Step |
Hyp |
Ref |
Expression |
1 |
|
fourierswlem.t |
|
2 |
|
fourierswlem.f |
|
3 |
|
fourierswlem.x |
|
4 |
|
fourierswlem.y |
|
5 |
|
simpr |
|
6 |
|
2z |
|
7 |
6
|
a1i |
|
8 |
|
pirp |
|
9 |
|
mod0 |
|
10 |
3 8 9
|
mp2an |
|
11 |
10
|
biimpi |
|
12 |
11
|
adantr |
|
13 |
|
divides |
|
14 |
7 12 13
|
syl2anc |
|
15 |
5 14
|
mpbid |
|
16 |
|
2cnd |
|
17 |
|
picn |
|
18 |
17
|
a1i |
|
19 |
|
zcn |
|
20 |
16 18 19
|
mulassd |
|
21 |
18 19
|
mulcld |
|
22 |
16 21
|
mulcomd |
|
23 |
20 22
|
eqtrd |
|
24 |
23
|
adantr |
|
25 |
18 19 16
|
mulassd |
|
26 |
25
|
adantr |
|
27 |
|
id |
|
28 |
27
|
eqcomd |
|
29 |
28
|
adantl |
|
30 |
3
|
recni |
|
31 |
30
|
a1i |
|
32 |
17
|
a1i |
|
33 |
19
|
adantr |
|
34 |
|
2cnd |
|
35 |
33 34
|
mulcld |
|
36 |
|
pire |
|
37 |
|
pipos |
|
38 |
36 37
|
gt0ne0ii |
|
39 |
38
|
a1i |
|
40 |
31 32 35 39
|
divmuld |
|
41 |
29 40
|
mpbid |
|
42 |
24 26 41
|
3eqtrrd |
|
43 |
1
|
a1i |
|
44 |
42 43
|
oveq12d |
|
45 |
16 18
|
mulcld |
|
46 |
|
2ne0 |
|
47 |
46
|
a1i |
|
48 |
38
|
a1i |
|
49 |
16 18 47 48
|
mulne0d |
|
50 |
19 45 49
|
divcan3d |
|
51 |
50
|
adantr |
|
52 |
44 51
|
eqtrd |
|
53 |
|
simpl |
|
54 |
52 53
|
eqeltrd |
|
55 |
54
|
ex |
|
56 |
55
|
a1i |
|
57 |
56
|
rexlimdv |
|
58 |
15 57
|
mpd |
|
59 |
|
2re |
|
60 |
59 36
|
remulcli |
|
61 |
1 60
|
eqeltri |
|
62 |
|
2pos |
|
63 |
59 36 62 37
|
mulgt0ii |
|
64 |
63 1
|
breqtrri |
|
65 |
61 64
|
elrpii |
|
66 |
|
mod0 |
|
67 |
3 65 66
|
mp2an |
|
68 |
58 67
|
sylibr |
|
69 |
68
|
orcd |
|
70 |
|
odd2np1 |
|
71 |
10 70
|
sylbi |
|
72 |
71
|
biimpa |
|
73 |
16 19
|
mulcld |
|
74 |
73
|
adantr |
|
75 |
|
1cnd |
|
76 |
17
|
a1i |
|
77 |
74 75 76
|
adddird |
|
78 |
16 19
|
mulcomd |
|
79 |
78
|
oveq1d |
|
80 |
19 16 18
|
mulassd |
|
81 |
1
|
eqcomi |
|
82 |
81
|
a1i |
|
83 |
82
|
oveq2d |
|
84 |
79 80 83
|
3eqtrd |
|
85 |
17
|
mulid2i |
|
86 |
85
|
a1i |
|
87 |
84 86
|
oveq12d |
|
88 |
87
|
adantr |
|
89 |
1 45
|
eqeltrid |
|
90 |
19 89
|
mulcld |
|
91 |
90 18
|
addcomd |
|
92 |
91
|
adantr |
|
93 |
77 88 92
|
3eqtrrd |
|
94 |
|
peano2cn |
|
95 |
73 94
|
syl |
|
96 |
95 18
|
mulcomd |
|
97 |
96
|
adantr |
|
98 |
|
id |
|
99 |
98
|
eqcomd |
|
100 |
99
|
adantl |
|
101 |
30
|
a1i |
|
102 |
95
|
adantr |
|
103 |
38
|
a1i |
|
104 |
101 76 102 103
|
divmuld |
|
105 |
100 104
|
mpbid |
|
106 |
93 97 105
|
3eqtrrd |
|
107 |
106
|
oveq1d |
|
108 |
|
modcyc |
|
109 |
36 65 108
|
mp3an12 |
|
110 |
109
|
adantr |
|
111 |
36
|
a1i |
|
112 |
65
|
a1i |
|
113 |
|
0re |
|
114 |
113 36 37
|
ltleii |
|
115 |
114
|
a1i |
|
116 |
|
2timesgt |
|
117 |
8 116
|
ax-mp |
|
118 |
117 1
|
breqtrri |
|
119 |
118
|
a1i |
|
120 |
|
modid |
|
121 |
111 112 115 119 120
|
syl22anc |
|
122 |
107 110 121
|
3eqtrd |
|
123 |
122
|
ex |
