Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem35.p |
|
2 |
|
etransclem35.m |
|
3 |
|
etransclem35.f |
|
4 |
|
etransclem35.c |
|
5 |
|
etransclem35.d |
|
6 |
|
reelprrecn |
|
7 |
6
|
a1i |
|
8 |
|
reopn |
|
9 |
|
eqid |
|
10 |
9
|
tgioo2 |
|
11 |
8 10
|
eleqtri |
|
12 |
11
|
a1i |
|
13 |
|
nnm1nn0 |
|
14 |
1 13
|
syl |
|
15 |
|
etransclem5 |
|
16 |
|
0red |
|
17 |
7 12 1 2 3 14 15 4 16
|
etransclem31 |
|
18 |
|
nfv |
|
19 |
|
nfcv |
|
20 |
4 14
|
etransclem16 |
|
21 |
|
simpr |
|
22 |
4 14
|
etransclem12 |
|
23 |
22
|
adantr |
|
24 |
21 23
|
eleqtrd |
|
25 |
|
rabid |
|
26 |
24 25
|
sylib |
|
27 |
26
|
simprd |
|
28 |
27
|
eqcomd |
|
29 |
28
|
fveq2d |
|
30 |
29
|
oveq1d |
|
31 |
|
nfcv |
|
32 |
|
fzfid |
|
33 |
|
nn0ex |
|
34 |
|
fzssnn0 |
|
35 |
|
mapss |
|
36 |
33 34 35
|
mp2an |
|
37 |
26
|
simpld |
|
38 |
36 37
|
sseldi |
|
39 |
31 32 38
|
mccl |
|
40 |
30 39
|
eqeltrd |
|
41 |
40
|
nnzd |
|
42 |
1
|
adantr |
|
43 |
2
|
adantr |
|
44 |
|
elmapi |
|
45 |
37 44
|
syl |
|
46 |
|
0zd |
|
47 |
42 43 45 46
|
etransclem10 |
|
48 |
|
fzfid |
|
49 |
1
|
ad2antrr |
|
50 |
45
|
adantr |
|
51 |
|
fz1ssfz0 |
|
52 |
51
|
sseli |
|
53 |
52
|
adantl |
|
54 |
|
0zd |
|
55 |
49 50 53 54
|
etransclem3 |
|
56 |
48 55
|
fprodzcl |
|
57 |
47 56
|
zmulcld |
|
58 |
41 57
|
zmulcld |
|
59 |
58
|
zcnd |
|
60 |
|
nn0uz |
|
61 |
14 60
|
eleqtrdi |
|
62 |
|
eluzfz2 |
|
63 |
61 62
|
syl |
|
64 |
|
eluzfz1 |
|
65 |
61 64
|
syl |
|
66 |
63 65
|
ifcld |
|
67 |
66
|
adantr |
|
68 |
67 5
|
fmptd |
|
69 |
|
ovex |
|
70 |
|
ovex |
|
71 |
69 70
|
elmap |
|
72 |
68 71
|
sylibr |
|
73 |
2 60
|
eleqtrdi |
|
74 |
|
fzsscn |
|
75 |
68
|
ffvelrnda |
|
76 |
74 75
|
sseldi |
|
77 |
|
fveq2 |
|
78 |
73 76 77
|
fsum1p |
|
79 |
5
|
a1i |
|
80 |
|
simpr |
|
81 |
80
|
iftrued |
|
82 |
|
eluzfz1 |
|
83 |
73 82
|
syl |
|
84 |
79 81 83 14
|
fvmptd |
|
85 |
|
0p1e1 |
|
86 |
85
|
oveq1i |
|
87 |
86
|
sumeq1i |
|
88 |
87
|
a1i |
|
89 |
5
|
fvmpt2 |
|
90 |
52 66 89
|
syl2anr |
|
91 |
|
0red |
|
92 |
|
1red |
|
93 |
|
elfzelz |
|
94 |
93
|
zred |
|
95 |
|
0lt1 |
|
96 |
95
|
a1i |
|
97 |
|
elfzle1 |
|
98 |
91 92 94 96 97
|
ltletrd |
|
99 |
91 98
|
gtned |
|
100 |
99
|
neneqd |
|
101 |
100
|
iffalsed |
|
102 |
101
|
adantl |
|
103 |
90 102
|
eqtrd |
|
104 |
103
|
sumeq2dv |
|
105 |
|
fzfi |
|
106 |
105
|
olci |
|
107 |
|
sumz |
|
108 |
106 107
|
mp1i |
|
109 |
88 104 108
|
3eqtrd |
|
110 |
84 109
|
oveq12d |
|
111 |
1
|
nncnd |
|
112 |
|
1cnd |
|
113 |
111 112
|
subcld |
|
114 |
113
|
addid1d |
|
115 |
78 110 114
