Step |
Hyp |
Ref |
Expression |
1 |
|
ioodvbdlimc2lem.a |
|
2 |
|
ioodvbdlimc2lem.b |
|
3 |
|
ioodvbdlimc2lem.altb |
|
4 |
|
ioodvbdlimc2lem.f |
|
5 |
|
ioodvbdlimc2lem.dmdv |
|
6 |
|
ioodvbdlimc2lem.dvbd |
|
7 |
|
ioodvbdlimc2lem.y |
|
8 |
|
ioodvbdlimc2lem.m |
|
9 |
|
ioodvbdlimc2lem.s |
|
10 |
|
ioodvbdlimc2lem.r |
|
11 |
|
ioodvbdlimc2lem.n |
|
12 |
|
ioodvbdlimc2lem.ch |
|
13 |
|
uzssz |
|
14 |
|
zssre |
|
15 |
13 14
|
sstri |
|
16 |
15
|
a1i |
|
17 |
2 1
|
resubcld |
|
18 |
1 2
|
posdifd |
|
19 |
3 18
|
mpbid |
|
20 |
19
|
gt0ne0d |
|
21 |
17 20
|
rereccld |
|
22 |
|
0red |
|
23 |
17 19
|
recgt0d |
|
24 |
22 21 23
|
ltled |
|
25 |
|
flge0nn0 |
|
26 |
21 24 25
|
syl2anc |
|
27 |
|
peano2nn0 |
|
28 |
26 27
|
syl |
|
29 |
8 28
|
eqeltrid |
|
30 |
29
|
nn0zd |
|
31 |
|
eqid |
|
32 |
31
|
uzsup |
|
33 |
30 32
|
syl |
|
34 |
4
|
adantr |
|
35 |
1
|
rexrd |
|
36 |
35
|
adantr |
|
37 |
2
|
rexrd |
|
38 |
37
|
adantr |
|
39 |
2
|
adantr |
|
40 |
|
eluzelre |
|
41 |
40
|
adantl |
|
42 |
|
0red |
|
43 |
|
0red |
|
44 |
|
1red |
|
45 |
43 44
|
readdcld |
|
46 |
45
|
adantl |
|
47 |
43
|
ltp1d |
|
48 |
47
|
adantl |
|
49 |
|
eluzel2 |
|
50 |
49
|
zred |
|
51 |
50
|
adantl |
|
52 |
21
|
flcld |
|
53 |
52
|
zred |
|
54 |
|
1red |
|
55 |
26
|
nn0ge0d |
|
56 |
22 53 54 55
|
leadd1dd |
|
57 |
56 8
|
breqtrrdi |
|
58 |
57
|
adantr |
|
59 |
|
eluzle |
|
60 |
59
|
adantl |
|
61 |
46 51 41 58 60
|
letrd |
|
62 |
42 46 41 48 61
|
ltletrd |
|
63 |
62
|
gt0ne0d |
|
64 |
41 63
|
rereccld |
|
65 |
39 64
|
resubcld |
|
66 |
1
|
adantr |
|
67 |
29
|
nn0red |
|
68 |
22 54
|
readdcld |
|
69 |
53 54
|
readdcld |
|
70 |
22
|
ltp1d |
|
71 |
22 68 69 70 56
|
ltletrd |
|
72 |
71 8
|
breqtrrdi |
|
73 |
72
|
gt0ne0d |
|
74 |
67 73
|
rereccld |
|
75 |
74
|
adantr |
|
76 |
39 75
|
resubcld |
|
77 |
8
|
eqcomi |
|
78 |
77
|
oveq2i |
|
79 |
78 74
|
eqeltrid |
|
80 |
21 23
|
elrpd |
|
81 |
69 71
|
elrpd |
|
82 |
|
1rp |
|
83 |
82
|
a1i |
|
84 |
|
fllelt |
|
85 |
21 84
|
syl |
|
86 |
85
|
simprd |
|
87 |
80 81 83 86
|
ltdiv2dd |
|
88 |
17
|
recnd |
|
89 |
88 20
|
recrecd |
|
90 |
87 89
|
breqtrd |
|
91 |
79 17 2 90
|
ltsub2dd |
|
92 |
2
|
recnd |
|
93 |
1
|
recnd |
|
94 |
92 93
|
nncand |
|
95 |
78
|
oveq2i |
|
96 |
95
|
a1i |
|
97 |
91 94 96
|
3brtr3d |
|
98 |
97
|
adantr |
|
99 |
67 72
|
elrpd |
|
100 |
99
|
adantr |
|
101 |
