Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|
2 |
|
poimirlem22.s |
|
3 |
|
poimirlem22.1 |
|
4 |
|
poimirlem22.2 |
|
5 |
|
poimirlem22.3 |
|
6 |
|
poimirlem21.4 |
|
7 |
|
fveq2 |
|
8 |
7
|
breq2d |
|
9 |
8
|
ifbid |
|
10 |
|
2fveq3 |
|
11 |
|
2fveq3 |
|
12 |
11
|
imaeq1d |
|
13 |
12
|
xpeq1d |
|
14 |
11
|
imaeq1d |
|
15 |
14
|
xpeq1d |
|
16 |
13 15
|
uneq12d |
|
17 |
10 16
|
oveq12d |
|
18 |
9 17
|
csbeq12dv |
|
19 |
18
|
mpteq2dv |
|
20 |
19
|
eqeq2d |
|
21 |
20 2
|
elrab2 |
|
22 |
21
|
simprbi |
|
23 |
4 22
|
syl |
|
24 |
|
elrabi |
|
25 |
24 2
|
eleq2s |
|
26 |
4 25
|
syl |
|
27 |
|
xp1st |
|
28 |
26 27
|
syl |
|
29 |
|
xp1st |
|
30 |
28 29
|
syl |
|
31 |
|
elmapfn |
|
32 |
30 31
|
syl |
|
33 |
32
|
adantr |
|
34 |
|
1ex |
|
35 |
|
fnconstg |
|
36 |
34 35
|
ax-mp |
|
37 |
|
c0ex |
|
38 |
|
fnconstg |
|
39 |
37 38
|
ax-mp |
|
40 |
36 39
|
pm3.2i |
|
41 |
|
xp2nd |
|
42 |
28 41
|
syl |
|
43 |
|
fvex |
|
44 |
|
f1oeq1 |
|
45 |
43 44
|
elab |
|
46 |
42 45
|
sylib |
|
47 |
|
dff1o3 |
|
48 |
47
|
simprbi |
|
49 |
46 48
|
syl |
|
50 |
|
imain |
|
51 |
49 50
|
syl |
|
52 |
|
elfznn0 |
|
53 |
52
|
nn0red |
|
54 |
53
|
ltp1d |
|
55 |
|
fzdisj |
|
56 |
54 55
|
syl |
|
57 |
56
|
imaeq2d |
|
58 |
|
ima0 |
|
59 |
57 58
|
eqtrdi |
|
60 |
51 59
|
sylan9req |
|
61 |
|
fnun |
|
62 |
40 60 61
|
sylancr |
|
63 |
|
imaundi |
|
64 |
|
nn0p1nn |
|
65 |
52 64
|
syl |
|
66 |
|
nnuz |
|
67 |
65 66
|
eleqtrdi |
|
68 |
67
|
adantl |
|
69 |
1
|
nncnd |
|
70 |
|
npcan1 |
|
71 |
69 70
|
syl |
|
72 |
71
|
adantr |
|
73 |
|
elfzuz3 |
|
74 |
|
peano2uz |
|
75 |
73 74
|
syl |
|
76 |
75
|
adantl |
|
77 |
72 76
|
eqeltrrd |
|
78 |
|
fzsplit2 |
|
79 |
68 77 78
|
syl2anc |
|
80 |
79
|
imaeq2d |
|
81 |
|
f1ofo |
|
82 |
|
foima |
|
83 |
46 81 82
|
3syl |
|
84 |
83
|
adantr |
|
85 |
80 84
|
eqtr3d |
|
86 |
63 85
|
eqtr3id |
|
87 |
86
|
fneq2d |
|
88 |
62 87
|
mpbid |
|
89 |
|
ovexd |
|
90 |
|
inidm |
|
91 |
|
eqidd |
|
92 |
|
eqidd |
|
93 |
33 88 89 89 90 91 92
|
offval |
|
94 |
|
elmapi |
|
95 |
30 94
|
syl |
|
96 |
95
|
ffvelrnda |
|
97 |
|
elfzonn0 |
|
98 |
96 97
|
syl |
|
99 |
98
|
nn0cnd |
|
100 |
99
|
adantlr |
|
101 |
|
ax-1cn |
|
102 |
|
0cn |
|
103 |
101 102
|
ifcli |
|
104 |
103
|
a1i |
|
105 |
|
snssi |
|
106 |
101 105
|
ax-mp |
|
107 |
|
snssi |
|
108 |
102 107
|
ax-mp |
|
109 |
106 108
|
unssi |
|
110 |
34
|
fconst |
