Step |
Hyp |
Ref |
Expression |
1 |
|
eulerth.1 |
|
2 |
|
eulerth.2 |
|
3 |
|
eulerth.3 |
|
4 |
|
eulerth.4 |
|
5 |
|
eulerth.5 |
|
6 |
1
|
simp1d |
|
7 |
6
|
phicld |
|
8 |
7
|
nnred |
|
9 |
8
|
leidd |
|
10 |
7
|
adantr |
|
11 |
|
breq1 |
|
12 |
11
|
anbi2d |
|
13 |
|
oveq2 |
|
14 |
|
fveq2 |
|
15 |
13 14
|
oveq12d |
|
16 |
15
|
oveq1d |
|
17 |
|
fveq2 |
|
18 |
17
|
oveq1d |
|
19 |
16 18
|
eqeq12d |
|
20 |
14
|
oveq2d |
|
21 |
20
|
eqeq1d |
|
22 |
19 21
|
anbi12d |
|
23 |
12 22
|
imbi12d |
|
24 |
|
breq1 |
|
25 |
24
|
anbi2d |
|
26 |
|
oveq2 |
|
27 |
|
fveq2 |
|
28 |
26 27
|
oveq12d |
|
29 |
28
|
oveq1d |
|
30 |
|
fveq2 |
|
31 |
30
|
oveq1d |
|
32 |
29 31
|
eqeq12d |
|
33 |
27
|
oveq2d |
|
34 |
33
|
eqeq1d |
|
35 |
32 34
|
anbi12d |
|
36 |
25 35
|
imbi12d |
|
37 |
|
breq1 |
|
38 |
37
|
anbi2d |
|
39 |
|
oveq2 |
|
40 |
|
fveq2 |
|
41 |
39 40
|
oveq12d |
|
42 |
41
|
oveq1d |
|
43 |
|
fveq2 |
|
44 |
43
|
oveq1d |
|
45 |
42 44
|
eqeq12d |
|
46 |
40
|
oveq2d |
|
47 |
46
|
eqeq1d |
|
48 |
45 47
|
anbi12d |
|
49 |
38 48
|
imbi12d |
|
50 |
|
breq1 |
|
51 |
50
|
anbi2d |
|
52 |
|
oveq2 |
|
53 |
|
fveq2 |
|
54 |
52 53
|
oveq12d |
|
55 |
54
|
oveq1d |
|
56 |
|
fveq2 |
|
57 |
56
|
oveq1d |
|
58 |
55 57
|
eqeq12d |
|
59 |
53
|
oveq2d |
|
60 |
59
|
eqeq1d |
|
61 |
58 60
|
anbi12d |
|
62 |
51 61
|
imbi12d |
|
63 |
1
|
simp2d |
|
64 |
|
f1of |
|
65 |
4 64
|
syl |
|
66 |
|
nnuz |
|
67 |
7 66
|
eleqtrdi |
|
68 |
|
eluzfz1 |
|
69 |
67 68
|
syl |
|
70 |
69 3
|
eleqtrrdi |
|
71 |
65 70
|
ffvelrnd |
|
72 |
|
oveq1 |
|
73 |
72
|
eqeq1d |
|
74 |
73 2
|
elrab2 |
|
75 |
71 74
|
sylib |
|
76 |
75
|
simpld |
|
77 |
|
elfzoelz |
|
78 |
76 77
|
syl |
|
79 |
63 78
|
zmulcld |
|
80 |
79
|
zred |
|
81 |
6
|
nnrpd |
|
82 |
|
modabs2 |
|
83 |
80 81 82
|
syl2anc |
|
84 |
|
1z |
|
85 |
|
fveq2 |
|
86 |
85
|
oveq2d |
|
87 |
86
|
oveq1d |
|
88 |
|
ovex |
|
89 |
87 5 88
|
fvmpt |
|
90 |
70 89
|
syl |
|
91 |
84 90
|
seq1i |
|
92 |
91
|
oveq1d |
|
93 |
63
|
zcnd |
|
94 |
93
|
exp1d |
|
95 |
|
seq1 |
|
96 |
84 95
|
ax-mp |
|
97 |
96
|
a1i |
|
98 |
94 97
|
oveq12d |
|
99 |
98
|
oveq1d |
|
100 |
83 92 99
|
3eqtr4rd |
|
101 |
96
|
oveq2i |
|
102 |
6
|
nnzd |
|
103 |
102 78
|
gcdcomd |
|
104 |
75
|
simprd |
|
105 |
103 104
|
eqtrd |
|
106 |
101 105
|
syl5eq |
|
107 |
100 106
|
jca |
|
108 |
107
|
adantr |
|
109 |
|
nnre |
|
110 |
109
|
adantr |
|
111 |
110
|
lep1d |
|
112 |
|
peano2re |
|
113 |
110 112
|
syl |
|
114 |
8
|
adantl |
|
115 |
|
letr |
|
116 |
110 113 114 115
|
syl3anc |