41 62
|
elrpd |
|
102 |
|
1red |
|
103 |
|
0le1 |
|
104 |
103
|
a1i |
|
105 |
100 101 102 104 60
|
lediv2ad |
|
106 |
64 75 39 105
|
lesub2dd |
|
107 |
66 76 65 98 106
|
ltletrd |
|
108 |
101
|
rpreccld |
|
109 |
39 108
|
ltsubrpd |
|
110 |
36 38 65 107 109
|
eliood |
|
111 |
34 110
|
ffvelrnd |
|
112 |
111 9
|
fmptd |
|
113 |
1 2 3 4 5 6
|
dvbdfbdioo |
|
114 |
67
|
adantr |
|
115 |
|
simpr |
|
116 |
9
|
fvmpt2 |
|
117 |
115 111 116
|
syl2anc |
|
118 |
117
|
fveq2d |
|
119 |
118
|
adantlr |
|
120 |
|
simplr |
|
121 |
110
|
adantlr |
|
122 |
|
2fveq3 |
|
123 |
122
|
breq1d |
|
124 |
123
|
rspccva |
|
125 |
120 121 124
|
syl2anc |
|
126 |
119 125
|
eqbrtrd |
|
127 |
126
|
a1d |
|
128 |
127
|
ralrimiva |
|
129 |
|
breq1 |
|
130 |
129
|
imbi1d |
|
131 |
130
|
ralbidv |
|
132 |
131
|
rspcev |
|
133 |
114 128 132
|
syl2anc |
|
134 |
133
|
ex |
|
135 |
134
|
reximdv |
|
136 |
113 135
|
mpd |
|
137 |
16 33 112 136
|
limsupre |
|
138 |
137
|
recnd |
|
139 |
|
eluzelre |
|
140 |
139
|
adantl |
|
141 |
|
0red |
|
142 |
52
|
peano2zd |
|
143 |
8 142
|
eqeltrid |
|
144 |
143
|
adantr |
|
145 |
144
|
zred |
|
146 |
145
|
adantr |
|
147 |
72
|
ad2antrr |
|
148 |
|
ioomidp |
|
149 |
1 2 3 148
|
syl3anc |
|
150 |
|
ne0i |
|
151 |
149 150
|
syl |
|
152 |
|
ioossre |
|
153 |
152
|
a1i |
|
154 |
|
dvfre |
|
155 |
4 153 154
|
syl2anc |
|
156 |
5
|
feq2d |
|
157 |
155 156
|
mpbid |
|
158 |
157
|
ffvelrnda |
|
159 |
158
|
recnd |
|
160 |
159
|
abscld |
|
161 |
|
eqid |
|
162 |
|
eqid |
|
163 |
151 160 6 161 162
|
suprnmpt |
|
164 |
163
|
simpld |
|
165 |
7 164
|
eqeltrid |
|
166 |
165
|
adantr |
|
167 |
|
rpre |
|
168 |
167
|
rehalfcld |
|
169 |
168
|
adantl |
|
170 |
167
|
recnd |
|
171 |
170
|
adantl |
|
172 |
|
2cnd |
|
173 |
|
rpne0 |
|
174 |
173
|
adantl |
|
175 |
|
2ne0 |
|
176 |
175
|
a1i |
|
177 |
171 172 174 176
|
divne0d |
|
178 |
166 169 177
|
redivcld |
|
179 |
178
|
flcld |
|
180 |
179
|
peano2zd |
|
181 |
180 144
|
ifcld |
|
182 |
11 181
|
eqeltrid |
|
183 |
182
|
zred |
|
184 |
183
|
adantr |
|
185 |
180
|
zred |
|
186 |
|
max1 |
|
187 |
145 185 186
|
syl2anc |
|
188 |
187 11
|
breqtrrdi |
|
189 |
188
|
adantr |
|
190 |
|
eluzle |
|
191 |
190
|
adantl |
|
192 |
146 184 140 189 191
|
letrd |
|
193 |
141 146 140 147 192
|
ltletrd |
|
194 |
193
|
gt0ne0d |
|
195 |
140 194
|
rereccld |
|
196 |
140 193
|