|
111 |
37
|
fconst |
|
112 |
110 111
|
pm3.2i |
|
113 |
|
simpr |
|
114 |
1
|
nnzd |
|
115 |
|
1z |
|
116 |
|
peano2z |
|
117 |
115 116
|
ax-mp |
|
118 |
114 117
|
jctil |
|
119 |
|
elfzelz |
|
120 |
119 115
|
jctir |
|
121 |
|
fzsubel |
|
122 |
118 120 121
|
syl2an |
|
123 |
113 122
|
mpbid |
|
124 |
101 101
|
pncan3oi |
|
125 |
124
|
oveq1i |
|
126 |
123 125
|
eleqtrdi |
|
127 |
126
|
ralrimiva |
|
128 |
|
simpr |
|
129 |
|
peano2zm |
|
130 |
114 129
|
syl |
|
131 |
130 115
|
jctil |
|
132 |
|
elfzelz |
|
133 |
132 115
|
jctir |
|
134 |
|
fzaddel |
|
135 |
131 133 134
|
syl2an |
|
136 |
128 135
|
mpbid |
|
137 |
71
|
oveq2d |
|
138 |
137
|
adantr |
|
139 |
136 138
|
eleqtrd |
|
140 |
119
|
zcnd |
|
141 |
132
|
zcnd |
|
142 |
|
subadd2 |
|
143 |
101 142
|
mp3an2 |
|
144 |
|
eqcom |
|
145 |
|
eqcom |
|
146 |
143 144 145
|
3bitr4g |
|
147 |
140 141 146
|
syl2anr |
|
148 |
147
|
ralrimiva |
|
149 |
148
|
adantl |
|
150 |
|
reu6i |
|
151 |
139 149 150
|
syl2anc |
|
152 |
151
|
ralrimiva |
|
153 |
|
eqid |
|
154 |
153
|
f1ompt |
|
155 |
127 152 154
|
sylanbrc |
|
156 |
|
f1osng |
|
157 |
34 1 156
|
sylancr |
|
158 |
1
|
nnred |
|
159 |
158
|
ltm1d |
|
160 |
130
|
zred |
|
161 |
160 158
|
ltnled |
|
162 |
159 161
|
mpbid |
|
163 |
|
elfzle2 |
|
164 |
162 163
|
nsyl |
|
165 |
|
disjsn |
|
166 |
164 165
|
sylibr |
|
167 |
|
1re |
|
168 |
167
|
ltp1i |
|
169 |
117
|
zrei |
|
170 |
167 169
|
ltnlei |
|
171 |
168 170
|
mpbi |
|
172 |
|
elfzle1 |
|
173 |
171 172
|
mto |
|
174 |
|
disjsn |
|
175 |
173 174
|
mpbir |
|
176 |
|
f1oun |
|
177 |
175 176
|
mpanr1 |
|
178 |
155 157 166 177
|
syl21anc |
|
179 |
|
eleq1 |
|
180 |
173 179
|
mtbiri |
|
181 |
180
|
necon2ai |
|
182 |
|
ifnefalse |
|
183 |
181 182
|
syl |
|
184 |
183
|
mpteq2ia |
|
185 |
184
|
uneq1i |
|
186 |
34
|
a1i |
|
187 |
1 66
|
eleqtrdi |
|
188 |
|
fzpred |
|
189 |
187 188
|
syl |
|
190 |
|
uncom |
|
191 |
189 190
|
eqtr2di |
|
192 |
|
iftrue |
|
193 |
192
|
adantl |
|
194 |
186 1 191 193
|
fmptapd |
|
195 |
185 194
|
eqtr3id |
|
196 |
71 187
|
eqeltrd |
|
197 |
|
uzid |
|
198 |
|
peano2uz |
|
199 |
130 197 198
|
3syl |
|
200 |
71 199
|
eqeltrrd |
|
201 |
|
fzsplit2 |
|
202 |
196 200 201
|
syl2anc |
|
203 |
71
|
oveq1d |
|
204 |
|
fzsn |
|
205 |
114 204
|
syl |
|
206 |
203 205
|
eqtrd |
|
207 |
206
|
uneq2d |
|
208 |
202 207
|
eqtr2d |
|
209 |
195 191 208
