Step |
Hyp |
Ref |
Expression |
1 |
|
dvnprodlem3.s |
|
2 |
|
dvnprodlem3.x |
|
3 |
|
dvnprodlem3.t |
|
4 |
|
dvnprodlem3.h |
|
5 |
|
dvnprodlem3.n |
|
6 |
|
dvnprodlem3.dvnh |
|
7 |
|
dvnprodlem3.f |
|
8 |
|
dvnprodlem3.d |
|
9 |
|
dvnprodlem3.c |
|
10 |
|
prodeq1 |
|
11 |
10
|
mpteq2dv |
|
12 |
11
|
oveq2d |
|
13 |
12
|
fveq1d |
|
14 |
|
fveq2 |
|
15 |
14
|
fveq1d |
|
16 |
15
|
sumeq1d |
|
17 |
|
prodeq1 |
|
18 |
17
|
oveq2d |
|
19 |
|
prodeq1 |
|
20 |
18 19
|
oveq12d |
|
21 |
20
|
sumeq2sdv |
|
22 |
16 21
|
eqtrd |
|
23 |
22
|
mpteq2dv |
|
24 |
13 23
|
eqeq12d |
|
25 |
24
|
ralbidv |
|
26 |
|
prodeq1 |
|
27 |
26
|
mpteq2dv |
|
28 |
27
|
oveq2d |
|
29 |
28
|
fveq1d |
|
30 |
|
fveq2 |
|
31 |
30
|
fveq1d |
|
32 |
31
|
sumeq1d |
|
33 |
|
prodeq1 |
|
34 |
33
|
oveq2d |
|
35 |
|
prodeq1 |
|
36 |
34 35
|
oveq12d |
|
37 |
36
|
sumeq2sdv |
|
38 |
32 37
|
eqtrd |
|
39 |
38
|
mpteq2dv |
|
40 |
29 39
|
eqeq12d |
|
41 |
40
|
ralbidv |
|
42 |
|
prodeq1 |
|
43 |
42
|
mpteq2dv |
|
44 |
43
|
oveq2d |
|
45 |
44
|
fveq1d |
|
46 |
|
fveq2 |
|
47 |
46
|
fveq1d |
|
48 |
47
|
sumeq1d |
|
49 |
|
prodeq1 |
|
50 |
49
|
oveq2d |
|
51 |
|
prodeq1 |
|
52 |
50 51
|
oveq12d |
|
53 |
52
|
sumeq2sdv |
|
54 |
48 53
|
eqtrd |
|
55 |
54
|
mpteq2dv |
|
56 |
45 55
|
eqeq12d |
|
57 |
56
|
ralbidv |
|
58 |
|
prodeq1 |
|
59 |
58
|
mpteq2dv |
|
60 |
7
|
a1i |
|
61 |
60
|
eqcomd |
|
62 |
59 61
|
eqtrd |
|
63 |
62
|
oveq2d |
|
64 |
63
|
fveq1d |
|
65 |
|
fveq2 |
|
66 |
65
|
fveq1d |
|
67 |
66
|
sumeq1d |
|
68 |
|
prodeq1 |
|
69 |
68
|
oveq2d |
|
70 |
|
prodeq1 |
|
71 |
69 70
|
oveq12d |
|
72 |
71
|
sumeq2sdv |
|
73 |
67 72
|
eqtrd |
|
74 |
73
|
mpteq2dv |
|
75 |
64 74
|
eqeq12d |
|
76 |
75
|
ralbidv |
|
77 |
|
prod0 |
|
78 |
77
|
mpteq2i |
|
79 |
78
|
oveq2i |
|
80 |
79
|
a1i |
|
81 |
|
id |
|
82 |
80 81
|
fveq12d |
|
83 |
82
|
adantl |
|
84 |
|
recnprss |
|
85 |
1 84
|
syl |
|
86 |
|
1cnd |
|
87 |
86
|
fmpttd |
|
88 |
|
1re |
|
89 |
88
|
rgenw |
|
90 |
|
dmmptg |
|
91 |
89 90
|
ax-mp |
|
92 |
91
|
a1i |
|
93 |
92
|
feq2d |
|
94 |
87 93
|
mpbird |
|
95 |
|
restsspw |
|
96 |
95 2
|
sseldi |
|
97 |
|
elpwi |
|
98 |
96 97
|
syl |
|
99 |
92 98
|
eqsstrd |
|
100 |
94 99
|
jca |
|
101 |
|
cnex |
|
102 |
101
|
a1i |
|
103 |
|
elpm2g |
|
104 |
102 1 103