recgt0d |
|
197 |
195 196
|
elrpd |
|
198 |
197
|
adantr |
|
199 |
12
|
biimpi |
|
200 |
|
simp-5l |
|
201 |
199 200
|
syl |
|
202 |
201 4
|
syl |
|
203 |
199
|
simplrd |
|
204 |
202 203
|
ffvelrnd |
|
205 |
204
|
recnd |
|
206 |
201 112
|
syl |
|
207 |
|
simp-5r |
|
208 |
199 207
|
syl |
|
209 |
|
eluz2 |
|
210 |
144 182 188 209
|
syl3anbrc |
|
211 |
201 208 210
|
syl2anc |
|
212 |
|
uzss |
|
213 |
211 212
|
syl |
|
214 |
|
simp-4r |
|
215 |
199 214
|
syl |
|
216 |
213 215
|
sseldd |
|
217 |
206 216
|
ffvelrnd |
|
218 |
217
|
recnd |
|
219 |
201 138
|
syl |
|
220 |
205 218 219
|
npncand |
|
221 |
220
|
eqcomd |
|
222 |
221
|
fveq2d |
|
223 |
204 217
|
resubcld |
|
224 |
201 137
|
syl |
|
225 |
217 224
|
resubcld |
|
226 |
223 225
|
readdcld |
|
227 |
226
|
recnd |
|
228 |
227
|
abscld |
|
229 |
223
|
recnd |
|
230 |
229
|
abscld |
|
231 |
225
|
recnd |
|
232 |
231
|
abscld |
|
233 |
230 232
|
readdcld |
|
234 |
208
|
rpred |
|
235 |
229 231
|
abstrid |
|
236 |
234
|
rehalfcld |
|
237 |
201 216 117
|
syl2anc |
|
238 |
237
|
oveq2d |
|
239 |
238
|
fveq2d |
|
240 |
239 230
|
eqeltrrd |
|
241 |
201 165
|
syl |
|
242 |
152 203
|
sseldi |
|
243 |
201 216 65
|
syl2anc |
|
244 |
242 243
|
resubcld |
|
245 |
241 244
|
remulcld |
|
246 |
201 1
|
syl |
|
247 |
201 2
|
syl |
|
248 |
201 5
|
syl |
|
249 |
163
|
simprd |
|
250 |
7
|
breq2i |
|
251 |
250
|
ralbii |
|
252 |
249 251
|
sylibr |
|
253 |
201 252
|
syl |
|
254 |
|
2fveq3 |
|
255 |
254
|
breq1d |
|
256 |
255
|
cbvralvw |
|
257 |
253 256
|
sylibr |
|
258 |
201 216 110
|
syl2anc |
|
259 |
243
|
rexrd |
|
260 |
201 37
|
syl |
|
261 |
15 216
|
sseldi |
|
262 |
201 216 63
|
syl2anc |
|
263 |
261 262
|
rereccld |
|
264 |
247 242
|
resubcld |
|
265 |
242 247
|
resubcld |
|
266 |
265
|
recnd |
|
267 |
266
|
abscld |
|
268 |
264
|
leabsd |
|
269 |
201 92
|
syl |
|
270 |
242
|
recnd |
|
271 |
269 270
|
abssubd |
|
272 |
268 271
|
breqtrd |
|
273 |
199
|
simprd |
|
274 |
264 267 263 272 273
|
lelttrd |
|
275 |
247 242 263 274
|
ltsub23d |
|
276 |
201 35
|
syl |
|
277 |
|
iooltub |
|
278 |
276 260 203 277
|
syl3anc |
|
279 |
259 260 242 275 278
|
eliood |
|
280 |
246 247 202 248 241 257 258 279
|
dvbdfbdioolem1 |
|
281 |
280
|
simpld |
|
282 |
201 216 64
|
syl2anc |
|
283 |
241 282
|
remulcld |
|
284 |
157 149
|
ffvelrnd |
|
285 |
284
|
recnd |
|
286 |
285
