Step |
Hyp |
Ref |
Expression |
1 |
|
hspmbllem2.h |
|
2 |
|
hspmbllem2.x |
|
3 |
|
hspmbllem2.k |
|
4 |
|
hspmbllem2.y |
|
5 |
|
hspmbllem2.e |
|
6 |
|
hspmbllem2.c |
|
7 |
|
hspmbllem2.d |
|
8 |
|
hspmbllem2.a |
|
9 |
|
hspmbllem2.g |
|
10 |
|
hspmbllem2.r |
|
11 |
|
hspmbllem2.i |
|
12 |
|
hspmbllem2.f |
|
13 |
|
hspmbllem2.l |
|
14 |
|
hspmbllem2.t |
|
15 |
|
hspmbllem2.s |
|
16 |
11 12
|
readdcld |
|
17 |
5
|
rpred |
|
18 |
10 17
|
readdcld |
|
19 |
|
nfv |
|
20 |
|
nnex |
|
21 |
20
|
a1i |
|
22 |
|
icossicc |
|
23 |
2
|
adantr |
|
24 |
6
|
ffvelrnda |
|
25 |
|
elmapi |
|
26 |
24 25
|
syl |
|
27 |
7
|
ffvelrnda |
|
28 |
|
elmapi |
|
29 |
27 28
|
syl |
|
30 |
13 23 26 29
|
hoidmvcl |
|
31 |
22 30
|
sselid |
|
32 |
19 21 31
|
sge0clmpt |
|
33 |
|
ne0i |
|
34 |
3 33
|
syl |
|
35 |
34
|
adantr |
|
36 |
13 23 35 26 29
|
hoidmvn0val |
|
37 |
36
|
mpteq2dva |
|
38 |
37
|
fveq2d |
|
39 |
38 9
|
eqbrtrd |
|
40 |
18 32 39
|
ge0lere |
|
41 |
4
|
adantr |
|
42 |
14 41 23 29
|
hsphoif |
|
43 |
13 23 26 42
|
hoidmvcl |
|
44 |
22 43
|
sselid |
|
45 |
19 21 44
|
sge0clmpt |
|
46 |
|
oveq2 |
|
47 |
|
eqeq1 |
|
48 |
|
prodeq1 |
|
49 |
47 48
|
ifbieq2d |
|
50 |
46 46 49
|
mpoeq123dv |
|
51 |
50
|
cbvmptv |
|
52 |
13 51
|
eqtri |
|
53 |
|
diffi |
|
54 |
2 53
|
syl |
|
55 |
|
snfi |
|
56 |
55
|
a1i |
|
57 |
|
unfi |
|
58 |
54 56 57
|
syl2anc |
|
59 |
58
|
adantr |
|
60 |
|
snidg |
|
61 |
3 60
|
syl |
|
62 |
|
elun2 |
|
63 |
61 62
|
syl |
|
64 |
|
neldifsnd |
|
65 |
63 64
|
eldifd |
|
66 |
65
|
adantr |
|
67 |
|
eqid |
|
68 |
|
eqid |
|
69 |
|
uncom |
|
70 |
69
|
a1i |
|
71 |
3
|
snssd |
|
72 |
|
undif |
|
73 |
71 72
|
sylib |
|
74 |
70 73
|
eqtrd |
|
75 |
74
|
adantr |
|
76 |
75
|
feq2d |
|
77 |
26 76
|
mpbird |
|
78 |
75
|
feq2d |
|
79 |
29 78
|
mpbird |
|
80 |
52 59 66 67 41 68 77 79
|
hsphoidmvle |
|
81 |
74
|
fveq2d |
|
82 |
|
eqidd |
|
83 |
14
|
a1i |
|
84 |
74
|
oveq2d |
|
85 |
74
|
mpteq1d |
|
86 |
84 85
|
mpteq12dv |
|
87 |
86
|
eqcomd |
|
88 |
87
|
mpteq2dv |
|
89 |
83 88
|
eqtr2d |
|
90 |
89
|
fveq1d |
|
91 |
90
|
fveq1d |
|
92 |
81 82 91
|
oveq123d |
|
93 |
92
|
adantr |
|
94 |
81
|
adantr |
|
95 |
94
|
oveqd |
|
96 |
93 95
|
breq12d |
|
97 |
80 96
|
mpbid |
|
98 |
19 21 44 31 97
|
sge0lempt |
|
99 |
40 45 98
|
ge0lere |
|
100 |
15 41 23 26
|
hoidifhspf |
|
101 |
13 23 100 29
|
hoidmvcl |
|
102 |
101
|
fmpttd |
|
103 |
22
|
a1i |
|
104 |
102 103
|
fssd |
|
105 |
21 104
|
sge0cl |
|
106 |
22 101
|