|
syl2anc |
|
105 |
100 104
|
mpbird |
|
106 |
|
dvn0 |
|
107 |
85 105 106
|
syl2anc |
|
108 |
107
|
adantr |
|
109 |
|
fveq2 |
|
110 |
109
|
adantl |
|
111 |
|
oveq2 |
|
112 |
|
elmapfn |
|
113 |
|
fn0 |
|
114 |
112 113
|
sylib |
|
115 |
|
velsn |
|
116 |
114 115
|
sylibr |
|
117 |
115
|
biimpi |
|
118 |
|
id |
|
119 |
|
f0 |
|
120 |
|
ovex |
|
121 |
|
0ex |
|
122 |
120 121
|
elmap |
|
123 |
119 122
|
mpbir |
|
124 |
123
|
a1i |
|
125 |
118 124
|
eqeltrd |
|
126 |
117 125
|
syl |
|
127 |
116 126
|
impbii |
|
128 |
127
|
ax-gen |
|
129 |
|
dfcleq |
|
130 |
128 129
|
mpbir |
|
131 |
130
|
a1i |
|
132 |
111 131
|
eqtrd |
|
133 |
|
rabeq |
|
134 |
132 133
|
syl |
|
135 |
|
sumeq1 |
|
136 |
135
|
eqeq1d |
|
137 |
136
|
rabbidv |
|
138 |
134 137
|
eqtrd |
|
139 |
138
|
mpteq2dv |
|
140 |
|
0elpw |
|
141 |
140
|
a1i |
|
142 |
|
nn0ex |
|
143 |
142
|
mptex |
|
144 |
143
|
a1i |
|
145 |
8 139 141 144
|
fvmptd3 |
|
146 |
|
eqeq2 |
|
147 |
146
|
rabbidv |
|
148 |
147
|
adantl |
|
149 |
|
0nn0 |
|
150 |
149
|
a1i |
|
151 |
|
p0ex |
|
152 |
151
|
rabex |
|
153 |
152
|
a1i |
|
154 |
145 148 150 153
|
fvmptd |
|
155 |
154
|
adantr |
|
156 |
|
snidg |
|
157 |
121 156
|
ax-mp |
|
158 |
|
eqid |
|
159 |
157 158
|
pm3.2i |
|
160 |
|
sum0 |
|
161 |
160
|
a1i |
|
162 |
161
|
eqeq1d |
|
163 |
162
|
elrab |
|
164 |
159 163
|
mpbir |
|
165 |
164
|
n0ii |
|
166 |
|
eqid |
|
167 |
|
rabrsn |
|
168 |
166 167
|
ax-mp |
|
169 |
165 168
|
mtpor |
|
170 |
169
|
a1i |
|
171 |
|
iftrue |
|
172 |
171
|
adantl |
|
173 |
170 172
|
eqtr4d |
|
174 |
110 155 173
|
3eqtrd |
|
175 |
174 172
|
eqtrd |
|
176 |
175
|
sumeq1d |
|
177 |
|
fveq2 |
|
178 |
|
fac0 |
|
179 |
178
|
a1i |
|
180 |
177 179
|
eqtrd |
|
181 |
180
|
oveq1d |
|
182 |
|
prod0 |
|
183 |
182
|
oveq2i |
|
184 |
|
1div1e1 |
|
185 |
183 184
|
eqtri |
|
186 |
181 185
|
eqtrdi |
|
187 |
|
prod0 |
|
188 |
187
|
a1i |
|
189 |
186 188
|
oveq12d |
|
190 |
189
|
ad2antlr |
|
191 |
|
1t1e1 |
|
192 |
191
|
a1i |
|
193 |
190 192
|
eqtrd |
|
194 |
193
|
sumeq2dv |
|
195 |
|
ax-1cn |
|
196 |
|
eqidd |
|
197 |
196
|
sumsn |
|
198 |
121 195 197
|
mp2an |
|
199 |
198
|
a1i |
|
200 |
176 194 199
|
3eqtrd |
|
201 |
200
|
mpteq2dv |
|
202 |
201
|
eqcomd |
|
203 |
83 108 202
|
3eqtrd |
|
204 |
203
|
