Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem93.1 |
|
2 |
|
fourierdlem93.2 |
|
3 |
|
fourierdlem93.3 |
|
4 |
|
fourierdlem93.4 |
|
5 |
|
fourierdlem93.5 |
|
6 |
|
fourierdlem93.6 |
|
7 |
|
fourierdlem93.7 |
|
8 |
|
fourierdlem93.8 |
|
9 |
|
fourierdlem93.9 |
|
10 |
1
|
fourierdlem2 |
|
11 |
3 10
|
syl |
|
12 |
4 11
|
mpbid |
|
13 |
12
|
simprd |
|
14 |
13
|
simplld |
|
15 |
14
|
eqcomd |
|
16 |
13
|
simplrd |
|
17 |
16
|
eqcomd |
|
18 |
15 17
|
oveq12d |
|
19 |
18
|
itgeq1d |
|
20 |
|
0zd |
|
21 |
|
nnuz |
|
22 |
3 21
|
eleqtrdi |
|
23 |
|
1e0p1 |
|
24 |
23
|
a1i |
|
25 |
24
|
fveq2d |
|
26 |
22 25
|
eleqtrd |
|
27 |
1 3 4
|
fourierdlem15 |
|
28 |
|
pire |
|
29 |
28
|
renegcli |
|
30 |
|
iccssre |
|
31 |
29 28 30
|
mp2an |
|
32 |
31
|
a1i |
|
33 |
27 32
|
fssd |
|
34 |
13
|
simprd |
|
35 |
34
|
r19.21bi |
|
36 |
6
|
adantr |
|
37 |
|
simpr |
|
38 |
18
|
eqcomd |
|
39 |
38
|
adantr |
|
40 |
37 39
|
eleqtrd |
|
41 |
36 40
|
ffvelrnd |
|
42 |
33
|
adantr |
|
43 |
|
elfzofz |
|
44 |
43
|
adantl |
|
45 |
42 44
|
ffvelrnd |
|
46 |
|
fzofzp1 |
|
47 |
46
|
adantl |
|
48 |
42 47
|
ffvelrnd |
|
49 |
6
|
feqmptd |
|
50 |
49
|
adantr |
|
51 |
50
|
reseq1d |
|
52 |
|
ioossicc |
|
53 |
52
|
a1i |
|
54 |
29
|
rexri |
|
55 |
54
|
a1i |
|
56 |
28
|
rexri |
|
57 |
56
|
a1i |
|
58 |
27
|
ad2antrr |
|
59 |
|
simplr |
|
60 |
|
simpr |
|
61 |
55 57 58 59 60
|
fourierdlem1 |
|
62 |
61
|
ralrimiva |
|
63 |
|
dfss3 |
|
64 |
62 63
|
sylibr |
|
65 |
53 64
|
sstrd |
|
66 |
65
|
resmptd |
|
67 |
51 66
|
eqtrd |
|
68 |
67
|
eqcomd |
|
69 |
68 7
|
eqeltrd |
|
70 |
67
|
oveq1d |
|
71 |
9 70
|
eleqtrd |
|
72 |
67
|
oveq1d |
|
73 |
8 72
|
eleqtrd |
|
74 |
45 48 69 71 73
|
iblcncfioo |
|
75 |
6
|
ad2antrr |
|
76 |
75 61
|
ffvelrnd |
|
77 |
45 48 74 76
|
ibliooicc |
|
78 |
20 26 33 35 41 77
|
itgspltprt |
|
79 |
|
fvres |
|
80 |
79
|
eqcomd |
|
81 |
80
|
adantl |
|
82 |
81
|
itgeq2dv |
|
83 |
|
eqid |
|
84 |
6
|
adantr |
|
85 |
84 64
|
fssresd |
|
86 |
53
|
resabs1d |
|
87 |
86 7
|
eqeltrd |
|
88 |
86
|
oveq1d |
|
89 |
45 48 35 85
|
limcicciooub |
|
90 |
88 89
|
eqtr3d |
|
91 |
9 90
|
eleqtrd |
|
92 |
86
|
eqcomd |
|
93 |
92
|
oveq1d |
|
94 |
45 48 35 85
|
limciccioolb |
|
95 |
93 94
|
eqtrd |
|
96 |
8 95
|
eleqtrd |
|
97 |
5
|
adantr |
|
98 |
83 45 48 35 85 87 91 96 97
