Step |
Hyp |
Ref |
Expression |
1 |
|
elrgspn.b |
|
2 |
|
elrgspn.m |
|
3 |
|
elrgspn.x |
|
4 |
|
elrgspn.n |
|
5 |
|
elrgspn.f |
|
6 |
|
elrgspn.r |
|
7 |
|
elrgspn.a |
|
8 |
|
elrgspnlem1.1 |
|
9 |
1 2 3 4 5 6 7 8
|
elrgspnlem1 |
|
10 |
|
eqeq2 |
|
11 |
|
eqeq2 |
|
12 |
|
simpr |
|
13 |
12
|
fveq2d |
|
14 |
|
eqid |
|
15 |
|
simpr |
|
16 |
15
|
iftrued |
|
17 |
|
wrd0 |
|
18 |
17
|
a1i |
|
19 |
|
1zzd |
|
20 |
14 16 18 19
|
fvmptd2 |
|
21 |
20
|
ad2antrr |
|
22 |
13 21
|
eqtrd |
|
23 |
12
|
oveq2d |
|
24 |
|
eqid |
|
25 |
2 24
|
ringidval |
|
26 |
25
|
gsum0 |
|
27 |
23 26
|
eqtrdi |
|
28 |
22 27
|
oveq12d |
|
29 |
1 24
|
ringidcl |
|
30 |
6 29
|
syl |
|
31 |
1 3
|
mulg1 |
|
32 |
30 31
|
syl |
|
33 |
32
|
ad2antrr |
|
34 |
28 33
|
eqtrd |
|
35 |
|
eqeq1 |
|
36 |
35
|
notbid |
|
37 |
36
|
biimparc |
|
38 |
37
|
adantll |
|
39 |
38
|
iffalsed |
|
40 |
|
simplr |
|
41 |
|
0zd |
|
42 |
14 39 40 41
|
fvmptd2 |
|
43 |
42
|
oveq1d |
|
44 |
2
|
ringmgp |
|
45 |
6 44
|
syl |
|
46 |
|
sswrd |
|
47 |
7 46
|
syl |
|
48 |
47
|
sselda |
|
49 |
2 1
|
mgpbas |
|
50 |
49
|
gsumwcl |
|
51 |
45 48 50
|
syl2an2r |
|
52 |
|
eqid |
|
53 |
1 52 3
|
mulg0 |
|
54 |
51 53
|
syl |
|
55 |
54
|
adantr |
|
56 |
43 55
|
eqtrd |
|
57 |
10 11 34 56
|
ifbothda |
|
58 |
57
|
mpteq2dva |
|
59 |
58
|
oveq2d |
|
60 |
6
|
ringcmnd |
|
61 |
60
|
cmnmndd |
|
62 |
1
|
fvexi |
|
63 |
62
|
a1i |
|
64 |
63 7
|
ssexd |
|
65 |
|
wrdexg |
|
66 |
64 65
|
syl |
|
67 |
|
eqid |
|
68 |
30 1
|
eleqtrdi |
|
69 |
52 61 66 18 67 68
|
gsummptif1n0 |
|
70 |
59 69
|
eqtrd |
|
71 |
|
eqid |
|
72 |
|
fveq1 |
|
73 |
72
|
oveq1d |
|
74 |
73
|
mpteq2dv |
|
75 |
74
|
oveq2d |
|
76 |
75
|
eqeq2d |
|
77 |
|
breq1 |
|
78 |
|
zex |
|
79 |
78
|
a1i |
|
80 |
|
1zzd |
|
81 |
|
0zd |
|
82 |
80 81
|
ifclda |
|
83 |
82
|
fmpttd |
|
84 |
79 66 83
|
elmapdd |
|
85 |
66
|
mptexd |
|
86 |
83
|
ffund |
|
87 |
|
0zd |
|
88 |
|
snfi |
|
89 |
88
|
a1i |
|
90 |
|
eldifsni |
|
91 |
90
|
adantl |
|
92 |
91
|
neneqd |
|
93 |
92
|
iffalsed |
|
94 |
93 66
|
suppss2 |
|
95 |
|
suppssfifsupp |
|
96 |
85 86 87 89 94 95
|
syl32anc |
|
97 |
77 84 96
|
elrabd |
|
98 |
97 5
|
eleqtrrdi |
|
99 |
|
eqidd |
|
100 |
76 98 99
|
rspcedvdw |
|
101 |
|
ovexd |
|
102 |
71 100 101
|
elrnmptd |
|
103 |
102 8
|