a1d |
|
205 |
79
|
fveq1i |
|
206 |
205
|
a1i |
|
207 |
1
|
adantr |
|
208 |
207
|
adantr |
|
209 |
2
|
adantr |
|
210 |
209
|
adantr |
|
211 |
195
|
a1i |
|
212 |
|
elfznn0 |
|
213 |
212
|
adantl |
|
214 |
|
neqne |
|
215 |
214
|
adantr |
|
216 |
213 215
|
jca |
|
217 |
|
elnnne0 |
|
218 |
216 217
|
sylibr |
|
219 |
218
|
adantll |
|
220 |
208 210 211 219
|
dvnmptconst |
|
221 |
145
|
ad2antrr |
|
222 |
|
eqeq2 |
|
223 |
222
|
rabbidv |
|
224 |
223
|
adantl |
|
225 |
|
eqidd |
|
226 |
|
id |
|
227 |
226
|
eqcomd |
|
228 |
160
|
a1i |
|
229 |
225 227 228
|
3eqtrd |
|
230 |
229
|
adantl |
|
231 |
230
|
adantll |
|
232 |
|
simpll |
|
233 |
231 232
|
pm2.65da |
|
234 |
233
|
ralrimiva |
|
235 |
|
rabeq0 |
|
236 |
234 235
|
sylibr |
|
237 |
236
|
adantr |
|
238 |
224 237
|
eqtrd |
|
239 |
238
|
adantll |
|
240 |
239
|
adantlr |
|
241 |
212
|
adantl |
|
242 |
121
|
a1i |
|
243 |
221 240 241 242
|
fvmptd |
|
244 |
243
|
sumeq1d |
|
245 |
|
sum0 |
|
246 |
245
|
a1i |
|
247 |
244 246
|
eqtr2d |
|
248 |
247
|
mpteq2dv |
|
249 |
206 220 248
|
3eqtrd |
|
250 |
249
|
ex |
|
251 |
204 250
|
pm2.61dan |
|
252 |
251
|
ralrimiv |
|
253 |
|
simpll |
|
254 |
|
fveq2 |
|
255 |
254
|
prodeq2ad |
|
256 |
|
fveq2 |
|
257 |
256
|
fveq1d |
|
258 |
257
|
cbvprodv |
|
259 |
258
|
a1i |
|
260 |
255 259
|
eqtrd |
|
261 |
260
|
cbvmptv |
|
262 |
261
|
oveq2i |
|
263 |
262
|
fveq1i |
|
264 |
|
fveq2 |
|
265 |
264
|
fveq2d |
|
266 |
265
|
cbvprodv |
|
267 |
266
|
oveq2i |
|
268 |
267
|
a1i |
|
269 |
|
fveq2 |
|
270 |
269
|
prodeq2ad |
|
271 |
256
|
oveq2d |
|
272 |
271 264
|
fveq12d |
|
273 |
272
|
fveq1d |
|
274 |
273
|
cbvprodv |
|
275 |
274
|
a1i |
|
276 |
270 275
|
eqtrd |
|
277 |
268 276
|
oveq12d |
|
278 |
277
|
sumeq2sdv |
|
279 |
|
fveq1 |
|
280 |
279
|
fveq2d |
|
281 |
280
|
prodeq2ad |
|
282 |
281
|
oveq2d |
|
283 |
279
|
fveq2d |
|
284 |
283
|
fveq1d |
|
285 |
284
|
prodeq2ad |
|
286 |
282 285
|
oveq12d |
|
287 |
286
|
cbvsumv |
|
288 |
287
|
a1i |
|
289 |
278 288
|
eqtrd |
|
290 |
289
|
cbvmptv |
|
291 |
263 290
|
eqeq12i |
|
292 |
291
|
ralbii |
|
293 |
292
|
biimpi |
|
294 |
293
|
ad2antlr |
|
295 |
|
simpr |
|
296 |
1
|
ad3antrrr |
|
297 |
2
|
ad3antrrr |
|
298 |
3
|
ad3antrrr |
|
299 |
|
simp-4l |
|
300 |
|
simpr |
|
301 |