|
fourierdlem82 |
|
99 |
45
|
adantr |
|
100 |
48
|
adantr |
|
101 |
5
|
ad2antrr |
|
102 |
99 101
|
resubcld |
|
103 |
100 101
|
resubcld |
|
104 |
|
simpr |
|
105 |
|
eliccre |
|
106 |
102 103 104 105
|
syl3anc |
|
107 |
101 106
|
readdcld |
|
108 |
|
elicc2 |
|
109 |
102 103 108
|
syl2anc |
|
110 |
104 109
|
mpbid |
|
111 |
110
|
simp2d |
|
112 |
99 101 106
|
lesubadd2d |
|
113 |
111 112
|
mpbid |
|
114 |
110
|
simp3d |
|
115 |
101 106 100
|
leaddsub2d |
|
116 |
114 115
|
mpbird |
|
117 |
99 100 107 113 116
|
eliccd |
|
118 |
|
fvres |
|
119 |
117 118
|
syl |
|
120 |
119
|
itgeq2dv |
|
121 |
82 98 120
|
3eqtrd |
|
122 |
121
|
sumeq2dv |
|
123 |
|
oveq2 |
|
124 |
123
|
fveq2d |
|
125 |
124
|
cbvitgv |
|
126 |
125
|
a1i |
|
127 |
2
|
a1i |
|
128 |
|
fveq2 |
|
129 |
128
|
oveq1d |
|
130 |
129
|
adantl |
|
131 |
3
|
nnzd |
|
132 |
|
0le0 |
|
133 |
132
|
a1i |
|
134 |
|
0red |
|
135 |
3
|
nnred |
|
136 |
3
|
nngt0d |
|
137 |
134 135 136
|
ltled |
|
138 |
20 131 20 133 137
|
elfzd |
|
139 |
14 29
|
eqeltrdi |
|
140 |
139 5
|
resubcld |
|
141 |
127 130 138 140
|
fvmptd |
|
142 |
14
|
oveq1d |
|
143 |
141 142
|
eqtr2d |
|
144 |
|
fveq2 |
|
145 |
144
|
oveq1d |
|
146 |
145
|
adantl |
|
147 |
135
|
leidd |
|
148 |
20 131 131 137 147
|
elfzd |
|
149 |
16 28
|
eqeltrdi |
|
150 |
149 5
|
resubcld |
|
151 |
127 146 148 150
|
fvmptd |
|
152 |
16
|
oveq1d |
|
153 |
151 152
|
eqtr2d |
|
154 |
143 153
|
oveq12d |
|
155 |
154
|
itgeq1d |
|
156 |
33
|
ffvelrnda |
|
157 |
5
|
adantr |
|
158 |
156 157
|
resubcld |
|
159 |
158 2
|
fmptd |
|
160 |
45 48 97 35
|
ltsub1dd |
|
161 |
44 158
|
syldan |
|
162 |
2
|
fvmpt2 |
|
163 |
44 161 162
|
syl2anc |
|
164 |
|
fveq2 |
|
165 |
164
|
oveq1d |
|
166 |
165
|
cbvmptv |
|
167 |
2 166
|
eqtri |
|
168 |
167
|
a1i |
|
169 |
|
fveq2 |
|
170 |
169
|
oveq1d |
|
171 |
170
|
adantl |
|
172 |
48 97
|
resubcld |
|
173 |
168 171 47 172
|
fvmptd |
|
174 |
160 163 173
|
3brtr4d |
|
175 |
|
frn |
|
176 |
6 175
|
syl |
|
177 |
176
|
adantr |
|
178 |
|
ffun |
|
179 |
6 178
|
syl |
|
180 |
179
|
adantr |
|
181 |
29
|
a1i |
|
182 |
28
|
a1i |
|
183 |
5
|
adantr |
|
184 |
141 140
|
eqeltrd |
|
185 |
184
|
adantr |
|
186 |
151 150
|
eqeltrd |
|
187 |
186
|
adantr |
|
188 |
|
simpr |
|
189 |
|
eliccre |
|
190 |
185 187 188 189
|