eleqtrrdi |
|
104 |
70 103
|
eqeltrrd |
|
105 |
|
simpllr |
|
106 |
|
simpr |
|
107 |
105 106
|
oveq12d |
|
108 |
|
eqid |
|
109 |
66
|
ad2antrr |
|
110 |
6
|
ad2antrr |
|
111 |
6
|
ringgrpd |
|
112 |
111
|
ad2antrr |
|
113 |
5
|
ssrab3 |
|
114 |
113
|
a1i |
|
115 |
114
|
sselda |
|
116 |
79 66
|
elmapd |
|
117 |
116
|
adantr |
|
118 |
115 117
|
mpbid |
|
119 |
118
|
ffvelcdmda |
|
120 |
51
|
adantlr |
|
121 |
1 3 112 119 120
|
mulgcld |
|
122 |
121
|
adantlr |
|
123 |
122
|
ralrimiva |
|
124 |
|
fveq2 |
|
125 |
|
oveq2 |
|
126 |
124 125
|
oveq12d |
|
127 |
126
|
eleq1d |
|
128 |
127
|
cbvralvw |
|
129 |
123 128
|
sylibr |
|
130 |
129
|
r19.21bi |
|
131 |
111
|
ad2antrr |
|
132 |
|
breq1 |
|
133 |
132 5
|
elrab2 |
|
134 |
133
|
simplbi |
|
135 |
134
|
adantl |
|
136 |
79 66
|
elmapd |
|
137 |
136
|
adantr |
|
138 |
135 137
|
mpbid |
|
139 |
138
|
ffvelcdmda |
|
140 |
51
|
adantlr |
|
141 |
1 3 131 139 140
|
mulgcld |
|
142 |
141
|
adantllr |
|
143 |
142
|
ralrimiva |
|
144 |
|
fveq2 |
|
145 |
|
oveq2 |
|
146 |
144 145
|
oveq12d |
|
147 |
146
|
eleq1d |
|
148 |
147
|
cbvralvw |
|
149 |
143 148
|
sylibr |
|
150 |
149
|
r19.21bi |
|
151 |
126
|
cbvmptv |
|
152 |
|
fvexd |
|
153 |
|
0zd |
|
154 |
66
|
adantr |
|
155 |
|
ssidd |
|
156 |
|
breq1 |
|
157 |
156 5
|
elrab2 |
|
158 |
157
|
simprbi |
|
159 |
158
|
adantl |
|
160 |
1 52 3
|
mulg0 |
|
161 |
160
|
adantl |
|
162 |
152 153 154 155 120 118 159 161
|
fisuppov1 |
|
163 |
162
|
adantr |
|
164 |
151 163
|
eqbrtrid |
|
165 |
146
|
cbvmptv |
|
166 |
162
|
ralrimiva |
|
167 |
|
fveq1 |
|
168 |
167
|
oveq1d |
|
169 |
168
|
mpteq2dv |
|
170 |
169
|
breq1d |
|
171 |
170
|
cbvralvw |
|
172 |
166 171
|
sylib |
|
173 |
172
|
r19.21bi |
|
174 |
173
|
adantlr |
|
175 |
165 174
|
eqbrtrid |
|
176 |
1 108 52 109 109 110 130 150 164 175
|
gsumdixp |
|
177 |
151
|
oveq2i |
|
178 |
165
|
oveq2i |
|
179 |
177 178
|
oveq12i |
|
180 |
179
|
a1i |
|
181 |
110
|
ad2antrr |
|
182 |
122
|
adantr |
|
183 |
111
|
ad4antr |
|
184 |
138
|
adantlr |
|
185 |
184
|
ffvelcdmda |
|
186 |
185
|
adantlr |
|
187 |
45
|
ad4antr |
|
188 |
47
|
adantr |
|
189 |
188
|
ad2antrr |
|
190 |
189
|
sselda |
|
191 |
49
|
gsumwcl |
|
192 |
187 190 191
|
syl2anc |
|
193 |
1 3 183 186 192
|
mulgcld |
|
194 |