|
f1oeq123d |
|
210 |
178 209
|
mpbid |
|
211 |
|
f1oco |
|
212 |
46 210 211
|
syl2anc |
|
213 |
|
dff1o3 |
|
214 |
213
|
simprbi |
|
215 |
|
imain |
|
216 |
212 214 215
|
3syl |
|
217 |
65
|
nnred |
|
218 |
217
|
ltp1d |
|
219 |
|
fzdisj |
|
220 |
218 219
|
syl |
|
221 |
220
|
imaeq2d |
|
222 |
|
ima0 |
|
223 |
221 222
|
eqtrdi |
|
224 |
216 223
|
sylan9req |
|
225 |
|
fun |
|
226 |
112 224 225
|
sylancr |
|
227 |
|
imaundi |
|
228 |
65
|
peano2nnd |
|
229 |
228 66
|
eleqtrdi |
|
230 |
229
|
adantl |
|
231 |
|
eluzp1p1 |
|
232 |
73 231
|
syl |
|
233 |
232
|
adantl |
|
234 |
72 233
|
eqeltrrd |
|
235 |
|
fzsplit2 |
|
236 |
230 234 235
|
syl2anc |
|
237 |
236
|
imaeq2d |
|
238 |
|
f1ofo |
|
239 |
|
foima |
|
240 |
212 238 239
|
3syl |
|
241 |
240
|
adantr |
|
242 |
237 241
|
eqtr3d |
|
243 |
227 242
|
eqtr3id |
|
244 |
243
|
feq2d |
|
245 |
226 244
|
mpbid |
|
246 |
245
|
ffvelrnda |
|
247 |
109 246
|
sselid |
|
248 |
100 104 247
|
subadd23d |
|
249 |
|
oveq2 |
|
250 |
249
|
eqeq1d |
|
251 |
|
oveq2 |
|
252 |
251
|
eqeq1d |
|
253 |
|
1m1e0 |
|
254 |
|
f1ofn |
|
255 |
46 254
|
syl |
|
256 |
255
|
adantr |
|
257 |
|
imassrn |
|
258 |
|
f1of |
|
259 |
210 258
|
syl |
|
260 |
259
|
frnd |
|
261 |
257 260
|
sstrid |
|
262 |
261
|
adantr |
|
263 |
|
eqidd |
|
264 |
|
eluzfz1 |
|
265 |
187 264
|
syl |
|
266 |
263 193 265 1
|
fvmptd |
|
267 |
266
|
adantr |
|
268 |
|
f1ofn |
|
269 |
210 268
|
syl |
|
270 |
269
|
adantr |
|
271 |
|
fzss2 |
|
272 |
234 271
|
syl |
|
273 |
|
eluzfz1 |
|
274 |
67 273
|
syl |
|
275 |
274
|
adantl |
|
276 |
|
fnfvima |
|
277 |
270 272 275 276
|
syl3anc |
|
278 |
267 277
|
eqeltrrd |
|
279 |
|
fnfvima |
|
280 |
256 262 278 279
|
syl3anc |
|
281 |
|
imaco |
|
282 |
280 281
|
eleqtrrdi |
|
283 |
|
fnconstg |
|
284 |
34 283
|
ax-mp |
|
285 |
|
fnconstg |
|
286 |
37 285
|
ax-mp |
|
287 |
|
fvun1 |
|
288 |
284 286 287
|
mp3an12 |
|
289 |
224 282 288
|
syl2anc |
|
290 |
34
|
fvconst2 |
|
291 |
282 290
|
syl |
|
292 |
289 291
|
eqtrd |
|
293 |
292
|
oveq1d |
|
294 |
|
fzss1 |
|
295 |
67 294
|
syl |
|
296 |
295
|
adantl |
|
297 |
|
eluzfz2 |
|
298 |
234 297
|
syl |
|
299 |
|
fnfvima |
|
300 |
256 296 298 299
|
syl3anc |
|
301 |
|
fvun2 |
|
302 |
36 39 301
|
mp3an12 |
|
303 |
60 300 302
|
syl2anc |
|
304 |
37
|
fvconst2 |
|
305 |
300 304
|
syl |
|
306 |
303 305
|
eqtrd |
|
307 |
253 293 306