syl3anc |
|
191 |
183 190
|
readdcld |
|
192 |
128
|
adantl |
|
193 |
192
|
oveq1d |
|
194 |
127 193 138 140
|
fvmptd |
|
195 |
194
|
oveq2d |
|
196 |
5
|
recnd |
|
197 |
139
|
recnd |
|
198 |
196 197
|
pncan3d |
|
199 |
195 198 14
|
3eqtrrd |
|
200 |
199
|
adantr |
|
201 |
|
elicc2 |
|
202 |
185 187 201
|
syl2anc |
|
203 |
188 202
|
mpbid |
|
204 |
203
|
simp2d |
|
205 |
185 190 183 204
|
leadd2dd |
|
206 |
200 205
|
eqbrtrd |
|
207 |
203
|
simp3d |
|
208 |
190 187 183 207
|
leadd2dd |
|
209 |
151
|
oveq2d |
|
210 |
149
|
recnd |
|
211 |
196 210
|
pncan3d |
|
212 |
209 211 16
|
3eqtrrd |
|
213 |
212
|
adantr |
|
214 |
208 213
|
breqtrrd |
|
215 |
181 182 191 206 214
|
eliccd |
|
216 |
|
fdm |
|
217 |
6 216
|
syl |
|
218 |
217
|
eqcomd |
|
219 |
218
|
adantr |
|
220 |
215 219
|
eleqtrd |
|
221 |
|
fvelrn |
|
222 |
180 220 221
|
syl2anc |
|
223 |
177 222
|
sseldd |
|
224 |
163 161
|
eqeltrd |
|
225 |
173 172
|
eqeltrd |
|
226 |
84 65
|
fssresd |
|
227 |
45
|
rexrd |
|
228 |
227
|
adantr |
|
229 |
48
|
rexrd |
|
230 |
229
|
adantr |
|
231 |
5
|
ad2antrr |
|
232 |
|
elioore |
|
233 |
232
|
adantl |
|
234 |
231 233
|
readdcld |
|
235 |
163
|
oveq2d |
|
236 |
196
|
adantr |
|
237 |
45
|
recnd |
|
238 |
236 237
|
pncan3d |
|
239 |
|
eqidd |
|
240 |
235 238 239
|
3eqtrrd |
|
241 |
240
|
adantr |
|
242 |
224
|
adantr |
|
243 |
|
simpr |
|
244 |
242
|
rexrd |
|
245 |
225
|
rexrd |
|
246 |
245
|
adantr |
|
247 |
|
elioo2 |
|
248 |
244 246 247
|
syl2anc |
|
249 |
243 248
|
mpbid |
|
250 |
249
|
simp2d |
|
251 |
242 233 231 250
|
ltadd2dd |
|
252 |
241 251
|
eqbrtrd |
|
253 |
225
|
adantr |
|
254 |
249
|
simp3d |
|
255 |
233 253 231 254
|
ltadd2dd |
|
256 |
173
|
oveq2d |
|
257 |
48
|
recnd |
|
258 |
236 257
|
pncan3d |
|
259 |
256 258
|
eqtrd |
|
260 |
259
|
adantr |
|
261 |
255 260
|
breqtrd |
|
262 |
228 230 234 252 261
|
eliood |
|
263 |
|
eqid |
|
264 |
262 263
|
fmptd |
|
265 |
|
fcompt |
|
266 |
226 264 265
|
syl2anc |
|
267 |
|
oveq2 |
|
268 |
267
|
cbvmptv |
|
269 |
268
|
fveq1i |
|
270 |
269
|
fveq2i |
|
271 |
270
|
mpteq2i |
|
272 |
271
|
a1i |
|
273 |
|
fveq2 |
|
274 |
273
|
fveq2d |
|
275 |
274
|
cbvmptv |
|
276 |
275
|
a1i |
|
277 |
|
eqidd |
|
278 |
|
oveq2 |
|
279 |
278
|
adantl |
|
280 |
277 279 243 234
|
fvmptd |
|
281 |
280
|
fveq2d |
