Step |
Hyp |
Ref |
Expression |
1 |
|
constr0.1 |
|
2 |
|
constrconj.1 |
|
3 |
|
constrconj.2 |
|
4 |
|
fveq2 |
|
5 |
4
|
eleq2d |
|
6 |
4 5
|
raleqbidv |
|
7 |
|
fveq2 |
|
8 |
7
|
eleq2d |
|
9 |
7 8
|
raleqbidv |
|
10 |
|
fveq2 |
|
11 |
10
|
eleq2d |
|
12 |
10 11
|
raleqbidv |
|
13 |
|
fveq2 |
|
14 |
13
|
eleq2d |
|
15 |
13 14
|
raleqbidv |
|
16 |
|
simpr |
|
17 |
16
|
fveq2d |
|
18 |
|
cj0 |
|
19 |
17 18
|
eqtrdi |
|
20 |
|
c0ex |
|
21 |
20
|
prid1 |
|
22 |
21
|
a1i |
|
23 |
19 22
|
eqeltrd |
|
24 |
1
|
constr0 |
|
25 |
24
|
a1i |
|
26 |
23 25
|
eleqtrrd |
|
27 |
|
simpr |
|
28 |
27
|
fveq2d |
|
29 |
|
1re |
|
30 |
|
cjre |
|
31 |
29 30
|
ax-mp |
|
32 |
28 31
|
eqtrdi |
|
33 |
|
1ex |
|
34 |
33
|
prid2 |
|
35 |
34
|
a1i |
|
36 |
32 35
|
eqeltrd |
|
37 |
24
|
a1i |
|
38 |
36 37
|
eleqtrrd |
|
39 |
24
|
eleq2i |
|
40 |
39
|
biimpi |
|
41 |
|
elpri |
|
42 |
40 41
|
syl |
|
43 |
26 38 42
|
mpjaodan |
|
44 |
43
|
rgen |
|
45 |
|
simpl |
|
46 |
|
eqid |
|
47 |
1 45 46
|
constrsuc |
|
48 |
47
|
biimpa |
|
49 |
48
|
simpld |
|
50 |
49
|
cjcld |
|
51 |
48
|
simprd |
|
52 |
|
fveq2 |
|
53 |
52
|
eleq1d |
|
54 |
|
simp-4r |
|
55 |
|
simplr |
|
56 |
53 54 55
|
rspcdva |
|
57 |
|
id |
|
58 |
|
oveq2 |
|
59 |
58
|
oveq2d |
|
60 |
57 59
|
oveq12d |
|
61 |
60
|
eqeq2d |
|
62 |
58
|
fveq2d |
|
63 |
62
|
oveq1d |
|
64 |
63
|
fveq2d |
|
65 |
64
|
neeq1d |
|
66 |
61 65
|
3anbi13d |
|
67 |
66
|
rexbidv |
|
68 |
67
|
2rexbidv |
|
69 |
68
|
2rexbidv |
|
70 |
69
|
adantl |
|
71 |
|
fveq2 |
|
72 |
71
|
eleq1d |
|
73 |
|
simp-5r |
|
74 |
|
simplr |
|
75 |
72 73 74
|
rspcdva |
|
76 |
|
oveq1 |
|
77 |
76
|
oveq2d |
|
78 |
77
|
oveq2d |
|
79 |
78
|
eqeq2d |
|
80 |
76
|
fveq2d |
|
81 |
80
|
oveq1d |
|
82 |
81
|
fveq2d |
|
83 |
82
|
neeq1d |
|
84 |
79 83
|
3anbi13d |
|
85 |
84
|
2rexbidv |
|
86 |
85
|
2rexbidv |
|
87 |
86
|
adantl |
|
88 |
|
fveq2 |
|
89 |
88
|
eleq1d |
|
90 |
|
simp-6r |
|
91 |
|
simplr |
|
92 |
89 90 91
|
rspcdva |
|
93 |
|
id |
|
94 |
|
oveq2 |
|
95 |
94
|
oveq2d |
|
96 |
93 95
|
oveq12d |
|
97 |
96
|
eqeq2d |
|
98 |
94
|
oveq2d |
|
99 |
98
|
fveq2d |
|
100 |
99
|
neeq1d |
|
101 |
97 100
|
3anbi23d |
|
102 |
101
|
rexbidv |
|
103 |
102
|
2rexbidv |
|
104 |
103
|
adantl |
|
105 |
|
fveq2 |
|