|
124 |
123
|
a1i |
|
125 |
124
|
rexlimdv |
|
126 |
72 125
|
mpd |
|
127 |
126
|
olcd |
|
128 |
69 127
|
pm2.61dan |
|
129 |
|
0xr |
|
130 |
36
|
rexri |
|
131 |
|
iocgtlb |
|
132 |
129 130 131
|
mp3an12 |
|
133 |
132
|
gt0ne0d |
|
134 |
133
|
neneqd |
|
135 |
|
pm2.53 |
|
136 |
135
|
imp |
|
137 |
128 134 136
|
syl2anr |
|
138 |
129
|
a1i |
|
139 |
130
|
a1i |
|
140 |
|
modcl |
|
141 |
3 65 140
|
mp2an |
|
142 |
141
|
rexri |
|
143 |
142
|
a1i |
|
144 |
|
id |
|
145 |
37 144
|
breqtrrid |
|
146 |
36
|
eqlei2 |
|
147 |
138 139 143 145 146
|
eliocd |
|
148 |
147
|
iftrued |
|
149 |
148
|
adantl |
|
150 |
|
oveq1 |
|
151 |
150
|
breq1d |
|
152 |
151
|
ifbid |
|
153 |
|
1ex |
|
154 |
|
negex |
|
155 |
153 154
|
ifex |
|
156 |
152 2 155
|
fvmpt |
|
157 |
3 156
|
ax-mp |
|
158 |
141
|
a1i |
|
159 |
|
id |
|
160 |
158 159
|
ltned |
|
161 |
160
|
necon2bi |
|
162 |
161
|
iffalsed |
|
163 |
157 162
|
syl5eq |
|
164 |
163
|
adantl |
|
165 |
149 164
|
oveq12d |
|
166 |
|
1pneg1e0 |
|
167 |
165 166
|
eqtrdi |
|
168 |
167
|
oveq1d |
|
169 |
168
|
adantll |
|
170 |
|
2cn |
|
171 |
170 46
|
div0i |
|
172 |
171
|
a1i |
|
173 |
|
iftrue |
|
174 |
4 173
|
syl5req |
|
175 |
174
|
ad2antlr |
|
176 |
169 172 175
|
3eqtrrd |
|
177 |
137 176
|
mpdan |
|
178 |
|
iftrue |
|
179 |
178
|
adantr |
|
180 |
141
|
a1i |
|
181 |
36
|
a1i |
|
182 |
|
iocleub |
|
183 |
129 130 182
|
mp3an12 |
|
184 |
183
|
adantr |
|
185 |
|
ax-1cn |
|
186 |
185 17
|
mulcomi |
|
187 |
85 186
|
eqtr3i |
|
188 |
187
|
oveq1i |
|
189 |
170 17
|
mulcomi |
|
190 |
1 189
|
eqtri |
|
191 |
190
|
oveq1i |
|
192 |
113 64
|
gtneii |
|
193 |
3 61 192
|
redivcli |
|
194 |
|
flcl |
|
195 |
193 194
|
ax-mp |
|
196 |
|
zcn |
|
197 |
195 196
|
ax-mp |
|
198 |
17 170 197
|
mulassi |
|
199 |
191 198
|
eqtri |
|
200 |
199
|
oveq2i |
|
201 |
170 197
|
mulcli |
|
202 |
17 185 201
|
adddii |
|
203 |
188 200 202
|
3eqtr4ri |
|
204 |
203
|
a1i |
|
205 |
|
id |
|
206 |
|
modval |
|
207 |
3 65 206
|
mp2an |
|
208 |
205 207
|
eqtrdi |
|
209 |
208
|
oveq1d |
|
210 |
30
|
a1i |
|
211 |
61
|
recni |
|
212 |
211 197
|
mulcli |
|
213 |
212
|
a1i |
|
214 |
210 213
|
npcand |
|
215 |
204 209 214
|
3eqtrrd |
|
216 |
215
|
oveq1d |
|
217 |
185 201
|
addcli |
|
218 |
217 17 38
|
divcan3i |
|
219 |
216 218
|
eqtrdi |
|
220 |
|
1z |
|
221 |
|
zmulcl |
|
222 |
6 195 221
|
mp2an |
|
223 |
|
zaddcl |
|
224 |
220 222 223
|
mp2an |
|
225 |
224
|
a1i |
|
226 |
219 225
|
eqeltrd |
|
227 |
226 10
|
sylibr |
|
228 |
227
|
necon3bi |
|
229 |
228
|
adantl |
|
230 |
180 181 184 229
|
leneltd |
|
231 |
|
iftrue |
|
232 |
157 231
|
syl5eq |
|
233 |
230 232
|
syl |
|
234 |
179 233
|
oveq12d |
|
235 |
234
|
oveq1d |
|
236 |
|
1p1e2 |
|
237 |
236
|
oveq1i |
|
238 |
|
2div2e1 |
|