|
117 |
111 116
|
mpand |
|
118 |
117
|
imdistanda |
|
119 |
118
|
imim1d |
|
120 |
63
|
adantr |
|
121 |
|
nnnn0 |
|
122 |
121
|
ad2antrl |
|
123 |
|
zexpcl |
|
124 |
120 122 123
|
syl2anc |
|
125 |
|
simprl |
|
126 |
125 66
|
eleqtrdi |
|
127 |
109
|
ad2antrl |
|
128 |
127 112
|
syl |
|
129 |
8
|
adantr |
|
130 |
127
|
lep1d |
|
131 |
|
simprr |
|
132 |
127 128 129 130 131
|
letrd |
|
133 |
|
nnz |
|
134 |
133
|
ad2antrl |
|
135 |
7
|
nnzd |
|
136 |
135
|
adantr |
|
137 |
|
eluz |
|
138 |
134 136 137
|
syl2anc |
|
139 |
132 138
|
mpbird |
|
140 |
|
fzss2 |
|
141 |
139 140
|
syl |
|
142 |
141 3
|
sseqtrrdi |
|
143 |
142
|
sselda |
|
144 |
65
|
ffvelrnda |
|
145 |
|
oveq1 |
|
146 |
145
|
eqeq1d |
|
147 |
146 2
|
elrab2 |
|
148 |
144 147
|
sylib |
|
149 |
148
|
simpld |
|
150 |
|
elfzoelz |
|
151 |
149 150
|
syl |
|
152 |
151
|
adantlr |
|
153 |
143 152
|
syldan |
|
154 |
|
zmulcl |
|
155 |
154
|
adantl |
|
156 |
126 153 155
|
seqcl |
|
157 |
124 156
|
zmulcld |
|
158 |
157
|
zred |
|
159 |
2
|
ssrab3 |
|
160 |
1 2 3 4 5
|
eulerthlem1 |
|
161 |
160
|
ffvelrnda |
|
162 |
159 161
|
sseldi |
|
163 |
|
elfzoelz |
|
164 |
162 163
|
syl |
|
165 |
164
|
adantlr |
|
166 |
143 165
|
syldan |
|
167 |
126 166 155
|
seqcl |
|
168 |
167
|
zred |
|
169 |
65
|
adantr |
|
170 |
|
peano2nn |
|
171 |
170
|
ad2antrl |
|
172 |
171
|
nnge1d |
|
173 |
171
|
nnzd |
|
174 |
|
elfz |
|
175 |
84 174
|
mp3an2 |
|
176 |
173 136 175
|
syl2anc |
|
177 |
172 131 176
|
mpbir2and |
|
178 |
177 3
|
eleqtrrdi |
|
179 |
169 178
|
ffvelrnd |
|
180 |
|
oveq1 |
|
181 |
180
|
eqeq1d |
|
182 |
181 2
|
elrab2 |
|
183 |
179 182
|
sylib |
|
184 |
183
|
simpld |
|
185 |
|
elfzoelz |
|
186 |
184 185
|
syl |
|
187 |
120 186
|
zmulcld |
|
188 |
81
|
adantr |
|
189 |
|
modmul1 |
|
190 |
189
|
3expia |
|
191 |
158 168 187 188 190
|
syl22anc |
|
192 |
124
|
zcnd |
|
193 |
156
|
zcnd |
|
194 |
93
|
adantr |
|
195 |
186
|
zcnd |
|
196 |
192 193 194 195
|
mul4d |
|
197 |
194 122
|
expp1d |
|
198 |
|
seqp1 |
|
199 |
126 198
|
syl |
|
200 |
197 199
|
oveq12d |
|
201 |
196 200
|
eqtr4d |
|
202 |
201
|
oveq1d |
|
203 |
187
|
zred |
|
204 |
203 188
|
modcld |
|
205 |
|
modabs2 |
|
206 |
203 188 205
|
syl2anc |
|
207 |
|
modmul1 |
|
208 |
204 203 167 188 206 207
|
syl221anc |
|
209 |
|
fveq2 |
|
210 |
209
|
oveq2d |
|
211 |
210
|
oveq1d |
|
212 |
|
ovex |
|
213 |
211 5 212
|
fvmpt |
|
214 |
178 213
|
syl |
|
215 |
214
|
oveq2d |
|
216 |
|
seqp1 |
|
217 |
126 216
|
syl |
|
218 |
204
|
recnd |
|
