Step |
Hyp |
Ref |
Expression |
1 |
|
cpmadugsum.a |
|
2 |
|
cpmadugsum.b |
|
3 |
|
cpmadugsum.p |
|
4 |
|
cpmadugsum.y |
|
5 |
|
cpmadugsum.t |
|
6 |
|
cpmadugsum.x |
|
7 |
|
cpmadugsum.e |
|
8 |
|
cpmadugsum.m |
|
9 |
|
cpmadugsum.r |
|
10 |
|
cpmadugsum.1 |
|
11 |
|
cpmadugsum.g |
|
12 |
|
cpmadugsum.s |
|
13 |
|
nnnn0 |
|
14 |
1 2 3 4 5 6 7 8 9 10
|
cpmadugsumlemB |
|
15 |
13 14
|
sylanr1 |
|
16 |
1 2 3 4 5 6 7 8 9 10
|
cpmadugsumlemC |
|
17 |
13 16
|
sylanr1 |
|
18 |
15 17
|
oveq12d |
|
19 |
|
nncn |
|
20 |
|
npcan1 |
|
21 |
20
|
eqcomd |
|
22 |
19 21
|
syl |
|
23 |
22
|
oveq2d |
|
24 |
23
|
mpteq1d |
|
25 |
24
|
oveq2d |
|
26 |
25
|
ad2antrl |
|
27 |
|
eqid |
|
28 |
|
crngring |
|
29 |
28
|
anim2i |
|
30 |
29
|
3adant3 |
|
31 |
3 4
|
pmatring |
|
32 |
30 31
|
syl |
|
33 |
|
ringcmn |
|
34 |
32 33
|
syl |
|
35 |
34
|
adantr |
|
36 |
|
nnm1nn0 |
|
37 |
36
|
ad2antrl |
|
38 |
|
simpll1 |
|
39 |
28
|
3ad2ant2 |
|
40 |
39
|
adantr |
|
41 |
40
|
adantr |
|
42 |
|
elmapi |
|
43 |
23
|
feq2d |
|
44 |
42 43
|
syl5ibcom |
|
45 |
44
|
impcom |
|
46 |
45
|
adantl |
|
47 |
46
|
ffvelcdmda |
|
48 |
|
elfznn0 |
|
49 |
48
|
adantl |
|
50 |
|
1nn0 |
|
51 |
50
|
a1i |
|
52 |
49 51
|
nn0addcld |
|
53 |
1 2 5 3 4 27 8 7 6
|
mat2pmatscmxcl |
|
54 |
38 41 47 52 53
|
syl22anc |
|
55 |
27 11 35 37 54
|
gsummptfzsplit |
|
56 |
|
ringmnd |
|
57 |
32 56
|
syl |
|
58 |
57
|
adantr |
|
59 |
|
ovexd |
|
60 |
|
simpl1 |
|
61 |
|
nn0fz0 |
|
62 |
13 61
|
sylib |
|
63 |
|
ffvelcdm |
|
64 |
42 62 63
|
syl2anr |
|
65 |
13
|
adantr |
|
66 |
50
|
a1i |
|
67 |
65 66
|
nn0addcld |
|
68 |
64 67
|
jca |
|
69 |
68
|
adantl |
|
70 |
1 2 5 3 4 27 8 7 6
|
mat2pmatscmxcl |
|
71 |
60 40 69 70
|
syl21anc |
|
72 |
|
oveq1 |
|
73 |
72
|
oveq1d |
|
74 |
|
2fveq3 |
|
75 |
73 74
|
oveq12d |
|
76 |
19 20
|
syl |
|
77 |
76
|
oveq1d |
|
78 |
77
|
oveq1d |
|
79 |
76
|
fveq2d |
|
80 |
79
|
fveq2d |
|
81 |
78 80
|
oveq12d |
|
82 |
81
|
ad2antrl |
|
83 |
75 82
|
sylan9eqr |
|
84 |
27 58 59 71 83
|
gsumsnd |
|
85 |
84
|
oveq2d |
|
86 |
26 55 85
|
3eqtrd |
|
87 |
13
|
ad2antrl |
|
88 |
3 4
|
pmatlmod |
|
89 |
29 88
|
syl |
|
90 |