|
3eqtrd |
|
116 |
|
fveq1 |
|
117 |
116
|
sumeq2sdv |
|
118 |
117
|
eqeq1d |
|
119 |
118
|
elrab |
|
120 |
72 115 119
|
sylanbrc |
|
121 |
120 22
|
eleqtrrd |
|
122 |
116
|
fveq2d |
|
123 |
122
|
prodeq2ad |
|
124 |
123
|
oveq2d |
|
125 |
|
fveq1 |
|
126 |
125
|
breq2d |
|
127 |
125
|
oveq2d |
|
128 |
127
|
fveq2d |
|
129 |
128
|
oveq2d |
|
130 |
127
|
oveq2d |
|
131 |
129 130
|
oveq12d |
|
132 |
126 131
|
ifbieq2d |
|
133 |
116
|
breq2d |
|
134 |
116
|
oveq2d |
|
135 |
134
|
fveq2d |
|
136 |
135
|
oveq2d |
|
137 |
134
|
oveq2d |
|
138 |
136 137
|
oveq12d |
|
139 |
133 138
|
ifbieq2d |
|
140 |
139
|
prodeq2ad |
|
141 |
132 140
|
oveq12d |
|
142 |
124 141
|
oveq12d |
|
143 |
18 19 20 59 121 142
|
fsumsplit1 |
|
144 |
34 75
|
sseldi |
|
145 |
144
|
faccld |
|
146 |
145
|
nncnd |
|
147 |
77
|
fveq2d |
|
148 |
73 146 147
|
fprod1p |
|
149 |
84
|
fveq2d |
|
150 |
86
|
prodeq1i |
|
151 |
150
|
a1i |
|
152 |
103
|
fveq2d |
|
153 |
|
fac0 |
|
154 |
152 153
|
eqtrdi |
|
155 |
154
|
prodeq2dv |
|
156 |
|
prod1 |
|
157 |
106 156
|
mp1i |
|
158 |
151 155 157
|
3eqtrd |
|
159 |
149 158
|
oveq12d |
|
160 |
14
|
faccld |
|
161 |
160
|
nncnd |
|
162 |
161
|
mulid1d |
|
163 |
148 159 162
|
3eqtrd |
|
164 |
163
|
oveq2d |
|
165 |
160
|
nnne0d |
|
166 |
161 165
|
dividd |
|
167 |
164 166
|
eqtrd |
|
168 |
14
|
nn0red |
|
169 |
84 168
|
eqeltrd |
|
170 |
169 168
|
lttri3d |
|
171 |
84 170
|
mpbid |
|
172 |
171
|
simprd |
|
173 |
172
|
iffalsed |
|
174 |
84
|
eqcomd |
|
175 |
113 174
|
subeq0bd |
|
176 |
175
|
fveq2d |
|
177 |
176 153
|
eqtrdi |
|
178 |
177
|
oveq2d |
|
179 |
161
|
div1d |
|
180 |
178 179
|
eqtrd |
|
181 |
175
|
oveq2d |
|
182 |
|
0cnd |
|
183 |
182
|
exp0d |
|
184 |
181 183
|
eqtrd |
|
185 |
180 184
|
oveq12d |
|
186 |
173 185 162
|
3eqtrd |
|
187 |
|
fzssre |
|
188 |
68
|
adantr |
|
189 |
52
|
adantl |
|
190 |
188 189
|
ffvelrnd |
|
191 |
187 190
|
sseldi |
|
192 |
1
|
nnred |
|
193 |
192
|
adantr |
|
194 |
1
|
nngt0d |
|
195 |
16 192 194
|
ltled |
|
196 |
195
|
adantr |
|
197 |
103 196
|
eqbrtrd |
|
198 |
191 193 197
|
lensymd |
|
199 |
198
|
iffalsed |
|
200 |
103
|
oveq2d |
|
201 |
111
|
adantr |
|
202 |
201
|
subid1d |
|
203 |
200 202
|
eqtrd |
|
204 |
203
|
fveq2d |
|
205 |
204
|
oveq2d |
|
206 |
1
|
nnnn0d |
|
207 |
206
|
faccld |
|
208 |
207
|
nncnd |
|
209 |
207
|
nnne0d |
|
210 |
208 209
|
dividd |
|
211 |
210
|
adantr |
|
212 |
205 211
|
eqtrd |
|
213 |
|
df-neg |
|
214 |
213
|
eqcomi |
|
215 |
214
|
a1i |
|
216 |
215 203
|
oveq12d |
|
217 |
212 216