106 |
105
|
eleq1d |
|
107 |
|
simp-7r |
|
108 |
|
simplr |
|
109 |
106 107 108
|
rspcdva |
|
110 |
|
oveq1 |
|
111 |
110
|
oveq2d |
|
112 |
111
|
oveq2d |
|
113 |
112
|
eqeq2d |
|
114 |
110
|
oveq2d |
|
115 |
114
|
fveq2d |
|
116 |
115
|
neeq1d |
|
117 |
113 116
|
3anbi23d |
|
118 |
117
|
2rexbidv |
|
119 |
118
|
adantl |
|
120 |
|
simpr1 |
|
121 |
120
|
fveq2d |
|
122 |
|
id |
|
123 |
1 122
|
constrsscn |
|
124 |
123
|
ad9antr |
|
125 |
|
simp-7r |
|
126 |
124 125
|
sseldd |
|
127 |
|
simpllr |
|
128 |
127
|
recnd |
|
129 |
|
simp-6r |
|
130 |
124 129
|
sseldd |
|
131 |
130 126
|
subcld |
|
132 |
128 131
|
mulcld |
|
133 |
126 132
|
cjaddd |
|
134 |
128 131
|
cjmuld |
|
135 |
127
|
cjred |
|
136 |
130 126
|
cjsubd |
|
137 |
135 136
|
oveq12d |
|
138 |
134 137
|
eqtrd |
|
139 |
138
|
oveq2d |
|
140 |
121 133 139
|
3eqtrd |
|
141 |
|
simpr2 |
|
142 |
141
|
fveq2d |
|
143 |
|
simp-5r |
|
144 |
124 143
|
sseldd |
|
145 |
|
simplr |
|
146 |
145
|
recnd |
|
147 |
|
simp-4r |
|
148 |
124 147
|
sseldd |
|
149 |
148 144
|
subcld |
|
150 |
146 149
|
mulcld |
|
151 |
144 150
|
cjaddd |
|
152 |
146 149
|
cjmuld |
|
153 |
145
|
cjred |
|
154 |
148 144
|
cjsubd |
|
155 |
153 154
|
oveq12d |
|
156 |
152 155
|
eqtrd |
|
157 |
156
|
oveq2d |
|
158 |
142 151 157
|
3eqtrd |
|
159 |
131
|
cjcjd |
|
160 |
159
|
oveq1d |
|
161 |
131
|
cjcld |
|
162 |
161 149
|
cjmuld |
|
163 |
130
|
cjcld |
|
164 |
126
|
cjcld |
|
165 |
163 164
|
cjsubd |
|
166 |
130
|
cjcjd |
|
167 |
126
|
cjcjd |
|
168 |
166 167
|
oveq12d |
|
169 |
165 168
|
eqtrd |
|
170 |
154
|
eqcomd |
|
171 |
169 170
|
oveq12d |
|
172 |
160 162 171
|
3eqtr4rd |
|
173 |
172
|
fveq2d |
|
174 |
161 149
|
mulcld |
|
175 |
174
|
imcjd |
|
176 |
173 175
|
eqtrd |
|
177 |
|
simpr3 |
|
178 |
174
|
imcld |
|
179 |
178
|
recnd |
|
180 |
179
|
negne0bd |
|
181 |
177 180
|
mpbid |
|
182 |
176 181
|
eqnetrd |
|
183 |
140 158 182
|
3jca |
|
184 |
183
|
ex |
|
185 |
184
|
reximdva |
|
186 |
185
|
reximdva |
|
187 |
186
|
imp |
|
188 |
109 119 187
|
rspcedvd |
|
189 |
188
|
r19.29an |
|
190 |
92 104 189
|
rspcedvd |
|
191 |
190
|
r19.29an |
|
192 |
75 87 191
|
rspcedvd |
|
193 |
192
|
r19.29an |
|
194 |
56 70 193
|
rspcedvd |
|
195 |
194
|
r19.29an |
|
196 |
|
id |
|