|
282 |
|
fvres |
|
283 |
262 282
|
syl |
|
284 |
281 283
|
eqtrd |
|
285 |
284
|
mpteq2dva |
|
286 |
272 276 285
|
3eqtrd |
|
287 |
266 286
|
eqtr2d |
|
288 |
|
eqid |
|
289 |
|
ssid |
|
290 |
289
|
a1i |
|
291 |
|
id |
|
292 |
290 291 290
|
constcncfg |
|
293 |
|
cncfmptid |
|
294 |
289 289 293
|
mp2an |
|
295 |
294
|
a1i |
|
296 |
292 295
|
addcncf |
|
297 |
236 296
|
syl |
|
298 |
|
ioosscn |
|
299 |
298
|
a1i |
|
300 |
|
ioosscn |
|
301 |
300
|
a1i |
|
302 |
288 297 299 301 262
|
cncfmptssg |
|
303 |
302 7
|
cncfco |
|
304 |
287 303
|
eqeltrd |
|
305 |
227
|
adantr |
|
306 |
229
|
adantr |
|
307 |
|
simpr |
|
308 |
|
vex |
|
309 |
263
|
elrnmpt |
|
310 |
308 309
|
ax-mp |
|
311 |
307 310
|
sylib |
|
312 |
|
nfv |
|
313 |
|
nfmpt1 |
|
314 |
313
|
nfrn |
|
315 |
314
|
nfcri |
|
316 |
312 315
|
nfan |
|
317 |
|
nfv |
|
318 |
|
simp3 |
|
319 |
5
|
3ad2ant1 |
|
320 |
232
|
3ad2ant2 |
|
321 |
319 320
|
readdcld |
|
322 |
318 321
|
eqeltrd |
|
323 |
322
|
3exp |
|
324 |
323
|
ad2antrr |
|
325 |
316 317 324
|
rexlimd |
|
326 |
311 325
|
mpd |
|
327 |
|
nfv |
|
328 |
252
|
3adant3 |
|
329 |
|
simp3 |
|
330 |
328 329
|
breqtrrd |
|
331 |
330
|
3exp |
|
332 |
331
|
adantr |
|
333 |
316 327 332
|
rexlimd |
|
334 |
311 333
|
mpd |
|
335 |
|
nfv |
|
336 |
261
|
3adant3 |
|
337 |
329 336
|
eqbrtrd |
|
338 |
337
|
3exp |
|
339 |
338
|
adantr |
|
340 |
316 335 339
|
rexlimd |
|
341 |
311 340
|
mpd |
|
342 |
305 306 326 334 341
|
eliood |
|
343 |
217
|
ineq2d |
|
344 |
343
|
adantr |
|
345 |
|
dmres |
|
346 |
345
|
a1i |
|
347 |
|
dfss |
|
348 |
65 347
|
sylib |
|
349 |
344 346 348
|
3eqtr4d |
|
350 |
349
|
adantr |
|
351 |
342 350
|
eleqtrrd |
|
352 |
326 341
|
ltned |
|
353 |
352
|
neneqd |
|
354 |
|
velsn |
|
355 |
353 354
|
sylnibr |
|
356 |
351 355
|
eldifd |
|
357 |
356
|
ralrimiva |
|
358 |
|
dfss3 |
|
359 |
357 358
|
sylibr |
|
360 |
|
eqid |
|
361 |
196
|
adantr |
|
362 |
|
simpr |
|
363 |
361 362
|
addcomd |
|
364 |
363
|
mpteq2dva |
|
365 |
|
eqid |
|
366 |
365
|
addccncf |
|
367 |
196 366
|
syl |
|
368 |
364 367
|
eqeltrd |
|
369 |
368
|
adantr |
|
370 |
224
|
rexrd |
|
371 |
|
iocssre |
|
372 |
370 225 371
|
syl2anc |
|
373 |
|
ax-resscn |
|
374 |
372 373
|
sstrdi |
|
375 |
289
|
a1i |
|
376 |
196
|
ad2antrr |
|
377 |
374
|
sselda |
|