1 108 181 182 193
|
ringcld |
|
195 |
194
|
anasss |
|
196 |
195
|
ralrimivva |
|
197 |
|
eqid |
|
198 |
197
|
fmpo |
|
199 |
196 198
|
sylib |
|
200 |
|
vex |
|
201 |
|
vex |
|
202 |
200 201
|
op1std |
|
203 |
202
|
fveq2d |
|
204 |
202
|
oveq2d |
|
205 |
203 204
|
oveq12d |
|
206 |
200 201
|
op2ndd |
|
207 |
206
|
fveq2d |
|
208 |
206
|
oveq2d |
|
209 |
207 208
|
oveq12d |
|
210 |
205 209
|
oveq12d |
|
211 |
210
|
mpompt |
|
212 |
66 66
|
xpexd |
|
213 |
212
|
ad2antrr |
|
214 |
213
|
mptexd |
|
215 |
|
fvexd |
|
216 |
110
|
adantr |
|
217 |
111
|
ad3antrrr |
|
218 |
118
|
ad2antrr |
|
219 |
|
xp1st |
|
220 |
219
|
adantl |
|
221 |
218 220
|
ffvelcdmd |
|
222 |
216 44
|
syl |
|
223 |
188
|
ad2antrr |
|
224 |
223 220
|
sseldd |
|
225 |
49
|
gsumwcl |
|
226 |
222 224 225
|
syl2anc |
|
227 |
1 3 217 221 226
|
mulgcld |
|
228 |
184
|
adantr |
|
229 |
|
xp2nd |
|
230 |
229
|
adantl |
|
231 |
228 230
|
ffvelcdmd |
|
232 |
223 230
|
sseldd |
|
233 |
49
|
gsumwcl |
|
234 |
222 232 233
|
syl2anc |
|
235 |
1 3 217 231 234
|
mulgcld |
|
236 |
1 108 216 227 235
|
ringcld |
|
237 |
236
|
fmpttd |
|
238 |
237
|
ffund |
|
239 |
159
|
fsuppimpd |
|
240 |
133
|
simprbi |
|
241 |
240
|
adantl |
|
242 |
241
|
fsuppimpd |
|
243 |
|
xpfi |
|
244 |
239 242 243
|
syl2an2r |
|
245 |
118
|
ffnd |
|
246 |
245
|
adantr |
|
247 |
246
|
ad2antrr |
|
248 |
109
|
ad2antrr |
|
249 |
|
0zd |
|
250 |
|
xp1st |
|
251 |
250
|
adantl |
|
252 |
247 248 249 251
|
fvdifsupp |
|
253 |
252
|
oveq1d |
|
254 |
45
|
ad4antr |
|
255 |
188
|
ad3antrrr |
|
256 |
251
|
eldifad |
|
257 |
255 256
|
sseldd |
|
258 |
254 257 225
|
syl2anc |
|
259 |
1 52 3
|
mulg0 |
|
260 |
258 259
|
syl |
|
261 |
253 260
|
eqtrd |
|
262 |
261
|
oveq1d |
|
263 |
110
|
ad2antrr |
|
264 |
111
|
ad4antr |
|
265 |
184
|
ad2antrr |
|
266 |
|
xp2nd |
|
267 |
266
|
adantl |
|
268 |
265 267
|
ffvelcdmd |
|
269 |
255 267
|
sseldd |
|
270 |
254 269 233
|
syl2anc |
|
271 |
1 3 264 268 270
|
mulgcld |
|
272 |
1 108 52 263 271
|
ringlzd |
|
273 |
262 272
|
eqtrd |
|
274 |
138
|
ffnd |
|
275 |
274
|
adantlr |
|
276 |
275
|
ad2antrr |
|
277 |
109
|
ad2antrr |
|
278 |
|
0zd |
|
279 |
|
xp2nd |
|
280 |
279
|
adantl |
|
281 |
276 277 278 280
|
fvdifsupp |
|
282 |
281
|
oveq1d |
|
283 |
45
|