197 |
|
oveq2 |
|
198 |
197
|
oveq2d |
|
199 |
196 198
|
oveq12d |
|
200 |
199
|
eqeq2d |
|
201 |
197
|
fveq2d |
|
202 |
201
|
oveq1d |
|
203 |
202
|
fveq2d |
|
204 |
203
|
neeq1d |
|
205 |
200 204
|
3anbi13d |
|
206 |
205
|
rexbidv |
|
207 |
206
|
2rexbidv |
|
208 |
207
|
2rexbidv |
|
209 |
208
|
cbvrexvw |
|
210 |
|
oveq1 |
|
211 |
210
|
oveq2d |
|
212 |
211
|
oveq2d |
|
213 |
212
|
eqeq2d |
|
214 |
210
|
fveq2d |
|
215 |
214
|
oveq1d |
|
216 |
215
|
fveq2d |
|
217 |
216
|
neeq1d |
|
218 |
213 217
|
3anbi13d |
|
219 |
218
|
2rexbidv |
|
220 |
219
|
2rexbidv |
|
221 |
220
|
cbvrexvw |
|
222 |
|
id |
|
223 |
|
oveq2 |
|
224 |
223
|
oveq2d |
|
225 |
222 224
|
oveq12d |
|
226 |
225
|
eqeq2d |
|
227 |
223
|
oveq2d |
|
228 |
227
|
fveq2d |
|
229 |
228
|
neeq1d |
|
230 |
226 229
|
3anbi23d |
|
231 |
230
|
rexbidv |
|
232 |
231
|
2rexbidv |
|
233 |
232
|
cbvrexvw |
|
234 |
|
oveq1 |
|
235 |
234
|
oveq2d |
|
236 |
235
|
oveq2d |
|
237 |
236
|
eqeq2d |
|
238 |
234
|
oveq2d |
|
239 |
238
|
fveq2d |
|
240 |
239
|
neeq1d |
|
241 |
237 240
|
3anbi23d |
|
242 |
241
|
2rexbidv |
|
243 |
242
|
cbvrexvw |
|
244 |
243
|
rexbii |
|
245 |
233 244
|
bitri |
|
246 |
245
|
rexbii |
|
247 |
221 246
|
bitri |
|
248 |
247
|
rexbii |
|
249 |
209 248
|
bitri |
|
250 |
195 249
|
sylibr |
|
251 |
250
|
ex |
|
252 |
|
simp-4r |
|
253 |
|
simplr |
|
254 |
53 252 253
|
rspcdva |
|
255 |
61
|
anbi1d |
|
256 |
255
|
rexbidv |
|
257 |
256
|
2rexbidv |
|
258 |
257
|
2rexbidv |
|
259 |
258
|
adantl |
|
260 |
|
simp-5r |
|
261 |
|
simplr |
|
262 |
72 260 261
|
rspcdva |
|
263 |
79
|
anbi1d |
|
264 |
263
|
2rexbidv |
|
265 |
264
|
2rexbidv |
|
266 |
265
|
adantl |
|
267 |
|
simp-6r |
|
268 |
|
simplr |
|
269 |
89 267 268
|
rspcdva |
|
270 |
|
oveq2 |
|
271 |
270
|
fveq2d |
|
272 |
271
|
eqeq1d |
|
273 |
272
|
anbi2d |
|
274 |
273
|
rexbidv |
|
275 |
274
|
2rexbidv |
|
276 |
275
|
adantl |
|
277 |
|
fveq2 |
|
278 |
277
|
eleq1d |
|
279 |
|
simp-7r |
|
280 |
|
simplr |
|
281 |
278 279 280
|
rspcdva |
|
282 |
|
oveq1 |
|
283 |
282
|
fveq2d |
|
284 |
283
|
eqeq2d |
|
285 |
284
|
anbi2d |
|
286 |
285
|
2rexbidv |
|
287 |
286
|
adantl |
|
288 |
|
fveq2 |
|
289 |
288
|
eleq1d |
|
290 |
|
simp-8r |
|
291 |
|
simplr |
|
292 |