|
3eqtr4a |
|
308 |
|
fveq2 |
|
309 |
308
|
oveq1d |
|
310 |
|
fveq2 |
|
311 |
309 310
|
eqeq12d |
|
312 |
307 311
|
syl5ibrcom |
|
313 |
312
|
imp |
|
314 |
313
|
adantlr |
|
315 |
247
|
subid1d |
|
316 |
315
|
adantr |
|
317 |
|
eldifsn |
|
318 |
|
df-ne |
|
319 |
318
|
anbi2i |
|
320 |
317 319
|
bitri |
|
321 |
|
fnconstg |
|
322 |
37 321
|
ax-mp |
|
323 |
36 322
|
pm3.2i |
|
324 |
|
imain |
|
325 |
49 324
|
syl |
|
326 |
|
fzdisj |
|
327 |
54 326
|
syl |
|
328 |
327
|
imaeq2d |
|
329 |
328 58
|
eqtrdi |
|
330 |
325 329
|
sylan9req |
|
331 |
|
fnun |
|
332 |
323 330 331
|
sylancr |
|
333 |
|
imaundi |
|
334 |
202 207
|
eqtrd |
|
335 |
334
|
difeq1d |
|
336 |
|
difun2 |
|
337 |
335 336
|
eqtrdi |
|
338 |
|
difsn |
|
339 |
164 338
|
syl |
|
340 |
337 339
|
eqtrd |
|
341 |
340
|
adantr |
|
342 |
73
|
adantl |
|
343 |
|
fzsplit2 |
|
344 |
68 342 343
|
syl2anc |
|
345 |
341 344
|
eqtrd |
|
346 |
345
|
imaeq2d |
|
347 |
|
imadif |
|
348 |
49 347
|
syl |
|
349 |
|
elfz1end |
|
350 |
1 349
|
sylib |
|
351 |
|
fnsnfv |
|
352 |
255 350 351
|
syl2anc |
|
353 |
352
|
eqcomd |
|
354 |
83 353
|
difeq12d |
|
355 |
348 354
|
eqtrd |
|
356 |
355
|
adantr |
|
357 |
346 356
|
eqtr3d |
|
358 |
333 357
|
eqtr3id |
|
359 |
358
|
fneq2d |
|
360 |
332 359
|
mpbid |
|
361 |
|
disjdifr |
|
362 |
|
fnconstg |
|
363 |
34 362
|
ax-mp |
|
364 |
|
fvun1 |
|
365 |
363 364
|
mp3an2 |
|
366 |
|
fnconstg |
|
367 |
37 366
|
ax-mp |
|
368 |
|
fvun1 |
|
369 |
367 368
|
mp3an2 |
|
370 |
365 369
|
eqtr4d |
|
371 |
361 370
|
mpanr1 |
|
372 |
360 371
|
sylan |
|
373 |
320 372
|
sylan2br |
|
374 |
373
|
anassrs |
|
375 |
|
imaundi |
|
376 |
|
imaco |
|
377 |
|
imaco |
|
378 |
376 377
|
uneq12i |
|
379 |
375 378
|
eqtri |
|
380 |
|
fzpred |
|
381 |
67 380
|
syl |
|
382 |
|
uncom |
|
383 |
381 382
|
eqtrdi |
|
384 |
383
|
imaeq2d |
|
385 |
384
|
adantl |
|
386 |
|
elfzelz |
|
387 |
124
|
a1i |
|
388 |
|
zcn |
|
389 |
|
pncan1 |
|
390 |
388 389
|
syl |
|
391 |
387 390
|
oveq12d |
|
392 |
|
elfzelz |
|
393 |
392
|
zcnd |
|
394 |
|
pncan1 |
|
395 |
393 394
|
syl |
|
396 |
395
|
eleq1d |
|
397 |
396
|
ibir |
|
398 |
397
|
adantl |
|
399 |
|
peano2z |
|
400 |
399 117
|
jctil |
|
401 |
392
|
peano2zd |
|
402 |
401 115
|
jctir |
|
403 |
|
fzsubel |
|
404 |
400 402 403
|
syl2an |
|
405 |
398 404
|
mpbird |
|
406 |
395
|
eqcomd |
|
407 |
406
|
adantl |