sselid |
|
107 |
3
|
adantr |
|
108 |
13 23 26 29 107 15 41
|
hoidifhspdmvle |
|
109 |
19 21 106 31 108
|
sge0lempt |
|
110 |
40 105 109
|
ge0lere |
|
111 |
4
|
adantr |
|
112 |
2
|
adantr |
|
113 |
|
eleq1w |
|
114 |
113
|
anbi2d |
|
115 |
|
fveq2 |
|
116 |
115
|
feq1d |
|
117 |
114 116
|
imbi12d |
|
118 |
117 29
|
chvarvv |
|
119 |
14 111 112 118
|
hsphoif |
|
120 |
|
reex |
|
121 |
120
|
a1i |
|
122 |
121 2
|
jca |
|
123 |
122
|
adantr |
|
124 |
|
elmapg |
|
125 |
123 124
|
syl |
|
126 |
119 125
|
mpbird |
|
127 |
126
|
fmpttd |
|
128 |
|
simpl |
|
129 |
|
elinel1 |
|
130 |
129
|
adantl |
|
131 |
8
|
sselda |
|
132 |
|
eliun |
|
133 |
131 132
|
sylib |
|
134 |
128 130 133
|
syl2anc |
|
135 |
|
simpll |
|
136 |
|
elinel2 |
|
137 |
136
|
adantl |
|
138 |
137
|
adantr |
|
139 |
|
simpr |
|
140 |
|
ixpfn |
|
141 |
140
|
adantl |
|
142 |
|
nfv |
|
143 |
|
nfcv |
|
144 |
|
nfixp1 |
|
145 |
143 144
|
nfel |
|
146 |
142 145
|
nfan |
|
147 |
26
|
3adant3 |
|
148 |
|
simp3 |
|
149 |
147 148
|
ffvelrnd |
|
150 |
149
|
rexrd |
|
151 |
150
|
ad5ant135 |
|
152 |
42
|
3adant3 |
|
153 |
152 148
|
ffvelrnd |
|
154 |
153
|
rexrd |
|
155 |
154
|
ad5ant135 |
|
156 |
|
iftrue |
|
157 |
|
ioossre |
|
158 |
157
|
a1i |
|
159 |
156 158
|
eqsstrd |
|
160 |
|
iffalse |
|
161 |
|
ssid |
|
162 |
161
|
a1i |
|
163 |
160 162
|
eqsstrd |
|
164 |
159 163
|
pm2.61i |
|
165 |
|
simpr |
|
166 |
1 2 3 4
|
hspval |
|
167 |
166
|
adantr |
|
168 |
165 167
|
eleqtrd |
|
169 |
168
|
adantr |
|
170 |
|
simpr |
|
171 |
|
vex |
|
172 |
171
|
elixp |
|
173 |
172
|
biimpi |
|
174 |
173
|
simprd |
|
175 |
174
|
adantr |
|
176 |
|
simpr |
|
177 |
|
rspa |
|
178 |
175 176 177
|
syl2anc |
|
179 |
169 170 178
|
syl2anc |
|
180 |
164 179
|
sselid |
|
181 |
180
|
rexrd |
|
182 |
181
|
ad4ant14 |
|
183 |
150
|
ad4ant124 |
|
184 |
29
|
3adant3 |
|
185 |
184 148
|
ffvelrnd |
|
186 |
185
|
rexrd |
|
187 |
186
|
ad4ant124 |
|
188 |
171
|
elixp |
|
189 |
188
|
biimpi |
|
190 |
189
|
simprd |
|
191 |
190
|
adantr |
|
192 |
|
simpr |
|
193 |
|
rspa |
|
194 |
191 192 193
|
syl2anc |
|
195 |
194
|
adantll |
|
196 |
|
icogelb |
|
197 |
183 187 195 196
|
syl3anc |
|
198 |
197
|
adantl3r |
|
199 |
|
icoltub |
|
200 |
183 187 195 199
|
syl3anc |
|
201 |
200
|
adantl3r |
|
202 |
201
|
ad2antrr |
|
203 |
|
simpll |
|
204 |
|
simpr |
|
205 |
203 204
|
jca |
|
206 |
205
|
3ad2ant1 |
|
207 |
|
simp2 |
|
208 |
|
simp3 |
|
209 |
|
fveq2 |
|
210 |
209
|
breq1d |
|
211 |