|
abscld |
|
287 |
285
|
absge0d |
|
288 |
|
2fveq3 |
|
289 |
7
|
eqcomi |
|
290 |
289
|
a1i |
|
291 |
288 290
|
breq12d |
|
292 |
291
|
rspcva |
|
293 |
149 249 292
|
syl2anc |
|
294 |
22 286 165 287 293
|
letrd |
|
295 |
201 294
|
syl |
|
296 |
282
|
recnd |
|
297 |
|
sub31 |
|
298 |
270 269 296 297
|
syl3anc |
|
299 |
242 247
|
posdifd |
|
300 |
278 299
|
mpbid |
|
301 |
264 300
|
elrpd |
|
302 |
282 301
|
ltsubrpd |
|
303 |
298 302
|
eqbrtrd |
|
304 |
244 282 303
|
ltled |
|
305 |
244 282 241 295 304
|
lemul2ad |
|
306 |
283
|
adantr |
|
307 |
236
|
adantr |
|
308 |
|
oveq1 |
|
309 |
296
|
mul02d |
|
310 |
308 309
|
sylan9eqr |
|
311 |
208
|
rphalfcld |
|
312 |
311
|
rpgt0d |
|
313 |
312
|
adantr |
|
314 |
310 313
|
eqbrtrd |
|
315 |
306 307 314
|
ltled |
|
316 |
241
|
adantr |
|
317 |
295
|
adantr |
|
318 |
|
neqne |
|
319 |
318
|
adantl |
|
320 |
316 317 319
|
ne0gt0d |
|
321 |
283
|
adantr |
|
322 |
15 211
|
sseldi |
|
323 |
|
0red |
|
324 |
201 208 145
|
syl2anc |
|
325 |
201 72
|
syl |
|
326 |
201 208 188
|
syl2anc |
|
327 |
323 324 322 325 326
|
ltletrd |
|
328 |
327
|
gt0ne0d |
|
329 |
322 328
|
rereccld |
|
330 |
241 329
|
remulcld |
|
331 |
330
|
adantr |
|
332 |
236
|
adantr |
|
333 |
282
|
adantr |
|
334 |
329
|
adantr |
|
335 |
241
|
adantr |
|
336 |
295
|
adantr |
|
337 |
322 327
|
elrpd |
|
338 |
201 216 101
|
syl2anc |
|
339 |
|
1red |
|
340 |
103
|
a1i |
|
341 |
215 190
|
syl |
|
342 |
337 338 339 340 341
|
lediv2ad |
|
343 |
342
|
adantr |
|
344 |
333 334 335 336 343
|
lemul2ad |
|
345 |
234
|
recnd |
|
346 |
|
2cnd |
|
347 |
208
|
rpne0d |
|
348 |
175
|
a1i |
|
349 |
345 346 347 348
|
divne0d |
|
350 |
241 236 349
|
redivcld |
|
351 |
350
|
adantr |
|
352 |
|
simpr |
|
353 |
312
|
adantr |
|
354 |
335 332 352 353
|
divgt0d |
|
355 |
351 354
|
elrpd |
|
356 |
355
|
rprecred |
|
357 |
337
|
adantr |
|
358 |
|
1red |
|
359 |
103
|
a1i |
|
360 |
350
|
flcld |
|
361 |
360
|
peano2zd |
|
362 |
361
|
zred |
|
363 |
201 143
|
syl |
|
364 |
361 363
|
ifcld |
|
365 |
11 364
|
eqeltrid |
|
366 |
365
|
zred |
|
367 |
|
flltp1 |
|
368 |
350 367
|
syl |
|
369 |
201 67
|
syl |
|
370 |
|
max2 |
|
371 |
369 362 370
|
syl2anc |
|
372 |
371 11
|
breqtrrdi |
|
373 |
350 362 366 368 372
|
ltletrd |
|
374 |
350 322 373
|
ltled |
|
375 |
374
|
adantr |
|
376 |
355 357 358 359 375
|
lediv2ad |