|
408 |
|
oveq1 |
|
409 |
408
|
rspceeqv |
|
410 |
405 407 409
|
syl2anc |
|
411 |
410
|
ex |
|
412 |
|
simpr |
|
413 |
|
elfzelz |
|
414 |
413 115
|
jctir |
|
415 |
|
fzsubel |
|
416 |
400 414 415
|
syl2an |
|
417 |
412 416
|
mpbid |
|
418 |
|
eleq1 |
|
419 |
417 418
|
syl5ibrcom |
|
420 |
419
|
rexlimdva |
|
421 |
411 420
|
impbid |
|
422 |
|
eqid |
|
423 |
422
|
elrnmpt |
|
424 |
423
|
elv |
|
425 |
421 424
|
bitr4di |
|
426 |
425
|
eqrdv |
|
427 |
391 426
|
eqtr3d |
|
428 |
386 427
|
syl |
|
429 |
428
|
adantl |
|
430 |
|
df-ima |
|
431 |
|
uzid |
|
432 |
|
peano2uz |
|
433 |
115 431 432
|
mp2b |
|
434 |
|
fzss1 |
|
435 |
433 434
|
ax-mp |
|
436 |
435 272
|
sstrid |
|
437 |
436
|
resmptd |
|
438 |
|
elfzle1 |
|
439 |
171 438
|
mto |
|
440 |
|
eleq1 |
|
441 |
439 440
|
mtbiri |
|
442 |
441
|
necon2ai |
|
443 |
442 182
|
syl |
|
444 |
443
|
mpteq2ia |
|
445 |
437 444
|
eqtrdi |
|
446 |
445
|
rneqd |
|
447 |
430 446
|
syl5eq |
|
448 |
429 447
|
eqtr4d |
|
449 |
448
|
imaeq2d |
|
450 |
266
|
sneqd |
|
451 |
|
fnsnfv |
|
452 |
269 265 451
|
syl2anc |
|
453 |
450 452
|
eqtr3d |
|
454 |
453
|
imaeq2d |
|
455 |
352 454
|
eqtrd |
|
456 |
455
|
adantr |
|
457 |
449 456
|
uneq12d |
|
458 |
379 385 457
|
3eqtr4a |
|
459 |
458
|
xpeq1d |
|
460 |
|
xpundir |
|
461 |
459 460
|
eqtrdi |
|
462 |
|
imaco |
|
463 |
|
df-ima |
|
464 |
|
fzss1 |
|
465 |
230 464
|
syl |
|
466 |
465
|
resmptd |
|
467 |
|
1red |
|
468 |
65
|
nnzd |
|
469 |
468
|
peano2zd |
|
470 |
469
|
zred |
|
471 |
65
|
nnge1d |
|
472 |
467 217 470 471 218
|
lelttrd |
|
473 |
467 470
|
ltnled |
|
474 |
472 473
|
mpbid |
|
475 |
|
elfzle1 |
|
476 |
474 475
|
nsyl |
|
477 |
|
eleq1 |
|
478 |
477
|
notbid |
|
479 |
476 478
|
syl5ibrcom |
|
480 |
479
|
necon2ad |
|
481 |
480
|
imp |
|
482 |
481 182
|
syl |
|
483 |
482
|
mpteq2dva |
|
484 |
483
|
adantl |
|
485 |
466 484
|
eqtrd |
|
486 |
485
|
rneqd |
|
487 |
463 486
|
syl5eq |
|
488 |
|
eqid |
|
489 |
488
|
elrnmpt |
|
490 |
489
|
elv |
|
491 |
|
simpr |
|
492 |
114 469
|
anim12ci |
|
493 |
|
elfzelz |
|
494 |
493 115
|
jctir |
|
495 |
|
fzsubel |
|
496 |
492 494 495
|
syl2an |
|
497 |
491 496
|
mpbid |
|
498 |
|
eleq1 |
|
499 |
497 498
|
syl5ibrcom |
|
500 |
499
|
rexlimdva |
|
501 |
|
elfzelz |
|
502 |
501
|
zcnd |
|
503 |
502 394
|
syl |
|
504 |
503
|
eleq1d |
|
505 |
504
|
ibir |
|
506 |
505
|
adantl |