378 |
376 377
|
addcld |
|
379 |
360 369 374 375 378
|
cncfmptssg |
|
380 |
|
eqid |
|
381 |
|
eqid |
|
382 |
380
|
cnfldtop |
|
383 |
|
unicntop |
|
384 |
383
|
restid |
|
385 |
382 384
|
ax-mp |
|
386 |
385
|
eqcomi |
|
387 |
380 381 386
|
cncfcn |
|
388 |
374 375 387
|
syl2anc |
|
389 |
379 388
|
eleqtrd |
|
390 |
380
|
cnfldtopon |
|
391 |
390
|
a1i |
|
392 |
|
resttopon |
|
393 |
391 374 392
|
syl2anc |
|
394 |
|
cncnp |
|
395 |
393 391 394
|
syl2anc |
|
396 |
389 395
|
mpbid |
|
397 |
396
|
simprd |
|
398 |
|
ubioc1 |
|
399 |
370 245 174 398
|
syl3anc |
|
400 |
|
fveq2 |
|
401 |
400
|
eleq2d |
|
402 |
401
|
rspccva |
|
403 |
397 399 402
|
syl2anc |
|
404 |
|
ioounsn |
|
405 |
370 245 174 404
|
syl3anc |
|
406 |
259
|
eqcomd |
|
407 |
406
|
ad2antrr |
|
408 |
|
iftrue |
|
409 |
408
|
adantl |
|
410 |
|
oveq2 |
|
411 |
410
|
adantl |
|
412 |
407 409 411
|
3eqtr4d |
|
413 |
|
iffalse |
|
414 |
413
|
adantl |
|
415 |
|
eqidd |
|
416 |
|
oveq2 |
|
417 |
416
|
adantl |
|
418 |
|
velsn |
|
419 |
418
|
notbii |
|
420 |
|
elun |
|
421 |
420
|
biimpi |
|
422 |
421
|
orcomd |
|
423 |
422
|
ord |
|
424 |
419 423
|
syl5bir |
|
425 |
424
|
imp |
|
426 |
425
|
adantll |
|
427 |
5
|
ad2antrr |
|
428 |
|
elioore |
|
429 |
428
|
adantl |
|
430 |
|
elsni |
|
431 |
430
|
adantl |
|
432 |
225
|
adantr |
|
433 |
431 432
|
eqeltrd |
|
434 |
429 433
|
jaodan |
|
435 |
420 434
|
sylan2b |
|
436 |
427 435
|
readdcld |
|
437 |
436
|
adantr |
|
438 |
415 417 426 437
|
fvmptd |
|
439 |
414 438
|
eqtrd |
|
440 |
412 439
|
pm2.61dan |
|
441 |
405 440
|
mpteq12dva |
|
442 |
405
|
oveq2d |
|
443 |
442
|
oveq1d |
|
444 |
443
|
fveq1d |
|
445 |
403 441 444
|
3eltr4d |
|
446 |
|
eqid |
|
447 |
|
eqid |
|
448 |
264 301
|
fssd |
|
449 |
225
|
recnd |
|
450 |
446 380 447 448 299 449
|
ellimc |
|
451 |
445 450
|
mpbird |
|
452 |
359 451 9
|
limccog |
|
453 |
266 286
|
eqtrd |
|
454 |
453
|
oveq1d |
|
455 |
452 454
|
eleqtrd |
|
456 |
45
|
adantr |
|
457 |
456 334
|
gtned |
|
458 |
457
|
neneqd |
|
459 |
|
velsn |
|
460 |
458 459
|
sylnibr |
|
461 |
351 460
|
eldifd |
|
462 |
461
|
ralrimiva |
|
463 |
|
dfss3 |
|
464 |
462 463
|
sylibr |
|
465 |
|
icossre |
|
466 |
224 245 465
|
syl2anc |
|
467 |
466 373
|
sstrdi |
|
468 |
196
|
ad2antrr |
|
469 |
467
|
sselda |
|
470 |
468 469