ad4antr |
|
284 |
188
|
ad3antrrr |
|
285 |
280
|
eldifad |
|
286 |
284 285
|
sseldd |
|
287 |
283 286 233
|
syl2anc |
|
288 |
1 52 3
|
mulg0 |
|
289 |
287 288
|
syl |
|
290 |
282 289
|
eqtrd |
|
291 |
290
|
oveq2d |
|
292 |
110
|
ad2antrr |
|
293 |
111
|
ad4antr |
|
294 |
118
|
adantr |
|
295 |
294
|
ad2antrr |
|
296 |
|
xp1st |
|
297 |
296
|
adantl |
|
298 |
295 297
|
ffvelcdmd |
|
299 |
284 297
|
sseldd |
|
300 |
283 299 225
|
syl2anc |
|
301 |
1 3 293 298 300
|
mulgcld |
|
302 |
1 108 52 292 301
|
ringrzd |
|
303 |
291 302
|
eqtrd |
|
304 |
|
simpr |
|
305 |
|
difxp |
|
306 |
304 305
|
eleqtrdi |
|
307 |
|
elun |
|
308 |
306 307
|
sylib |
|
309 |
273 303 308
|
mpjaodan |
|
310 |
309 213
|
suppss2 |
|
311 |
244 310
|
ssfid |
|
312 |
214 215 238 311
|
isfsuppd |
|
313 |
211 312
|
eqbrtrrid |
|
314 |
60
|
ad2antrr |
|
315 |
7
|
ad2antrr |
|
316 |
1 52 199 313 314 315
|
gsumwrd2dccat |
|
317 |
126
|
oveq1d |
|
318 |
|
fveq2 |
|
319 |
|
oveq2 |
|
320 |
318 319
|
oveq12d |
|
321 |
320
|
oveq2d |
|
322 |
317 321
|
cbvmpov |
|
323 |
322
|
oveq2i |
|
324 |
323
|
a1i |
|
325 |
|
pfxcctswrd |
|
326 |
325
|
adantll |
|
327 |
326
|
oveq2d |
|
328 |
327
|
oveq2d |
|
329 |
328
|
mpteq2dva |
|
330 |
329
|
oveq2d |
|
331 |
|
df-ov |
|
332 |
188
|
sselda |
|
333 |
332
|
ad4ant13 |
|
334 |
187 333 50
|
syl2anc |
|
335 |
1 3 108
|
mulgass3 |
|
336 |
181 186 334 192 335
|
syl13anc |
|
337 |
336
|
oveq2d |
|
338 |
119
|
ad4ant13 |
|
339 |
1 3 108
|
mulgass2 |
|
340 |
181 338 334 193 339
|
syl13anc |
|
341 |
1 108 181 334 192
|
ringcld |
|
342 |
1 3
|
mulgass |
|
343 |
183 338 186 341 342
|
syl13anc |
|
344 |
337 340 343
|
3eqtr4d |
|
345 |
2 108
|
mgpplusg |
|
346 |
49 345
|
gsumccat |
|
347 |
187 333 190 346
|
syl3anc |
|
348 |
347
|
oveq2d |
|
349 |
344 348
|
eqtr4d |
|
350 |
349
|
adantllr |
|
351 |
350
|
adantllr |
|
352 |
351
|
3impa |
|
353 |
352
|
mpoeq3dva |
|
354 |
|
fveq2 |
|
355 |
|
fveq2 |
|
356 |
354 355
|
oveqan12d |
|
357 |
|
oveq12 |
|
358 |
357
|
oveq2d |
|
359 |
356 358
|
oveq12d |
|
360 |
359
|
adantl |
|
361 |
|
pfxcl |
|
362 |
361
|
ad2antlr |
|
363 |
|
swrdcl |
|
364 |
363
|
ad2antlr |
|
365 |
|
ovexd |
|
366 |
353 360 362 364 365
|
ovmpod |
|
367 |
331 366
|
eqtr3id |
|
368 |
367
|
mpteq2dva |
|
369 |