|
507 |
501
|
peano2zd |
|
508 |
507 115
|
jctir |
|
509 |
|
fzsubel |
|
510 |
492 508 509
|
syl2an |
|
511 |
506 510
|
mpbird |
|
512 |
503
|
eqcomd |
|
513 |
512
|
adantl |
|
514 |
408
|
rspceeqv |
|
515 |
511 513 514
|
syl2anc |
|
516 |
515
|
ex |
|
517 |
500 516
|
impbid |
|
518 |
490 517
|
syl5bb |
|
519 |
518
|
eqrdv |
|
520 |
65
|
nncnd |
|
521 |
|
pncan1 |
|
522 |
520 521
|
syl |
|
523 |
522
|
oveq1d |
|
524 |
523
|
adantl |
|
525 |
487 519 524
|
3eqtrd |
|
526 |
525
|
imaeq2d |
|
527 |
462 526
|
syl5eq |
|
528 |
527
|
xpeq1d |
|
529 |
461 528
|
uneq12d |
|
530 |
|
un23 |
|
531 |
529 530
|
eqtrdi |
|
532 |
531
|
fveq1d |
|
533 |
532
|
ad2antrr |
|
534 |
|
imaundi |
|
535 |
|
fzsplit2 |
|
536 |
232 200 535
|
syl2anr |
|
537 |
206
|
uneq2d |
|
538 |
537
|
adantr |
|
539 |
536 538
|
eqtrd |
|
540 |
539
|
imaeq2d |
|
541 |
352
|
adantr |
|
542 |
541
|
uneq2d |
|
543 |
534 540 542
|
3eqtr4a |
|
544 |
543
|
xpeq1d |
|
545 |
|
xpundir |
|
546 |
544 545
|
eqtrdi |
|
547 |
546
|
uneq2d |
|
548 |
|
unass |
|
549 |
547 548
|
eqtr4di |
|
550 |
549
|
fveq1d |
|
551 |
550
|
ad2antrr |
|
552 |
374 533 551
|
3eqtr4d |
|
553 |
316 552
|
eqtrd |
|
554 |
250 252 314 553
|
ifbothda |
|
555 |
554
|
oveq2d |
|
556 |
248 555
|
eqtr2d |
|
557 |
556
|
mpteq2dva |
|
558 |
93 557
|
eqtrd |
|
559 |
53
|
adantl |
|
560 |
160
|
adantr |
|
561 |
158
|
adantr |
|
562 |
|
elfzle2 |
|
563 |
562
|
adantl |
|
564 |
159
|
adantr |
|
565 |
559 560 561 563 564
|
lelttrd |
|
566 |
6
|
adantr |
|
567 |
565 566
|
breqtrrd |
|
568 |
567
|
iftrued |
|
569 |
568
|
csbeq1d |
|
570 |
|
vex |
|
571 |
|
oveq2 |
|
572 |
571
|
imaeq2d |
|
573 |
572
|
xpeq1d |
|
574 |
|
oveq1 |
|
575 |
574
|
oveq1d |
|
576 |
575
|
imaeq2d |
|
577 |
576
|
xpeq1d |
|
578 |
573 577
|
uneq12d |
|
579 |
578
|
oveq2d |
|
580 |
570 579
|
csbie |
|
581 |
569 580
|
eqtrdi |
|
582 |
|
ovexd |
|
583 |
|
fvexd |
|
584 |
|
eqidd |
|
585 |
245
|
ffnd |
|
586 |
|
nfcv |
|
587 |
|
nfmpt1 |
|
588 |
586 587
|
nfco |
|
589 |
|
nfcv |
|
590 |
588 589
|
nfima |
|
591 |
|
nfcv |
|
592 |
590 591
|
nfxp |
|
593 |
|
nfcv |
|
594 |
588 593
|
nfima |
|
595 |
|
nfcv |
|
596 |
594 595
|
nfxp |
|
597 |
592 596
|
nfun |
|
598 |
597
|
dffn5f |
|
599 |
585 598
|
sylib |
|
600 |
89 582 583 584 599
|
offval2 |
|
601 |
558 581 600
|
3eqtr4rd |
|
602 |
601
|
mpteq2dva |
|
603 |
23 602
|
eqtr4d |
|