289 290 291
|
rspcdva |
|
293 |
|
oveq2 |
|
294 |
293
|
fveq2d |
|
295 |
294
|
eqeq2d |
|
296 |
295
|
anbi2d |
|
297 |
296
|
rexbidv |
|
298 |
297
|
adantl |
|
299 |
|
simprl |
|
300 |
299
|
fveq2d |
|
301 |
123
|
ad9antr |
|
302 |
|
simp-7r |
|
303 |
301 302
|
sseldd |
|
304 |
|
simplr |
|
305 |
304
|
recnd |
|
306 |
|
simp-6r |
|
307 |
301 306
|
sseldd |
|
308 |
307 303
|
subcld |
|
309 |
305 308
|
mulcld |
|
310 |
303 309
|
cjaddd |
|
311 |
305 308
|
cjmuld |
|
312 |
304
|
cjred |
|
313 |
307 303
|
cjsubd |
|
314 |
312 313
|
oveq12d |
|
315 |
311 314
|
eqtrd |
|
316 |
315
|
oveq2d |
|
317 |
300 310 316
|
3eqtrd |
|
318 |
|
simprr |
|
319 |
49
|
ad7antr |
|
320 |
|
simp-5r |
|
321 |
301 320
|
sseldd |
|
322 |
319 321
|
subcld |
|
323 |
322
|
abscjd |
|
324 |
|
simp-4r |
|
325 |
301 324
|
sseldd |
|
326 |
|
simpllr |
|
327 |
301 326
|
sseldd |
|
328 |
325 327
|
subcld |
|
329 |
328
|
abscjd |
|
330 |
318 323 329
|
3eqtr4d |
|
331 |
319 321
|
cjsubd |
|
332 |
331
|
fveq2d |
|
333 |
325 327
|
cjsubd |
|
334 |
333
|
fveq2d |
|
335 |
330 332 334
|
3eqtr3d |
|
336 |
317 335
|
jca |
|
337 |
336
|
ex |
|
338 |
337
|
reximdva |
|
339 |
338
|
imp |
|
340 |
292 298 339
|
rspcedvd |
|
341 |
340
|
r19.29an |
|
342 |
281 287 341
|
rspcedvd |
|
343 |
342
|
r19.29an |
|
344 |
269 276 343
|
rspcedvd |
|
345 |
344
|
r19.29an |
|
346 |
262 266 345
|
rspcedvd |
|
347 |
346
|
r19.29an |
|
348 |
254 259 347
|
rspcedvd |
|
349 |
348
|
r19.29an |
|
350 |
200
|
anbi1d |
|
351 |
350
|
rexbidv |
|
352 |
351
|
2rexbidv |
|
353 |
352
|
2rexbidv |
|
354 |
353
|
cbvrexvw |
|
355 |
213
|
anbi1d |
|
356 |
355
|
2rexbidv |
|
357 |
356
|
2rexbidv |
|
358 |
357
|
cbvrexvw |
|
359 |
|
oveq2 |
|
360 |
359
|
fveq2d |
|
361 |
360
|
eqeq1d |
|
362 |
361
|
anbi2d |
|
363 |
362
|
rexbidv |
|
364 |
363
|
2rexbidv |
|
365 |
364
|
cbvrexvw |
|
366 |
|
oveq1 |
|
367 |
366
|
fveq2d |
|
368 |
367
|
eqeq2d |
|
369 |
368
|
anbi2d |
|
370 |
369
|
2rexbidv |
|
371 |
370
|
cbvrexvw |
|
372 |
|
oveq2 |
|
373 |
372
|
fveq2d |
|
374 |
373
|
eqeq2d |
|
375 |
374
|
anbi2d |
|
376 |
375
|
rexbidv |
|
377 |
376
|
cbvrexvw |
|
378 |
377
|
rexbii |
|
379 |
371 378
|
bitri |
|
380 |
379
|
rexbii |
|
381 |
365 380
|
bitri |
|
382 |
381
|
rexbii |
|
383 |