|
addcld |
|
471 |
360 369 467 375 470
|
cncfmptssg |
|
472 |
|
eqid |
|
473 |
380 472 386
|
cncfcn |
|
474 |
467 375 473
|
syl2anc |
|
475 |
471 474
|
eleqtrd |
|
476 |
|
resttopon |
|
477 |
391 467 476
|
syl2anc |
|
478 |
|
cncnp |
|
479 |
477 391 478
|
syl2anc |
|
480 |
475 479
|
mpbid |
|
481 |
480
|
simprd |
|
482 |
|
lbico1 |
|
483 |
370 245 174 482
|
syl3anc |
|
484 |
|
fveq2 |
|
485 |
484
|
eleq2d |
|
486 |
485
|
rspccva |
|
487 |
481 483 486
|
syl2anc |
|
488 |
|
uncom |
|
489 |
|
snunioo |
|
490 |
370 245 174 489
|
syl3anc |
|
491 |
488 490
|
syl5eq |
|
492 |
|
iftrue |
|
493 |
492
|
adantl |
|
494 |
240
|
adantr |
|
495 |
|
oveq2 |
|
496 |
495
|
eqcomd |
|
497 |
496
|
adantl |
|
498 |
493 494 497
|
3eqtrd |
|
499 |
498
|
adantlr |
|
500 |
|
iffalse |
|
501 |
500
|
adantl |
|
502 |
|
eqidd |
|
503 |
416
|
adantl |
|
504 |
|
velsn |
|
505 |
504
|
notbii |
|
506 |
|
elun |
|
507 |
506
|
biimpi |
|
508 |
507
|
orcomd |
|
509 |
508
|
ord |
|
510 |
505 509
|
syl5bir |
|
511 |
510
|
imp |
|
512 |
511
|
adantll |
|
513 |
5
|
ad2antrr |
|
514 |
|
elsni |
|
515 |
514
|
adantl |
|
516 |
224
|
adantr |
|
517 |
515 516
|
eqeltrd |
|
518 |
429 517
|
jaodan |
|
519 |
506 518
|
sylan2b |
|
520 |
513 519
|
readdcld |
|
521 |
520
|
adantr |
|
522 |
502 503 512 521
|
fvmptd |
|
523 |
501 522
|
eqtrd |
|
524 |
499 523
|
pm2.61dan |
|
525 |
491 524
|
mpteq12dva |
|
526 |
491
|
oveq2d |
|
527 |
526
|
oveq1d |
|
528 |
527
|
fveq1d |
|
529 |
487 525 528
|
3eltr4d |
|
530 |
|
eqid |
|
531 |
|
eqid |
|
532 |
224
|
recnd |
|
533 |
530 380 531 448 299 532
|
ellimc |
|
534 |
529 533
|
mpbird |
|
535 |
464 534 8
|
limccog |
|
536 |
453
|
oveq1d |
|
537 |
535 536
|
eleqtrd |
|
538 |
224 225 304 455 537
|
iblcncfioo |
|
539 |
6
|
ad2antrr |
|
540 |
54
|
a1i |
|
541 |
56
|
a1i |
|
542 |
27
|
ad2antrr |
|
543 |
|
simplr |
|
544 |
|
simpr |
|
545 |
163 173
|
oveq12d |
|
546 |
545
|
adantr |
|
547 |
544 546
|
eleqtrd |
|
548 |
547 117
|
syldan |
|
549 |
540 541 542 543 548
|
fourierdlem1 |
|
550 |
539 549
|
ffvelrnd |
|
551 |
224 225 538 550
|
ibliooicc |
|
552 |
20 26 159 174 223 551
|
itgspltprt |
|
553 |
545
|
itgeq1d |
|
554 |
553
|
sumeq2dv |
|
555 |
552 554
|
eqtrd |
|
556 |
126 155 555
|
3eqtrd |
|
557 |
122 556
|
eqtr4d |
|
558 |
19 78 557
|
3eqtrd |
|