89
|
3adant3 |
|
91 |
90
|
adantr |
|
92 |
91
|
adantr |
|
93 |
|
eqid |
|
94 |
|
eqid |
|
95 |
93 94
|
mgpbas |
|
96 |
3
|
ply1ring |
|
97 |
28 96
|
syl |
|
98 |
97
|
3ad2ant2 |
|
99 |
93
|
ringmgp |
|
100 |
98 99
|
syl |
|
101 |
100
|
adantr |
|
102 |
101
|
adantr |
|
103 |
|
elfznn0 |
|
104 |
103
|
adantl |
|
105 |
6 3 94
|
vr1cl |
|
106 |
28 105
|
syl |
|
107 |
106
|
3ad2ant2 |
|
108 |
107
|
adantr |
|
109 |
108
|
adantr |
|
110 |
95 7 102 104 109
|
mulgnn0cld |
|
111 |
3
|
ply1crng |
|
112 |
111
|
anim2i |
|
113 |
112
|
3adant3 |
|
114 |
4
|
matsca2 |
|
115 |
113 114
|
syl |
|
116 |
115
|
eqcomd |
|
117 |
116
|
fveq2d |
|
118 |
117
|
eleq2d |
|
119 |
118
|
adantr |
|
120 |
119
|
adantr |
|
121 |
110 120
|
mpbird |
|
122 |
32
|
adantr |
|
123 |
122
|
adantr |
|
124 |
|
simpll1 |
|
125 |
40
|
adantr |
|
126 |
|
simpll3 |
|
127 |
5 1 2 3 4
|
mat2pmatbas |
|
128 |
124 125 126 127
|
syl3anc |
|
129 |
87
|
adantr |
|
130 |
|
simprr |
|
131 |
130
|
anim1i |
|
132 |
1 2 3 4 5
|
m2pmfzmap |
|
133 |
124 125 129 131 132
|
syl31anc |
|
134 |
27 9
|
ringcl |
|
135 |
123 128 133 134
|
syl3anc |
|
136 |
|
eqid |
|
137 |
|
eqid |
|
138 |
27 136 8 137
|
lmodvscl |
|
139 |
92 121 135 138
|
syl3anc |
|
140 |
27 11 35 87 139
|
gsummptfzsplitl |
|
141 |
|
0nn0 |
|
142 |
141
|
a1i |
|
143 |
|
eqid |
|
144 |
95 143 7
|
mulg0 |
|
145 |
107 144
|
syl |
|
146 |
145
|
adantr |
|
147 |
146
|
oveq1d |
|
148 |
|
eqid |
|
149 |
93 148
|
ringidval |
|
150 |
149
|
a1i |
|
151 |
150
|
eqcomd |
|
152 |
151
|
oveq1d |
|
153 |
115
|
adantr |
|
154 |
153
|
fveq2d |
|
155 |
154
|
oveq1d |
|
156 |
28 127
|
syl3an2 |
|
157 |
156
|
adantr |
|
158 |
|
simpl |
|
159 |
|
elnn0uz |
|
160 |
13 159
|
sylib |
|
161 |
|
eluzfz1 |
|
162 |
160 161
|
syl |
|
163 |
162
|
adantl |
|
164 |
158 163
|
ffvelcdmd |
|
165 |
164
|
ex |
|
166 |
42 165
|
syl |
|
167 |
166
|
impcom |
|
168 |
167
|
adantl |
|
169 |
5 1 2 3 4
|
mat2pmatbas |
|
170 |
60 40 168 169
|
syl3anc |
|
171 |
27 9
|
ringcl |
|
172 |
122 157 170 171
|
syl3anc |
|
173 |
|
eqid |
|
174 |
27 136 8 173
|
lmodvs1 |
|
175 |
91 172 174
|
syl2anc |
|
176 |
155 175
|
eqtrd |