368
|
oveq2d |
|
370 |
|
eqid |
|
371 |
|
fveq2 |
|
372 |
371
|
oveq2d |
|
373 |
|
fvoveq1 |
|
374 |
|
id |
|
375 |
371
|
opeq2d |
|
376 |
374 375
|
oveq12d |
|
377 |
376
|
fveq2d |
|
378 |
373 377
|
oveq12d |
|
379 |
378
|
adantr |
|
380 |
372 379
|
sumeq12dv |
|
381 |
|
simpr |
|
382 |
|
fzfid |
|
383 |
294
|
ad2antrr |
|
384 |
383 362
|
ffvelcdmd |
|
385 |
184
|
ad2antrr |
|
386 |
385 364
|
ffvelcdmd |
|
387 |
384 386
|
zmulcld |
|
388 |
387
|
zcnd |
|
389 |
382 388
|
fsumcl |
|
390 |
370 380 381 389
|
fvmptd3 |
|
391 |
390
|
oveq1d |
|
392 |
111
|
ad3antrrr |
|
393 |
45
|
ad3antrrr |
|
394 |
315 46
|
syl |
|
395 |
394
|
sselda |
|
396 |
49
|
gsumwcl |
|
397 |
393 395 396
|
syl2anc |
|
398 |
1 3 392 382 397 387
|
gsummulgc2 |
|
399 |
391 398
|
eqtr4d |
|
400 |
330 369 399
|
3eqtr4rd |
|
401 |
400
|
mpteq2dva |
|
402 |
401
|
oveq2d |
|
403 |
316 324 402
|
3eqtr4d |
|
404 |
176 180 403
|
3eqtr3d |
|
405 |
|
fveq1 |
|
406 |
405
|
oveq1d |
|
407 |
406
|
mpteq2dv |
|
408 |
407
|
oveq2d |
|
409 |
408
|
cbvmptv |
|
410 |
|
fveq1 |
|
411 |
410
|
oveq1d |
|
412 |
411
|
mpteq2dv |
|
413 |
412
|
oveq2d |
|
414 |
413
|
eqeq2d |
|
415 |
|
breq1 |
|
416 |
78
|
a1i |
|
417 |
|
fzfid |
|
418 |
294
|
ad2antrr |
|
419 |
|
pfxcl |
|
420 |
419
|
ad2antlr |
|
421 |
418 420
|
ffvelcdmd |
|
422 |
184
|
ad2antrr |
|
423 |
|
swrdcl |
|
424 |
423
|
ad2antlr |
|
425 |
422 424
|
ffvelcdmd |
|
426 |
421 425
|
zmulcld |
|
427 |
417 426
|
fsumzcl |
|
428 |
427
|
fmpttd |
|
429 |
416 109 428
|
elmapdd |
|
430 |
|
0zd |
|
431 |
428
|
ffund |
|
432 |
|
ccatfn |
|
433 |
|
fnfun |
|
434 |
432 433
|
ax-mp |
|
435 |
|
imafi |
|
436 |
434 244 435
|
sylancr |
|
437 |
|
fveq2 |
|
438 |
437
|
oveq2d |
|
439 |
|
fvoveq1 |
|
440 |
|
id |
|
441 |
437
|
opeq2d |
|
442 |
440 441
|
oveq12d |
|
443 |
442
|
fveq2d |
|
444 |
439 443
|
oveq12d |
|
445 |
444
|
adantr |
|
446 |
438 445
|
sumeq12dv |
|
447 |
|
oveq1 |
|
448 |
447
|
eqeq2d |
|
449 |
|
oveq2 |
|
450 |
449
|
eqeq2d |
|
451 |
246
|
ad4antr |
|
452 |
109
|
ad4antr |
|
453 |
|
0zd |
|
454 |
|
simpr |
|
455 |
454
|
eldifad |
|
456 |
455
|
adantr |
|
457 |
|
pfxcl |
|
458 |
456 457
|
syl |
|
459 |
458
|
ad2antrr |
|
460 |
|
simplr |
|
461 |
451 452 453 459 460
|
elsuppfnd |
|
462 |
275
|
ad4antr |
|