|
377 |
334 356 335 336 376
|
lemul2ad |
|
378 |
335
|
recnd |
|
379 |
351
|
recnd |
|
380 |
354
|
gt0ne0d |
|
381 |
378 379 380
|
divrecd |
|
382 |
332
|
recnd |
|
383 |
352
|
gt0ne0d |
|
384 |
349
|
adantr |
|
385 |
378 382 383 384
|
ddcand |
|
386 |
381 385
|
eqtr3d |
|
387 |
377 386
|
breqtrd |
|
388 |
321 331 332 344 387
|
letrd |
|
389 |
320 388
|
syldan |
|
390 |
315 389
|
pm2.61dan |
|
391 |
245 283 236 305 390
|
letrd |
|
392 |
240 245 236 281 391
|
letrd |
|
393 |
239 392
|
eqbrtrd |
|
394 |
|
simpllr |
|
395 |
199 394
|
syl |
|
396 |
230 232 236 236 393 395
|
leltaddd |
|
397 |
345
|
2halvesd |
|
398 |
396 397
|
breqtrd |
|
399 |
228 233 234 235 398
|
lelttrd |
|
400 |
222 399
|
eqbrtrd |
|
401 |
12 400
|
sylbir |
|
402 |
401
|
adantrl |
|
403 |
402
|
ex |
|
404 |
403
|
ralrimiva |
|
405 |
|
brimralrspcev |
|
406 |
198 404 405
|
syl2anc |
|
407 |
|
simpr |
|
408 |
407
|
iftrued |
|
409 |
|
uzid |
|
410 |
182 409
|
syl |
|
411 |
410
|
adantr |
|
412 |
408 411
|
eqeltrd |
|
413 |
412
|
adantlr |
|
414 |
|
iffalse |
|
415 |
414
|
adantl |
|
416 |
182
|
ad2antrr |
|
417 |
|
simplr |
|
418 |
416
|
zred |
|
419 |
417
|
zred |
|
420 |
|
simpr |
|
421 |
418 419
|
ltnled |
|
422 |
420 421
|
mpbird |
|
423 |
418 419 422
|
ltled |
|
424 |
|
eluz2 |
|
425 |
416 417 423 424
|
syl3anbrc |
|
426 |
415 425
|
eqeltrd |
|
427 |
413 426
|
pm2.61dan |
|
428 |
427
|
adantr |
|
429 |
|
simpr |
|
430 |
|
simpr |
|
431 |
182
|
adantr |
|
432 |
431 430
|
ifcld |
|
433 |
430
|
zred |
|
434 |
431
|
zred |
|
435 |
|
max1 |
|
436 |
433 434 435
|
syl2anc |
|
437 |
|
eluz2 |
|
438 |
430 432 436 437
|
syl3anbrc |
|
439 |
438
|
adantr |
|
440 |
|
fveq2 |
|
441 |
440
|
eleq1d |
|
442 |
440
|
fvoveq1d |
|
443 |
442
|
breq1d |
|
444 |
441 443
|
anbi12d |
|
445 |
444
|
rspccva |
|
446 |
429 439 445
|
syl2anc |
|
447 |
446
|
simprd |
|
448 |
|
fveq2 |
|
449 |
448
|
fvoveq1d |
|
450 |
449
|
breq1d |
|
451 |
450
|
rspcev |
|
452 |
428 447 451
|
syl2anc |
|
453 |
|
ax-resscn |
|
454 |
453
|
a1i |
|
455 |
4 454
|
fssd |
|
456 |
|
dvcn |
|
457 |
454 455 153 5 456
|
syl31anc |
|
458 |
|
cncffvrn |
|
459 |
454 457 458
|
syl2anc |
|
460 |
4 459
|
mpbird |
|
461 |
110 10
|
fmptd |
|
462 |
|
eqid |
|
463 |
|
climrel |
|
464 |
463
|
a1i |
|
465 |
|
fvex |
|
466 |
465
|
mptex |
|
467 |
466
|
a1i |
|
468 |
|
eqidd |
|
469 |
|