239 |
237 238
|
eqtr2i |
|
240 |
233 239
|
eqtr2di |
|
241 |
|
iffalse |
|
242 |
4 241
|
syl5req |
|
243 |
242
|
adantl |
|
244 |
235 240 243
|
3eqtrrd |
|
245 |
177 244
|
pm2.61dan |
|
246 |
133
|
necon2bi |
|
247 |
246
|
iffalsed |
|
248 |
|
id |
|
249 |
248 37
|
eqbrtrdi |
|
250 |
249
|
iftrued |
|
251 |
157 250
|
syl5eq |
|
252 |
247 251
|
oveq12d |
|
253 |
252
|
oveq1d |
|
254 |
|
neg1cn |
|
255 |
185 254 166
|
addcomli |
|
256 |
255
|
oveq1i |
|
257 |
256 171
|
eqtri |
|
258 |
257
|
a1i |
|
259 |
1
|
oveq2i |
|
260 |
|
2cnne0 |
|
261 |
17 38
|
pm3.2i |
|
262 |
|
divdiv1 |
|
263 |
30 260 261 262
|
mp3an |
|
264 |
30 170 17 46 38
|
divdiv32i |
|
265 |
259 263 264
|
3eqtr2i |
|
266 |
265
|
oveq2i |
|
267 |
30 17 38
|
divcli |
|
268 |
267 170 46
|
divcan2i |
|
269 |
266 268
|
eqtr2i |
|
270 |
6
|
a1i |
|
271 |
|
id |
|
272 |
270 271
|
zmulcld |
|
273 |
269 272
|
eqeltrid |
|
274 |
67 273
|
sylbi |
|
275 |
274 10
|
sylibr |
|
276 |
275
|
iftrued |
|
277 |
4 276
|
syl5req |
|
278 |
253 258 277
|
3eqtrrd |
|
279 |
278
|
adantl |
|
280 |
130
|
a1i |
|
281 |
61
|
rexri |
|
282 |
281
|
a1i |
|
283 |
141
|
a1i |
|
284 |
|
pm4.56 |
|
285 |
284
|
biimpi |
|
286 |
|
olc |
|
287 |
286
|
adantl |
|
288 |
129
|
a1i |
|
289 |
130
|
a1i |
|
290 |
142
|
a1i |
|
291 |
|
0red |
|
292 |
141
|
a1i |
|
293 |
|
modge0 |
|
294 |
3 65 293
|
mp2an |
|
295 |
294
|
a1i |
|
296 |
|
neqne |
|
297 |
291 292 295 296
|
leneltd |
|
298 |
297
|
adantl |
|
299 |
|
simpl |
|
300 |
288 289 290 298 299
|
eliocd |
|
301 |
300
|
orcd |
|
302 |
287 301
|
pm2.61dan |
|
303 |
285 302
|
nsyl |
|
304 |
36
|
a1i |
|
305 |
304 283
|
ltnled |
|
306 |
303 305
|
mpbird |
|
307 |
|
modlt |
|
308 |
3 65 307
|
mp2an |
|
309 |
308
|
a1i |
|
310 |
280 282 283 306 309
|
eliood |
|
311 |
129
|
a1i |
|
312 |
36
|
a1i |
|
313 |
142
|
a1i |
|
314 |
|
ioogtlb |
|
315 |
130 281 314
|
mp3an12 |
|
316 |
311 312 313 315
|
gtnelioc |
|
317 |
316
|
iffalsed |
|
318 |
141
|
a1i |
|
319 |
312 318 315
|
ltnsymd |
|
320 |
319
|
iffalsed |
|
321 |
157 320
|
syl5eq |
|
322 |
317 321
|
oveq12d |
|
323 |
322
|
oveq1d |
|
324 |
|
df-2 |
|
325 |
324
|
negeqi |
|
326 |
185 185
|
negdii |
|
327 |
325 326
|
eqtr2i |
|
328 |
327
|
oveq1i |
|
329 |
|
divneg |
|
330 |
170 170 46 329
|
mp3an |
|
331 |
238
|
negeqi |
|
332 |
328 330 331
|
3eqtr2i |
|
333 |
332
|
a1i |
|
334 |
4
|
a1i |
|
335 |
312 318
|
ltnled |
|
336 |
315 335
|
mpbid |
|
337 |
248 114
|
eqbrtrdi |
|
338 |
337
|
adantl |
|
339 |
128
|
orcanai |
|
340 |
339 146
|
syl |
|
341 |
338 340
|
pm2.61dan |
|
342 |
336 341
|
nsyl |
|
343 |
342
|
iffalsed |
|
344 |
334 343 321
|
3eqtrrd |
|
345 |
323 333 344
|
3eqtrrd |
|
346 |
310 345
|
syl |
|
347 |
279 346
|
pm2.61dan |
|
348 |
245 347
|
pm2.61i |
|