358 382
|
bitri |
|
384 |
383
|
rexbii |
|
385 |
354 384
|
bitri |
|
386 |
349 385
|
sylibr |
|
387 |
386
|
ex |
|
388 |
|
simp-4r |
|
389 |
|
simplr |
|
390 |
53 388 389
|
rspcdva |
|
391 |
|
neeq1 |
|
392 |
|
oveq2 |
|
393 |
392
|
fveq2d |
|
394 |
393
|
eqeq1d |
|
395 |
391 394
|
3anbi12d |
|
396 |
395
|
rexbidv |
|
397 |
396
|
2rexbidv |
|
398 |
397
|
2rexbidv |
|
399 |
398
|
adantl |
|
400 |
|
simp-5r |
|
401 |
|
simplr |
|
402 |
72 400 401
|
rspcdva |
|
403 |
|
oveq1 |
|
404 |
403
|
fveq2d |
|
405 |
404
|
eqeq2d |
|
406 |
405
|
3anbi2d |
|
407 |
406
|
2rexbidv |
|
408 |
407
|
2rexbidv |
|
409 |
408
|
adantl |
|
410 |
|
simp-6r |
|
411 |
|
simplr |
|
412 |
89 410 411
|
rspcdva |
|
413 |
|
oveq2 |
|
414 |
413
|
fveq2d |
|
415 |
414
|
eqeq2d |
|
416 |
415
|
3anbi2d |
|
417 |
416
|
rexbidv |
|
418 |
417
|
2rexbidv |
|
419 |
418
|
adantl |
|
420 |
|
simp-7r |
|
421 |
|
simplr |
|
422 |
106 420 421
|
rspcdva |
|
423 |
|
neeq2 |
|
424 |
|
oveq2 |
|
425 |
424
|
fveq2d |
|
426 |
425
|
eqeq1d |
|
427 |
423 426
|
3anbi13d |
|
428 |
427
|
2rexbidv |
|
429 |
428
|
adantl |
|
430 |
123
|
ad9antr |
|
431 |
|
simp-7r |
|
432 |
430 431
|
sseldd |
|
433 |
|
simp-4r |
|
434 |
430 433
|
sseldd |
|
435 |
|
simpr |
|
436 |
|
cj11 |
|
437 |
436
|
necon3bid |
|
438 |
437
|
biimpar |
|
439 |
432 434 435 438
|
syl21anc |
|
440 |
439
|
ex |
|
441 |
|
simpr |
|
442 |
49
|
ad7antr |
|
443 |
123
|
ad9antr |
|
444 |
|
simp-7r |
|
445 |
443 444
|
sseldd |
|
446 |
442 445
|
subcld |
|
447 |
446
|
abscjd |
|
448 |
|
simp-6r |
|
449 |
443 448
|
sseldd |
|
450 |
|
simp-5r |
|
451 |
443 450
|
sseldd |
|
452 |
449 451
|
subcld |
|
453 |
452
|
abscjd |
|
454 |
441 447 453
|
3eqtr4d |
|
455 |
442 445
|
cjsubd |
|
456 |
455
|
fveq2d |
|
457 |
449 451
|
cjsubd |
|
458 |
457
|
fveq2d |
|
459 |
454 456 458
|
3eqtr3d |
|
460 |
459
|
ex |
|
461 |
49
|
ad7antr |
|
462 |
123
|
ad9antr |
|
463 |
|
simp-4r |
|
464 |
462 463
|
sseldd |
|
465 |
461 464
|
subcld |
|
466 |
465
|
abscjd |
|
467 |
461 464
|
cjsubd |
|
468 |
467
|
fveq2d |
|
469 |
|
simpr |
|
470 |
466 468 469
|
3eqtr3d |
|
471 |
470
|
ex |
|
472 |
440 460 471
|
3anim123d |
|
473 |
472
|
reximdva |
|
474 |
473
|
reximdva |
|
475 |
474
|
imp |
|
476 |
422 429 475
|