210
|
biimpa |
|
212 |
211
|
iftrued |
|
213 |
209
|
eqcomd |
|
214 |
213
|
adantr |
|
215 |
212 214
|
eqtrd |
|
216 |
215
|
3adant1 |
|
217 |
|
breq2 |
|
218 |
|
id |
|
219 |
217 218
|
ifbieq2d |
|
220 |
219
|
ifeq2d |
|
221 |
220
|
mpteq2dv |
|
222 |
221
|
mpteq2dv |
|
223 |
|
ovex |
|
224 |
223
|
mptex |
|
225 |
224
|
a1i |
|
226 |
14 222 4 225
|
fvmptd3 |
|
227 |
226
|
adantr |
|
228 |
|
fveq1 |
|
229 |
228
|
breq1d |
|
230 |
229 228
|
ifbieq1d |
|
231 |
228 230
|
ifeq12d |
|
232 |
231
|
mpteq2dv |
|
233 |
232
|
adantl |
|
234 |
|
mptexg |
|
235 |
2 234
|
syl |
|
236 |
235
|
adantr |
|
237 |
227 233 27 236
|
fvmptd |
|
238 |
237
|
fveq1d |
|
239 |
238
|
3adant3 |
|
240 |
|
simpl |
|
241 |
|
simpr |
|
242 |
240 3
|
syl |
|
243 |
241 242
|
eqeltrd |
|
244 |
|
eqidd |
|
245 |
|
eleq1w |
|
246 |
|
fveq2 |
|
247 |
246
|
breq1d |
|
248 |
247 246
|
ifbieq1d |
|
249 |
245 246 248
|
ifbieq12d |
|
250 |
249
|
adantl |
|
251 |
|
simpr |
|
252 |
|
fvexd |
|
253 |
252 4
|
ifexd |
|
254 |
252 253
|
ifexd |
|
255 |
254
|
adantr |
|
256 |
244 250 251 255
|
fvmptd |
|
257 |
240 243 256
|
syl2anc |
|
258 |
|
eleq1 |
|
259 |
210 209
|
ifbieq1d |
|
260 |
258 209 259
|
ifbieq12d |
|
261 |
260
|
adantl |
|
262 |
257 261
|
eqtrd |
|
263 |
262
|
3adant2 |
|
264 |
|
neldifsnd |
|
265 |
264
|
iffalsed |
|
266 |
265
|
3ad2ant3 |
|
267 |
239 263 266
|
3eqtrrd |
|
268 |
267
|
3expa |
|
269 |
268
|
3adant3 |
|
270 |
216 269
|
eqtr3d |
|
271 |
206 207 208 270
|
syl3anc |
|
272 |
271
|
ad5ant145 |
|
273 |
202 272
|
breqtrd |
|
274 |
|
mnfxr |
|
275 |
274
|
a1i |
|
276 |
4
|
rexrd |
|
277 |
276
|
adantr |
|
278 |
277
|
3ad2ant1 |
|
279 |
179
|
3adant3 |
|
280 |
156
|
3ad2ant3 |
|
281 |
279 280
|
eleqtrd |
|
282 |
|
iooltub |
|
283 |
275 278 281 282
|
syl3anc |
|
284 |
283
|
3adant1r |
|
285 |
284
|
ad4ant123 |
|
286 |
|
simpr |
|
287 |
210
|
notbid |
|
288 |
287
|
adantr |
|
289 |
286 288
|
mpbid |
|
290 |
289
|
iffalsed |
|
291 |
|
eqidd |
|
292 |
290 291
|
eqtr2d |
|
293 |
292
|
adantll |
|
294 |
268
|
adantlr |
|
295 |
294
|
adantl3r |
|
296 |
295
|
adantr |
|
297 |
293 296
|
eqtrd |
|
298 |
285 297
|
breqtrd |
|
299 |
298
|
adantl3r |
|
300 |
273 299
|
pm2.61dan |
|
301 |
201
|
adantr |
|
302 |
237
|
3adant3 |
|
303 |
249
|
adantl |
|
304 |
255
|
3adant2 |
|
305 |
302 303 148 304
|
fvmptd |
|
306 |
305
|
3expa |
|
307 |
306
|
adantllr |
|
308 |
307
|
ad4ant13 |
|
309 |
|
simpl |
|
310 |
|
neqne |
|
311 |
|
nelsn |
|
312 |