299 300 4
|
syl2anc |
|
302 |
5
|
ad3antrrr |
|
303 |
|
simplll |
|
304 |
303
|
3ad2ant1 |
|
305 |
|
simp2 |
|
306 |
|
simp3 |
|
307 |
|
eleq1w |
|
308 |
307
|
3anbi3d |
|
309 |
|
fveq2 |
|
310 |
309
|
feq1d |
|
311 |
308 310
|
imbi12d |
|
312 |
311 6
|
chvarvv |
|
313 |
304 305 306 312
|
syl3anc |
|
314 |
|
simprl |
|
315 |
314
|
ad2antrr |
|
316 |
|
simprr |
|
317 |
316
|
ad2antrr |
|
318 |
262
|
eqcomi |
|
319 |
318
|
a1i |
|
320 |
|
id |
|
321 |
319 320
|
fveq12d |
|
322 |
290
|
eqcomi |
|
323 |
322
|
a1i |
|
324 |
|
fveq2 |
|
325 |
324
|
oveq1d |
|
326 |
325
|
oveq1d |
|
327 |
326
|
sumeq2sdv |
|
328 |
|
fveq2 |
|
329 |
328
|
sumeq1d |
|
330 |
327 329
|
eqtrd |
|
331 |
330
|
mpteq2dv |
|
332 |
323 331
|
eqtrd |
|
333 |
321 332
|
eqeq12d |
|
334 |
333
|
cbvralvw |
|
335 |
334
|
biimpi |
|
336 |
335
|
ad2antlr |
|
337 |
|
simpr |
|
338 |
|
fveq1 |
|
339 |
338
|
oveq2d |
|
340 |
|
reseq1 |
|
341 |
339 340
|
opeq12d |
|
342 |
341
|
cbvmptv |
|
343 |
296 297 298 301 302 313 8 315 317 336 337 342
|
dvnprodlem2 |
|
344 |
253 294 295 343
|
syl21anc |
|
345 |
344
|
ralrimiva |
|
346 |
|
fveq2 |
|
347 |
|
fveq2 |
|
348 |
347
|
oveq1d |
|
349 |
348
|
oveq1d |
|
350 |
349
|
sumeq2sdv |
|
351 |
|
fveq2 |
|
352 |
351
|
sumeq1d |
|
353 |
350 352
|
eqtrd |
|
354 |
353
|
mpteq2dv |
|
355 |
346 354
|
eqeq12d |
|
356 |
355
|
cbvralvw |
|
357 |
345 356
|
sylib |
|
358 |
357
|
ex |
|
359 |
25 41 57 76 252 358 3
|
findcard2d |
|
360 |
|
nn0uz |
|
361 |
5 360
|
eleqtrdi |
|
362 |
|
eluzfz2 |
|
363 |
361 362
|
syl |
|
364 |
|
fveq2 |
|
365 |
|
fveq2 |
|
366 |
365
|
sumeq1d |
|
367 |
|
fveq2 |
|
368 |
367
|
oveq1d |
|
369 |
368
|
oveq1d |
|
370 |
369
|
sumeq2sdv |
|
371 |
366 370
|
eqtrd |
|
372 |
371
|
mpteq2dv |
|
373 |
364 372
|
eqeq12d |
|
374 |
373
|
rspccva |
|
375 |
359 363 374
|
syl2anc |
|
376 |
|
oveq2 |
|
377 |
|
rabeq |
|
378 |
376 377
|
syl |
|
379 |
|
sumeq1 |
|
380 |
379
|
eqeq1d |
|
381 |
380
|
rabbidv |
|
382 |
378 381
|
eqtrd |
|
383 |
382
|
mpteq2dv |
|
384 |
|
pwidg |
|
385 |
3 384
|
syl |
|
386 |
142
|
mptex |
|
387 |
386
|
a1i |
|
388 |
8 383 385 387
|
fvmptd3 |
|
389 |
9
|
a1i |
|
390 |
388 389
|
eqtr4d |
|
391 |
390
|
fveq1d |
|
392 |
391
|
sumeq1d |
|
393 |
392
|
mpteq2dv |
|
394 |
375 393
|
eqtrd |
|