|
177 |
147 152 176
|
3eqtrd |
|
178 |
177 172
|
eqeltrd |
|
179 |
|
oveq1 |
|
180 |
|
2fveq3 |
|
181 |
180
|
oveq2d |
|
182 |
179 181
|
oveq12d |
|
183 |
182
|
adantl |
|
184 |
27 58 142 178 183
|
gsumsnd |
|
185 |
95 149 7
|
mulg0 |
|
186 |
107 185
|
syl |
|
187 |
186
|
adantr |
|
188 |
187
|
oveq1d |
|
189 |
184 188 176
|
3eqtrd |
|
190 |
189
|
oveq2d |
|
191 |
140 190
|
eqtrd |
|
192 |
86 191
|
oveq12d |
|
193 |
|
fzfid |
|
194 |
|
simpll1 |
|
195 |
40
|
adantr |
|
196 |
42
|
adantl |
|
197 |
196
|
adantr |
|
198 |
|
nnz |
|
199 |
|
fzoval |
|
200 |
198 199
|
syl |
|
201 |
200
|
eqcomd |
|
202 |
201
|
eleq2d |
|
203 |
|
elfzofz |
|
204 |
202 203
|
biimtrdi |
|
205 |
204
|
adantr |
|
206 |
205
|
imp |
|
207 |
197 206
|
ffvelcdmd |
|
208 |
207
|
adantll |
|
209 |
|
elfznn0 |
|
210 |
209
|
adantl |
|
211 |
50
|
a1i |
|
212 |
210 211
|
nn0addcld |
|
213 |
194 195 208 212 53
|
syl22anc |
|
214 |
213
|
ralrimiva |
|
215 |
27 35 193 214
|
gsummptcl |
|
216 |
27 11
|
cmncom |
|
217 |
35 215 71 216
|
syl3anc |
|
218 |
217
|
oveq1d |
|
219 |
|
ringgrp |
|
220 |
32 219
|
syl |
|
221 |
220
|
adantr |
|
222 |
|
fzfid |
|
223 |
91
|
adantr |
|
224 |
101
|
adantr |
|
225 |
|
elfznn |
|
226 |
225
|
nnnn0d |
|
227 |
226
|
adantl |
|
228 |
108
|
adantr |
|
229 |
95 7 224 227 228
|
mulgnn0cld |
|
230 |
115
|
fveq2d |
|
231 |
230
|
adantr |
|
232 |
231
|
adantr |
|
233 |
229 232
|
eleqtrd |
|
234 |
122
|
adantr |
|
235 |
157
|
adantr |
|
236 |
|
simpll1 |
|
237 |
40
|
adantr |
|
238 |
196
|
adantl |
|
239 |
238
|
adantr |
|
240 |
|
1eluzge0 |
|
241 |
|
fzss1 |
|
242 |
240 241
|
mp1i |
|
243 |
242
|
sseld |
|
244 |
243
|
ad2antrl |
|
245 |
244
|
imp |
|
246 |
239 245
|
ffvelcdmd |
|
247 |
5 1 2 3 4
|
mat2pmatbas |
|
248 |
236 237 246 247
|
syl3anc |
|
249 |
234 235 248 134
|
syl3anc |
|
250 |
223 233 249 138
|
syl3anc |
|
251 |
250
|
ralrimiva |
|
252 |
27 35 222 251
|
gsummptcl |
|
253 |
27 11 12
|
grpaddsubass |
|
254 |
221 71 215 252 253
|
syl13anc |
|
255 |
|
oveq1 |
|
256 |
255
|
oveq1d |
|
257 |
256
|
oveq1d |
|
258 |
255
|
fveq2d |
|
259 |
258
|
fveq2d |
|
260 |
257 259
|
oveq12d |