eqidd |
|
470 |
|
simpr |
|
471 |
2
|
adantr |
|
472 |
468 469 470 471
|
fvmptd |
|
473 |
31 30 467 92 472
|
climconst |
|
474 |
465
|
mptex |
|
475 |
10 474
|
eqeltri |
|
476 |
475
|
a1i |
|
477 |
|
1cnd |
|
478 |
|
elnnnn0b |
|
479 |
29 72 478
|
sylanbrc |
|
480 |
|
divcnvg |
|
481 |
477 479 480
|
syl2anc |
|
482 |
|
eqidd |
|
483 |
|
eqidd |
|
484 |
|
simpr |
|
485 |
2
|
adantr |
|
486 |
482 483 484 485
|
fvmptd |
|
487 |
486 485
|
eqeltrd |
|
488 |
487
|
recnd |
|
489 |
|
eqidd |
|
490 |
|
oveq2 |
|
491 |
490
|
adantl |
|
492 |
15 484
|
sseldi |
|
493 |
|
0red |
|
494 |
67
|
adantr |
|
495 |
72
|
adantr |
|
496 |
|
eluzle |
|
497 |
496
|
adantl |
|
498 |
493 494 492 495 497
|
ltletrd |
|
499 |
498
|
gt0ne0d |
|
500 |
492 499
|
rereccld |
|
501 |
489 491 484 500
|
fvmptd |
|
502 |
492
|
recnd |
|
503 |
502 499
|
reccld |
|
504 |
501 503
|
eqeltrd |
|
505 |
490
|
oveq2d |
|
506 |
|
ovex |
|
507 |
505 10 506
|
fvmpt |
|
508 |
507
|
adantl |
|
509 |
486 501
|
oveq12d |
|
510 |
508 509
|
eqtr4d |
|
511 |
31 30 473 476 481 488 504 510
|
climsub |
|
512 |
92
|
subid1d |
|
513 |
511 512
|
breqtrd |
|
514 |
|
releldm |
|
515 |
464 513 514
|
syl2anc |
|
516 |
|
fveq2 |
|
517 |
|
fveq2 |
|
518 |
517
|
oveq2d |
|
519 |
518
|
fveq2d |
|
520 |
519
|
breq1d |
|
521 |
516 520
|
raleqbidv |
|
522 |
521
|
cbvrabv |
|
523 |
|
fveq2 |
|
524 |
523
|
fvoveq1d |
|
525 |
524
|
breq1d |
|
526 |
525
|
cbvralvw |
|
527 |
526
|
rgenw |
|
528 |
|
rabbi |
|
529 |
527 528
|
mpbi |
|
530 |
522 529
|
eqtri |
|
531 |
530
|
infeq1i |
|
532 |
1 2 3 460 5 6 30 461 462 515 531
|
ioodvbdlimc1lem1 |
|
533 |
10
|
fvmpt2 |
|
534 |
115 65 533
|
syl2anc |
|
535 |
534
|
eqcomd |
|
536 |
535
|
fveq2d |
|
537 |
536
|
mpteq2dva |
|
538 |
9 537
|
syl5eq |
|
539 |
538
|
fveq2d |
|
540 |
532 538 539
|
3brtr4d |
|
541 |
465
|
mptex |
|
542 |
9 541
|
eqeltri |
|
543 |
542
|
a1i |
|
544 |
|
eqidd |
|
545 |
543 544
|
clim |
|
546 |
540 545
|
mpbid |
|
547 |
546
|
simprd |
|
548 |
547
|
adantr |
|
549 |
|
simpr |
|
550 |
549
|
rphalfcld |
|
551 |
|
breq2 |
|
552 |
551
|
anbi2d |
|
553 |
552
|
rexralbidv |
|
554 |
553
|
rspccva |
|
555 |
548 550 554
|
syl2anc |
|
556 |
452 555
|
r19.29a |
|
557 |
406 556
|
r19.29a |
|
558 |
557
|
ralrimiva |
|
559 |
|
ioosscn |
|
560 |
559
|
a1i |
|
561 |
455 560 92
|
ellimc3 |
|
562 |
138 558 561
|
mpbir2and |
|