 |
|
6 |
|
abelth.6 |
|
7 |
|
abelth.7 |
|
8 |
|
abelthlem6.1 |
|
9 |
8
|
eldifad |
|
10 |
|
oveq1 |
|
11 |
10
|
oveq2d |
|
12 |
11
|
sumeq2sdv |
|
13 |
|
sumex |
|
14 |
12 6 13
|
fvmpt |
|
15 |
9 14
|
syl |
|
16 |
|
nn0uz |
|
17 |
|
0zd |
|
18 |
|
fveq2 |
|
19 |
|
oveq2 |
|
20 |
18 19
|
oveq12d |
|
21 |
|
eqid |
|
22 |
|
ovex |
|
23 |
20 21 22
|
fvmpt |
|
24 |
23
|
adantl |
|
25 |
1
|
ffvelrnda |
|
26 |
5
|
ssrab3 |
|
27 |
26 9
|
sselid |
|
28 |
|
expcl |
|
29 |
27 28
|
sylan |
|
30 |
25 29
|
mulcld |
|
31 |
|
fveq2 |
|
32 |
31 19
|
oveq12d |
|
33 |
|
eqid |
|
34 |
|
ovex |
|
35 |
32 33 34
|
fvmpt |
|
36 |
35
|
adantl |
|
37 |
16 17 25
|
serf |
|
38 |
37
|
ffvelrnda |
|
39 |
38 29
|
mulcld |
|
40 |
1 2 3 4 5
|
abelthlem2 |
|
41 |
40
|
simprd |
|
42 |
41 8
|
sseldd |
|
43 |
1 2 3 4 5 6 7
|
abelthlem5 |
|
44 |
42 43
|
mpdan |
|
45 |
16 17 36 39 44
|
isumclim2 |
|
46 |
|
seqex |
|
47 |
46
|
a1i |
|
48 |
|
0nn0 |
|
49 |
48
|
a1i |
|
50 |
|
oveq1 |
|
51 |
50
|
oveq2d |
|
52 |
51
|
sumeq1d |
|
53 |
|
oveq2 |
|
54 |
52 53
|
oveq12d |
|
55 |
|
eqid |
|
56 |
|
ovex |
|
57 |
54 55 56
|
fvmpt |
|
58 |
57
|
adantl |
|
59 |
|
fzfid |
|
60 |
1
|
adantr |
|
61 |
|
elfznn0 |
|
62 |
|
ffvelrn |
|
63 |
60 61 62
|
syl2an |
|
64 |
59 63
|
fsumcl |
|
65 |
|
expcl |
|
66 |
27 65
|
sylan |
|
67 |
64 66
|
mulcld |
|
68 |
58 67
|
eqeltrd |
|
69 |
17
|
peano2zd |
|
70 |
|
nnuz |
|
71 |
|
1e0p1 |
|
72 |
71
|
fveq2i |
|
73 |
70 72
|
eqtri |
|
74 |
73
|
eleq2i |
|
75 |
|
nnm1nn0 |
|
76 |
75
|
adantl |
|
77 |
|
fveq2 |
|
78 |
|
oveq2 |
|
79 |
77 78
|
oveq12d |
|
80 |
79
|
oveq2d |
|
81 |
|
eqid |
|
82 |
|
ovex |
|
83 |
80 81 82
|
fvmpt |
|
84 |
76 83
|
syl |
|
85 |
|
ax-1cn |
|
86 |
|
nncn |
|
87 |
86
|
adantl |
|
88 |
|
nn0ex |
|
89 |
88
|
mptex |
|
90 |
89
|
shftval |
|
91 |
85 87 90
|
sylancr |
|
92 |
|
eqidd |
|
93 |
76 16
|
eleqtrdi |
|
94 |
1
|
adantr |
|
95 |
|
elfznn0 |
|
96 |
94 95 62
|
syl2an |
|
97 |
92 93 96
|
fsumser |
|
98 |
|
expm1t |
|
99 |
27 98
|
sylan |
|
100 |
27
|
adantr |
|
101 |
|
expcl |
|
102 |
27 75 101
|
syl2an |
|
103 |
100 102
|
mulcomd |
|
104 |
99 103
|
eqtr4d |
|
105 |
97 104
|
oveq12d |
|
106 |
|
nnnn0 |
|
107 |
106
|
adantl |
|
108 |
|
oveq1 |
|
109 |
108
|
oveq2d |
|
110 |
109
|
sumeq1d |
|
111 |
110 19
|
oveq12d |
|
112 |
|
ovex |
|
113 |
111 55 112
|
fvmpt |
|
114 |
107 113
|
syl |
|
115 |
|
ffvelrn |
|
116 |
37 75 115
|
syl2an |
|
117 |
100 116 102
|
mul12d |
|
118 |
105 114 117
|
3eqtr4d |
|