rspcedvd |
|
477 |
476
|
r19.29an |
|
478 |
412 419 477
|
rspcedvd |
|
479 |
478
|
r19.29an |
|
480 |
402 409 479
|
rspcedvd |
|
481 |
480
|
r19.29an |
|
482 |
390 399 481
|
rspcedvd |
|
483 |
482
|
r19.29an |
|
484 |
|
neeq1 |
|
485 |
|
oveq2 |
|
486 |
485
|
fveq2d |
|
487 |
486
|
eqeq1d |
|
488 |
484 487
|
3anbi12d |
|
489 |
488
|
rexbidv |
|
490 |
489
|
2rexbidv |
|
491 |
490
|
2rexbidv |
|
492 |
491
|
cbvrexvw |
|
493 |
|
oveq1 |
|
494 |
493
|
fveq2d |
|
495 |
494
|
eqeq2d |
|
496 |
495
|
3anbi2d |
|
497 |
496
|
2rexbidv |
|
498 |
497
|
2rexbidv |
|
499 |
498
|
cbvrexvw |
|
500 |
|
oveq2 |
|
501 |
500
|
fveq2d |
|
502 |
501
|
eqeq2d |
|
503 |
502
|
3anbi2d |
|
504 |
503
|
rexbidv |
|
505 |
504
|
2rexbidv |
|
506 |
505
|
cbvrexvw |
|
507 |
|
neeq2 |
|
508 |
|
oveq2 |
|
509 |
508
|
fveq2d |
|
510 |
509
|
eqeq1d |
|
511 |
507 510
|
3anbi13d |
|
512 |
511
|
2rexbidv |
|
513 |
512
|
cbvrexvw |
|
514 |
513
|
rexbii |
|
515 |
506 514
|
bitri |
|
516 |
515
|
rexbii |
|
517 |
499 516
|
bitri |
|
518 |
517
|
rexbii |
|
519 |
492 518
|
bitri |
|
520 |
483 519
|
sylibr |
|
521 |
520
|
ex |
|
522 |
251 387 521
|
3orim123d |
|
523 |
51 522
|
mpd |
|
524 |
50 523
|
jca |
|
525 |
45
|
adantr |
|
526 |
1 525 46
|
constrsuc |
|
527 |
524 526
|
mpbird |
|
528 |
527
|
ralrimiva |
|
529 |
|
fveq2 |
|
530 |
529
|
eleq1d |
|
531 |
530
|
cbvralvw |
|
532 |
528 531
|
sylibr |
|
533 |
532
|
ex |
|
534 |
|
simpr |
|
535 |
|
vex |
|
536 |
535
|
a1i |
|
537 |
|
simpll |
|
538 |
1 536 537
|
constrlim |
|
539 |
534 538
|
eleqtrd |
|
540 |
|
eliun |
|
541 |
539 540
|
sylib |
|
542 |
529
|
eleq1d |
|
543 |
|
fveq2 |
|
544 |
543
|
eleq2d |
|
545 |
543 544
|
raleqbidv |
|
546 |
|
simp-4r |
|
547 |
|
simplr |
|
548 |
545 546 547
|
rspcdva |
|
549 |
|
simpr |
|
550 |
542 548 549
|
rspcdva |
|
551 |
550
|
ex |
|
552 |
551
|
reximdva |
|
553 |
541 552
|
mpd |
|
554 |
|
eliun |
|
555 |
553 554
|
sylibr |
|
556 |
555 538
|
eleqtrrd |
|
557 |
556
|
ralrimiva |
|
558 |
529
|
eleq1d |
|
559 |
558
|
cbvralvw |
|
560 |
557 559
|
sylibr |
|
561 |
560
|
ex |
|
562 |
6 9 12 15 44 533 561
|
tfinds |
|
563 |
2 562
|
syl |
|
564 |
|
fveq2 |
|
565 |
564
|
eleq1d |
|
566 |
565
|
adantl |
|
567 |
3 566
|
rspcdv |
|
568 |
563 567
|
mpd |
|