310 311
|
syl |
|
313 |
312
|
adantl |
|
314 |
309 313
|
eldifd |
|
315 |
314
|
iftrued |
|
316 |
315
|
adantll |
|
317 |
308 316
|
eqtr2d |
|
318 |
301 317
|
breqtrd |
|
319 |
300 318
|
pm2.61dan |
|
320 |
151 155 182 198 319
|
elicod |
|
321 |
320
|
ex |
|
322 |
146 321
|
ralrimi |
|
323 |
141 322
|
jca |
|
324 |
171
|
elixp |
|
325 |
323 324
|
sylibr |
|
326 |
325
|
ex |
|
327 |
135 138 139 326
|
syl21anc |
|
328 |
327
|
reximdva |
|
329 |
134 328
|
mpd |
|
330 |
|
eliun |
|
331 |
329 330
|
sylibr |
|
332 |
331
|
ralrimiva |
|
333 |
|
dfss3 |
|
334 |
332 333
|
sylibr |
|
335 |
|
eqidd |
|
336 |
|
2fveq3 |
|
337 |
336
|
adantl |
|
338 |
|
id |
|
339 |
|
fvexd |
|
340 |
335 337 338 339
|
fvmptd |
|
341 |
340
|
fveq1d |
|
342 |
341
|
oveq2d |
|
343 |
342
|
ixpeq2dv |
|
344 |
343
|
iuneq2i |
|
345 |
334 344
|
sseqtrrdi |
|
346 |
2 6 127 345 13
|
ovnlecvr2 |
|
347 |
340
|
oveq2d |
|
348 |
347
|
mpteq2ia |
|
349 |
348
|
fveq2i |
|
350 |
349
|
a1i |
|
351 |
346 350
|
breqtrd |
|
352 |
6
|
ffvelrnda |
|
353 |
|
elmapi |
|
354 |
352 353
|
syl |
|
355 |
15 111 112 354
|
hoidifhspf |
|
356 |
|
elmapg |
|
357 |
122 356
|
syl |
|
358 |
357
|
adantr |
|
359 |
355 358
|
mpbird |
|
360 |
359
|
fmpttd |
|
361 |
|
simpl |
|
362 |
|
eldifi |
|
363 |
362
|
adantl |
|
364 |
361 363 133
|
syl2anc |
|
365 |
140
|
adantl |
|
366 |
|
nfv |
|
367 |
366 145
|
nfan |
|
368 |
100
|
3adant3 |
|
369 |
368 148
|
ffvelrnd |
|
370 |
369
|
rexrd |
|
371 |
370
|
ad5ant135 |
|
372 |
187
|
adantl3r |
|
373 |
149
|
3expa |
|
374 |
186
|
3expa |
|
375 |
|
icossre |
|
376 |
373 374 375
|
syl2anc |
|
377 |
376
|
adantlr |
|
378 |
377 195
|
sseldd |
|
379 |
378
|
rexrd |
|
380 |
379
|
adantl3r |
|
381 |
41
|
3adant3 |
|
382 |
23
|
3adant3 |
|
383 |
15 381 382 147 148
|
hoidifhspval3 |
|
384 |
383
|
ad5ant134 |
|
385 |
|
iftrue |
|
386 |
385
|
adantl |
|
387 |
384 386
|
eqtrd |
|
388 |
387
|
adantllr |
|
389 |
|
iftrue |
|
390 |
389
|
adantl |
|
391 |
197
|
adantl3r |
|
392 |
391
|
ad2antrr |
|
393 |
390 392
|
eqbrtrd |
|
394 |
|
iffalse |
|
395 |
394
|
adantl |
|
396 |
|
simpl1 |
|
397 |
|
simpr |
|
398 |
|
fveq2 |
|
399 |
398
|
breq2d |
|
400 |
399
|
notbid |
|
401 |
400
|
adantr |
|
402 |
397 401
|
mpbid |
|
403 |
402
|
3ad2antl3 |
|
404 |
398
|
eqcomd |
|
405 |
404
|
3ad2ant3 |
|
406 |
364
|
adantr |
|
407 |
|
id |
|
408 |
407
|
ad4ant13 |
|
409 |
|
simpr |
|
410 |
251
|
ad2antrr |
|
411 |
408 409 410 378
|
syl21anc |
|
412 |
411
|