119 |
84 91 118
|
3eqtr4d |
|
120 |
74 119
|
sylan2br |
|
121 |
69 120
|
seqfeq |
|
122 |
|
fveq2 |
|
123 |
122 53
|
oveq12d |
|
124 |
|
ovex |
|
125 |
123 33 124
|
fvmpt |
|
126 |
125
|
adantl |
|
127 |
37
|
ffvelrnda |
|
128 |
127 66
|
mulcld |
|
129 |
126 128
|
eqeltrd |
|
130 |
123
|
oveq2d |
|
131 |
|
ovex |
|
132 |
130 81 131
|
fvmpt |
|
133 |
132
|
adantl |
|
134 |
126
|
oveq2d |
|
135 |
133 134
|
eqtr4d |
|
136 |
16 17 27 45 129 135
|
isermulc2 |
|
137 |
|
0z |
|
138 |
|
1z |
|
139 |
89
|
isershft |
|
140 |
137 138 139
|
mp2an |
|
141 |
136 140
|
sylib |
|
142 |
121 141
|
eqbrtrrd |
|
143 |
16 49 68 142
|
clim2ser2 |
|
144 |
|
seq1 |
|
145 |
137 144
|
ax-mp |
|
146 |
|
oveq1 |
|
147 |
146
|
oveq2d |
|
148 |
|
risefall0lem |
|
149 |
147 148
|
eqtrdi |
|
150 |
149
|
sumeq1d |
|
151 |
|
sum0 |
|
152 |
150 151
|
eqtrdi |
|
153 |
|
oveq2 |
|
154 |
152 153
|
oveq12d |
|
155 |
|
ovex |
|
156 |
154 55 155
|
fvmpt |
|
157 |
48 156
|
ax-mp |
|
158 |
145 157
|
eqtri |
|
159 |
|
expcl |
|
160 |
27 48 159
|
sylancl |
|
161 |
160
|
mul02d |
|
162 |
158 161
|
syl5eq |
|
163 |
162
|
oveq2d |
|
164 |
16 17 36 39 44
|
isumcl |
|
165 |
27 164
|
mulcld |
|
166 |
165
|
addid1d |
|
167 |
163 166
|
eqtrd |
|
168 |
143 167
|
breqtrd |
|
169 |
16 17 129
|
serf |
|
170 |
169
|
ffvelrnda |
|
171 |
16 17 68
|
serf |
|
172 |
171
|
ffvelrnda |
|
173 |
|
simpr |
|
174 |
173 16
|
eleqtrdi |
|
175 |
|
simpl |
|
176 |
|
elfznn0 |
|
177 |
36 39
|
eqeltrd |
|
178 |
175 176 177
|
syl2an |
|
179 |
113
|
adantl |
|
180 |
|
fzfid |
|
181 |
1
|
adantr |
|
182 |
181 95 62
|
syl2an |
|
183 |
180 182
|
fsumcl |
|
184 |
183 29
|
mulcld |
|
185 |
179 184
|
eqeltrd |
|
186 |
175 176 185
|
syl2an |
|
187 |
|
eqidd |
|
188 |
|
simpr |
|
189 |
188 16
|
eleqtrdi |
|
190 |
|
elfznn0 |
|
191 |
181 190 62
|
syl2an |
|
192 |
187 189 191
|
fsumser |
|
193 |
|
fveq2 |
|
194 |
189 191 193
|
fsumm1 |
|
195 |
192 194
|
eqtr3d |
|
196 |
195
|
oveq1d |
|
197 |
183 25
|
pncan2d |
|
198 |
196 197
|
eqtr2d |
|
199 |
198
|
oveq1d |
|
200 |
38 183 29
|
subdird |
|
201 |
199 200
|
eqtrd |
|
202 |
36 179
|
oveq12d |
|
203 |
201 24 202
|
3eqtr4d |
|
204 |
175 176 203
|
syl2an |
|
205 |
174 178 186 204
|
sersub |
|
206 |
16 17 45 47 168 170 172 205
|
climsub |
|
207 |
|
1cnd |
|
208 |
207 27 164
|
subdird |
|
209 |
164
|
mulid2d |
|
210 |
209
|
oveq1d |
|
211 |
208 210
|
eqtrd |
|
212 |
206 211
|
breqtrrd |
|
213 |
16 17 24 30 212
|
isumclim |
|
214 |
15 213
|
eqtrd |
|