463 |
|
swrdcl |
|
464 |
456 463
|
syl |
|
465 |
464
|
ad2antrr |
|
466 |
|
simpr |
|
467 |
462 452 453 465 466
|
elsuppfnd |
|
468 |
456
|
ad2antrr |
|
469 |
|
simpllr |
|
470 |
|
pfxcctswrd |
|
471 |
468 469 470
|
syl2anc |
|
472 |
471
|
eqcomd |
|
473 |
448 450 461 467 472
|
2rspcedvdw |
|
474 |
|
fnov |
|
475 |
432 474
|
mpbi |
|
476 |
200
|
a1i |
|
477 |
|
ssv |
|
478 |
477
|
a1i |
|
479 |
|
ssv |
|
480 |
479
|
a1i |
|
481 |
475 476 478 480
|
elimampo |
|
482 |
481
|
mptru |
|
483 |
473 482
|
sylibr |
|
484 |
483
|
anasss |
|
485 |
454
|
ad3antrrr |
|
486 |
485
|
eldifbd |
|
487 |
486
|
anasss |
|
488 |
484 487
|
pm2.65da |
|
489 |
|
df-ne |
|
490 |
|
df-ne |
|
491 |
489 490
|
anbi12i |
|
492 |
491
|
notbii |
|
493 |
|
pm4.57 |
|
494 |
492 493
|
bitr2i |
|
495 |
488 494
|
sylibr |
|
496 |
294
|
ad2antrr |
|
497 |
496 458
|
ffvelcdmd |
|
498 |
497
|
zcnd |
|
499 |
184
|
ad2antrr |
|
500 |
499 464
|
ffvelcdmd |
|
501 |
500
|
zcnd |
|
502 |
498 501
|
mul0ord |
|
503 |
495 502
|
mpbird |
|
504 |
503
|
sumeq2dv |
|
505 |
|
fzssuz |
|
506 |
|
sumz |
|
507 |
506
|
orcs |
|
508 |
505 507
|
mp1i |
|
509 |
504 508
|
eqtrd |
|
510 |
446 509
|
sylan9eqr |
|
511 |
|
0zd |
|
512 |
370 510 455 511
|
fvmptd2 |
|
513 |
428 512
|
suppss |
|
514 |
436 513
|
ssfid |
|
515 |
429 430 431 514
|
isfsuppd |
|
516 |
415 429 515
|
elrabd |
|
517 |
516 5
|
eleqtrrdi |
|
518 |
|
fveq2 |
|
519 |
518 145
|
oveq12d |
|
520 |
519
|
cbvmptv |
|
521 |
520
|
oveq2i |
|
522 |
521
|
a1i |
|
523 |
414 517 522
|
rspcedvdw |
|
524 |
|
ovexd |
|
525 |
409 523 524
|
elrnmptd |
|
526 |
525 8
|
eleqtrrdi |
|
527 |
404 526
|
eqeltrd |
|
528 |
527
|
adantllr |
|
529 |
528
|
adantllr |
|
530 |
529
|
adantlr |
|
531 |
530
|
adantr |
|
532 |
107 531
|
eqeltrd |
|
533 |
8
|
eleq2i |
|
534 |
169
|
oveq2d |
|
535 |
534
|
cbvmptv |
|
536 |
535
|
elrnmpt |
|
537 |
536
|
elv |
|
538 |
533 537
|
sylbb |
|
539 |
538
|
adantl |
|
540 |
539
|
ad2antrr |
|
541 |
532 540
|
r19.29a |
|
542 |
8
|
eleq2i |
|
543 |
71
|
elrnmpt |
|
544 |
543
|
elv |
|
545 |
542 544
|
sylbb |
|
546 |
545
|
ad2antlr |
|
547 |
541 546
|
r19.29a |
|
548 |
547
|
anasss |
|
549 |
548
|
ralrimivva |
|
550 |
1 24 108
|
issubrg2 |
|
551 |
550
|
biimpar |
|
552 |
6 9 104 549 551
|
syl13anc |
|