rexlimdva2 |
|
413 |
412
|
adantlr |
|
414 |
406 413
|
mpd |
|
415 |
414
|
3adant3 |
|
416 |
405 415
|
eqeltrd |
|
417 |
416
|
adantr |
|
418 |
396 361 4
|
3syl |
|
419 |
417 418
|
ltnled |
|
420 |
403 419
|
mpbird |
|
421 |
365 364
|
r19.29a |
|
422 |
421
|
adantr |
|
423 |
274
|
a1i |
|
424 |
276
|
ad4antr |
|
425 |
414
|
ad4ant13 |
|
426 |
425
|
mnfltd |
|
427 |
398
|
adantl |
|
428 |
|
simpl |
|
429 |
427 428
|
eqbrtrd |
|
430 |
429
|
ad4ant24 |
|
431 |
423 424 425 426 430
|
eliood |
|
432 |
156
|
eqcomd |
|
433 |
432
|
adantl |
|
434 |
431 433
|
eleqtrd |
|
435 |
414
|
ad4ant13 |
|
436 |
160
|
eqcomd |
|
437 |
436
|
adantl |
|
438 |
435 437
|
eleqtrd |
|
439 |
434 438
|
pm2.61dan |
|
440 |
439
|
ralrimiva |
|
441 |
422 440
|
jca |
|
442 |
396 420 441
|
syl2anc |
|
443 |
442 172
|
sylibr |
|
444 |
166
|
eqcomd |
|
445 |
444
|
ad2antrr |
|
446 |
445
|
3ad2antl1 |
|
447 |
443 446
|
eleqtrd |
|
448 |
|
eldifn |
|
449 |
448
|
adantl |
|
450 |
449
|
3ad2ant1 |
|
451 |
450
|
adantr |
|
452 |
447 451
|
condan |
|
453 |
452
|
ad5ant145 |
|
454 |
453
|
adantr |
|
455 |
395 454
|
eqbrtrd |
|
456 |
393 455
|
pm2.61dan |
|
457 |
388 456
|
eqbrtrd |
|
458 |
383
|
ad5ant124 |
|
459 |
|
iffalse |
|
460 |
459
|
adantl |
|
461 |
458 460
|
eqtrd |
|
462 |
197
|
adantr |
|
463 |
461 462
|
eqbrtrd |
|
464 |
463
|
adantl4r |
|
465 |
457 464
|
pm2.61dan |
|
466 |
200
|
adantl3r |
|
467 |
371 372 380 465 466
|
elicod |
|
468 |
467
|
ex |
|
469 |
367 468
|
ralrimi |
|
470 |
365 469
|
jca |
|
471 |
171
|
elixp |
|
472 |
470 471
|
sylibr |
|
473 |
|
eqidd |
|
474 |
|
2fveq3 |
|
475 |
474
|
adantl |
|
476 |
|
fvexd |
|
477 |
473 475 338 476
|
fvmptd |
|
478 |
477
|
fveq1d |
|
479 |
478
|
oveq1d |
|
480 |
479
|
ixpeq2dv |
|
481 |
480
|
ad2antlr |
|
482 |
481
|
eleq2d |
|
483 |
472 482
|
mpbird |
|
484 |
483
|
ex |
|
485 |
484
|
reximdva |
|
486 |
364 485
|
mpd |
|
487 |
|
eliun |
|
488 |
486 487
|
sylibr |
|
489 |
488
|
ralrimiva |
|
490 |
|
dfss3 |
|
491 |
489 490
|
sylibr |
|
492 |
2 360 7 491 13
|
ovnlecvr2 |
|
493 |
477
|
oveq1d |
|
494 |
493
|
mpteq2ia |
|
495 |
494
|
fveq2i |
|
496 |
495
|
a1i |
|
497 |
492 496
|
breqtrd |
|
498 |
11 12 99 110 351 497
|
leadd12dd |
|
499 |
23 107 41 26 29 13 14 15
|
hspmbllem1 |
|
500 |
499
|
mpteq2dva |
|
501 |
500
|
fveq2d |
|
502 |
19 21 44 106
|
sge0xadd |
|
503 |
99 110
|
rexaddd |
|
504 |
501 502 503
|
3eqtrrd |
|
505 |
498 504
|
breqtrd |
|
506 |
16 40 18 505 39
|
letrd |
|