|
261 |
260
|
cbvmptv |
|
262 |
225
|
nncnd |
|
263 |
262
|
adantl |
|
264 |
|
npcan1 |
|
265 |
263 264
|
syl |
|
266 |
265
|
oveq1d |
|
267 |
266
|
oveq1d |
|
268 |
267
|
mpteq2dva |
|
269 |
261 268
|
eqtrid |
|
270 |
269
|
oveq2d |
|
271 |
270
|
ad2antrl |
|
272 |
271
|
oveq1d |
|
273 |
|
eqid |
|
274 |
|
1zzd |
|
275 |
|
0zd |
|
276 |
37
|
nn0zd |
|
277 |
|
oveq1 |
|
278 |
277
|
oveq1d |
|
279 |
|
2fveq3 |
|
280 |
278 279
|
oveq12d |
|
281 |
27 273 35 274 275 276 213 280
|
gsummptshft |
|
282 |
|
0p1e1 |
|
283 |
282
|
a1i |
|
284 |
76
|
ad2antrl |
|
285 |
283 284
|
oveq12d |
|
286 |
285
|
mpteq1d |
|
287 |
286
|
oveq2d |
|
288 |
281 287
|
eqtrd |
|
289 |
288
|
oveq1d |
|
290 |
|
ringabl |
|
291 |
32 290
|
syl |
|
292 |
291
|
adantr |
|
293 |
225
|
adantl |
|
294 |
|
nnz |
|
295 |
|
elfzm1b |
|
296 |
294 198 295
|
syl2an |
|
297 |
200
|
adantl |
|
298 |
297
|
eqcomd |
|
299 |
298
|
eleq2d |
|
300 |
|
elfzofz |
|
301 |
299 300
|
biimtrdi |
|
302 |
296 301
|
sylbid |
|
303 |
302
|
expimpd |
|
304 |
293 303
|
mpcom |
|
305 |
304
|
ex |
|
306 |
305
|
ad2antrl |
|
307 |
306
|
imp |
|
308 |
239 307
|
ffvelcdmd |
|
309 |
1 2 5 3 4 27 8 7 6
|
mat2pmatscmxcl |
|
310 |
236 237 308 227 309
|
syl22anc |
|
311 |
|
eqid |
|
312 |
|
eqid |
|
313 |
27 12 292 222 310 250 311 312
|
gsummptfidmsub |
|
314 |
272 289 313
|
3eqtr4d |
|
315 |
314
|
oveq2d |
|
316 |
221
|
adantr |
|
317 |
27 12
|
grpsubcl |
|
318 |
316 310 250 317
|
syl3anc |
|
319 |
318
|
ralrimiva |
|
320 |
27 35 222 319
|
gsummptcl |
|
321 |
27 11
|
cmncom |
|
322 |
35 71 320 321
|
syl3anc |
|
323 |
254 315 322
|
3eqtrd |
|
324 |
323
|
oveq1d |
|
325 |
27 11
|
mndcl |
|
326 |
58 71 215 325
|
syl3anc |
|
327 |
27 11 12 292 326 252 172
|
ablsubsub4 |
|
328 |
27 11 12
|
grpaddsubass |
|
329 |
221 320 71 172 328
|
syl13anc |
|
330 |
324 327 329
|
3eqtr3d |
|
331 |
5 1 2 3 4
|
mat2pmatbas |
|
332 |
236 237 308 331
|
syl3anc |
|
333 |
27 8 136 137 12 223 233 332 249
|
lmodsubdi |
|
334 |
333
|
eqcomd |
|
335 |
334
|
mpteq2dva |
|
336 |
335
|
oveq2d |
|
337 |
336
|
oveq1d |
|
338 |
218 330 337
|
3eqtrd |
|
339 |
18 192 338
|
3eqtrd |
|