|
oveq12d |
|
218 |
93
|
znegcld |
|
219 |
218
|
zcnd |
|
220 |
219
|
adantl |
|
221 |
206
|
adantr |
|
222 |
220 221
|
expcld |
|
223 |
222
|
mulid2d |
|
224 |
199 217 223
|
3eqtrd |
|
225 |
224
|
prodeq2dv |
|
226 |
186 225
|
oveq12d |
|
227 |
167 226
|
oveq12d |
|
228 |
|
fzfid |
|
229 |
|
zexpcl |
|
230 |
218 206 229
|
syl2anr |
|
231 |
228 230
|
fprodzcl |
|
232 |
231
|
zcnd |
|
233 |
161 232
|
mulcld |
|
234 |
233
|
mulid2d |
|
235 |
227 234
|
eqtrd |
|
236 |
|
eldifi |
|
237 |
83
|
adantr |
|
238 |
45 237
|
ffvelrnd |
|
239 |
236 238
|
sylan2 |
|
240 |
187 239
|
sseldi |
|
241 |
168
|
adantr |
|
242 |
|
elfzle2 |
|
243 |
239 242
|
syl |
|
244 |
240 241 243
|
lensymd |
|
245 |
244
|
iffalsed |
|
246 |
14
|
nn0zd |
|
247 |
246
|
adantr |
|
248 |
239
|
elfzelzd |
|
249 |
247 248
|
zsubcld |
|
250 |
45
|
ffnd |
|
251 |
250
|
adantr |
|
252 |
68
|
ffnd |
|
253 |
252
|
ad2antrr |
|
254 |
|
fveq2 |
|
255 |
254
|
adantl |
|
256 |
|
id |
|
257 |
256
|
eqcomd |
|
258 |
257
|
ad2antlr |
|
259 |
77
|
adantl |
|
260 |
84
|
adantr |
|
261 |
259 260
|
eqtr2d |
|
262 |
261
|
adantlr |
|
263 |
255 258 262
|
3eqtrd |
|
264 |
263
|
adantllr |
|
265 |
264
|
adantlr |
|
266 |
27
|
ad4antr |
|
267 |
168
|
ad5antr |
|
268 |
168
|
ad4antr |
|
269 |
45
|
adantr |
|
270 |
51
|
sseli |
|
271 |
270
|
adantl |
|
272 |
269 271
|
ffvelrnd |
|
273 |
34 272
|
sseldi |
|
274 |
48 273
|
fsumnn0cl |
|
275 |
274
|
nn0red |
|
276 |
275
|
ad3antrrr |
|
277 |
|
0red |
|
278 |
45
|
ffvelrnda |
|
279 |
187 278
|
sseldi |
|
280 |
279
|
ad2antrr |
|
281 |
|
nfv |
|
282 |
|
nfcv |
|
283 |
|
fzfid |
|
284 |
|
simp-4l |
|
285 |
74 272
|
sseldi |
|
286 |
284 285
|
sylancom |
|
287 |
|
1zzd |
|
288 |
|
elfzel2 |
|
289 |
288
|
adantr |
|
290 |
|
elfzelz |
|
291 |
290
|
adantr |
|
292 |
287 289 291
|
3jca |
|
293 |
|
elfznn0 |
|
294 |
293
|
adantr |
|
295 |
|
neqne |
|
296 |
295
|
adantl |
|
297 |
|
elnnne0 |
|
298 |
294 296 297
|
sylanbrc |
|
299 |
298
|
nnge1d |
|
300 |
|
elfzle2 |
|
301 |
300
|
adantr |
|
302 |
292 299 301
|
jca32 |
|
303 |
|
elfz2 |
|
304 |
302 303
|
sylibr |
|
305 |
304
|
adantr |
|
306 |
305
|
adantlll |
|
307 |
|
fveq2 |
|
308 |
281 282 283 286 306 307
|
fsumsplit1 |
|
309 |
308
|
eqcomd |
|
310 |
309 276
|
eqeltrd |
|
311 |
|
elfzle1 |
|
312 |
278 311
|
syl |
|
313 |
312
|
ad2antrr |
|
314 |
|
neqne |
|
315 |
314
|
adantl |
|
316 |
277 280 313 315
|
leneltd |
|
317 |
|
diffi |
|
318 |
105 317
|
mp1i |
|
319 |
|
eldifi |
|
320 |
319
|
adantl |
|
321 |
51 320
|
sseldi |
|
322 |
45
|
ffvelrnda |
|
323 |
187 322
|
sseldi |
|
324 |
321 323
|
syldan |
|
325 |
|
elfzle1 |
|
326 |
322 325
|
syl |
|
327 |
321 326
|
syldan |
|
328 |
318 324 327
|
fsumge0 |
|
329 |
328
|
adantr |
|
330 |
318 324
|
fsumrecl |
|
331 |
330
|
adantr |
|
332 |
279 331
|
addge01d |
|
333 |
329 332
|
mpbid |
|
334 |
333
|
ad2antrr |
|
335 |
277 280 310 316 334
|
ltletrd |
|
336 |
335 309
|
breqtrd |
|
337 |
276 336
|
elrpd |
|
338 |
268 337
|
ltaddrpd |
|
339 |
338
|
adantl3r |
|
340 |
|
fveq2 |
|
341 |
340
|
cbvsumv |
|
342 |
341
|
a1i |
|
343 |
73
|
ad5antr |
|
344 |
|
simp-5l |
|
345 |
74 322
|
sseldi |
|
346 |
344 345
|
sylancom |
|
347 |
|
fveq2 |
|
348 |
343 346 347
|
fsum1p |
|
349 |
257
|
ad4antlr |
|
350 |
86
|
sumeq1i |
|
351 |
350
|
a1i |
|
352 |
349 351
|
oveq12d |
|
353 |
342 348 352
|
3eqtrrd |
|
354 |
339 353
|
breqtrd |
|
355 |
267 354
|
gtned |
|
356 |
355
|
neneqd |
|
357 |
266 356
|
condan |
|
358 |
|
simpr |
|
359 |
34 67
|
sseldi |
|
360 |
5
|
fvmpt2 |
|
361 |
358 359 360
|
syl2anc |
|
362 |
361
|
adantr |
|
363 |
|
simpr |
|
364 |
363
|
iffalsed |
|
365 |
362 364
|
eqtr2d |
|
366 |
365
|
adantllr |
|
367 |
366
|
adantllr |
|
368 |
357 367
|
eqtrd |
|
369 |
265 368
|
pm2.61dan |
|
370 |
251 253 369
|
eqfnfvd |
|
371 |
236 370
|
sylanl2 |
|
372 |
|
eldifsni |
|
373 |
372
|
neneqd |
|
374 |
373
|
ad2antlr |
|
375 |
371 374
|
pm2.65da |
|
376 |
375
|
neqned |
|
377 |
240 241 243 376
|
leneltd |
|
378 |
240 241
|
posdifd |
|
379 |
377 378
|
mpbid |
|
380 |
|
elnnz |
|
381 |
249 379 380
|
sylanbrc |
|
382 |
381
|
0expd |
|
383 |
382
|
oveq2d |
|
384 |
161
|
adantr |
|
385 |
381
|
nnnn0d |
|
386 |
385
|
faccld |
|
387 |
386
|
nncnd |
|
388 |
386
|
nnne0d |
|
389 |
384 387 388
|
divcld |
|
390 |
389
|
mul01d |
|
391 |
245 383 390
|
3eqtrd |
|
392 |
391
|
oveq1d |
|
393 |
236 56
|
sylan2 |
|
394 |
393
|
zcnd |
|
395 |
394
|
mul02d |
|
396 |
392 395
|
eqtrd |
|
397 |
396
|
oveq2d |
|
398 |
|
fzfid |
|
399 |
34 278
|
sseldi |
|
400 |
236 399
|
sylanl2 |
|
401 |
400
|
faccld |
|
402 |
398 401
|
fprodnncl |
|
403 |
402
|
nncnd |
|
404 |
402
|
nnne0d |
|
405 |
384 403 404
|
divcld |
|
406 |
405
|
mul01d |
|
407 |
397 406
|
eqtrd |
|
408 |
407
|
sumeq2dv |
|
409 |
|
diffi |
|
410 |
20 409
|
syl |
|
411 |
410
|
olcd |
|
412 |
|
sumz |
|
413 |
411 412
|
syl |
|
414 |
408 413
|
eqtrd |
|
415 |
235 414
|
oveq12d |
|
416 |
233
|
addid1d |
|
417 |
|
nfv |
|
418 |
417 206 228 220
|
fprodexp |
|
419 |
418
|
oveq2d |
|
420 |
415 416 419
|
3eqtrd |
|
421 |
17 143 420
|
3eqtrd |
|