219 |
167
|
zcnd |
|
220 |
218 219
|
mulcomd |
|
221 |
215 217 220
|
3eqtr4d |
|
222 |
221
|
oveq1d |
|
223 |
187
|
zcnd |
|
224 |
219 223
|
mulcomd |
|
225 |
224
|
oveq1d |
|
226 |
208 222 225
|
3eqtr4rd |
|
227 |
202 226
|
eqeq12d |
|
228 |
191 227
|
sylibd |
|
229 |
102
|
adantr |
|
230 |
229 186
|
gcdcomd |
|
231 |
183
|
simprd |
|
232 |
230 231
|
eqtrd |
|
233 |
|
rpmul |
|
234 |
229 156 186 233
|
syl3anc |
|
235 |
232 234
|
mpan2d |
|
236 |
199
|
oveq2d |
|
237 |
236
|
eqeq1d |
|
238 |
235 237
|
sylibrd |
|
239 |
228 238
|
anim12d |
|
240 |
239
|
an12s |
|
241 |
240
|
ex |
|
242 |
241
|
a2d |
|
243 |
119 242
|
syld |
|
244 |
23 36 49 62 108 243
|
nnind |
|
245 |
10 244
|
mpcom |
|
246 |
9 245
|
mpdan |
|
247 |
246
|
simpld |
|
248 |
7
|
nnnn0d |
|
249 |
|
zexpcl |
|
250 |
63 248 249
|
syl2anc |
|
251 |
3
|
eleq2i |
|
252 |
251 151
|
sylan2br |
|
253 |
154
|
adantl |
|
254 |
67 252 253
|
seqcl |
|
255 |
250 254
|
zmulcld |
|
256 |
|
mulcl |
|
257 |
256
|
adantl |
|
258 |
|
mulcom |
|
259 |
258
|
adantl |
|
260 |
|
mulass |
|
261 |
260
|
adantl |
|
262 |
|
ssidd |
|
263 |
|
f1ocnv |
|
264 |
4 263
|
syl |
|
265 |
6
|
adantr |
|
266 |
63
|
adantr |
|
267 |
65
|
ffvelrnda |
|
268 |
267
|
adantrr |
|
269 |
159 268
|
sseldi |
|
270 |
|
elfzoelz |
|
271 |
269 270
|
syl |
|
272 |
266 271
|
zmulcld |
|
273 |
65
|
ffvelrnda |
|
274 |
273
|
adantrl |
|
275 |
159 274
|
sseldi |
|
276 |
|
elfzoelz |
|
277 |
275 276
|
syl |
|
278 |
266 277
|
zmulcld |
|
279 |
|
moddvds |
|
280 |
265 272 278 279
|
syl3anc |
|
281 |
|
fveq2 |
|
282 |
281
|
oveq2d |
|
283 |
282
|
oveq1d |
|
284 |
|
ovex |
|
285 |
283 5 284
|
fvmpt |
|
286 |
|
fveq2 |
|
287 |
286
|
oveq2d |
|
288 |
287
|
oveq1d |
|
289 |
|
ovex |
|
290 |
288 5 289
|
fvmpt |
|
291 |
285 290
|
eqeqan12d |
|
292 |
291
|
adantl |
|
293 |
93
|
adantr |
|
294 |
271
|
zcnd |
|
295 |
277
|
zcnd |
|
296 |
293 294 295
|
subdid |
|
297 |
296
|
breq2d |
|
298 |
280 292 297
|
3bitr4d |
|
299 |
102 63
|
gcdcomd |
|
300 |
1
|
simp3d |
|
301 |
299 300
|
eqtrd |
|
302 |
301
|
adantr |
|
303 |
102
|
adantr |
|
304 |
271 277
|
zsubcld |
|
305 |
|
coprmdvds |
|
306 |
303 266 304 305
|
syl3anc |
|
307 |
271
|
zred |
|
308 |
81
|
adantr |
|
309 |
|
elfzole1 |
|
310 |
269 309
|
syl |
|
311 |
|
elfzolt2 |
|
312 |
269 311
|
syl |
|
313 |
|
modid |
|
314 |
307 308 310 312 313
|
syl22anc |
|
315 |
277
|
zred |
|
316 |
|
elfzole1 |
|
317 |
275 316
|
syl |
|
318 |
|
elfzolt2 |
|
319 |
275 318
|
syl |
|
320 |
|
modid |
|
321 |
315 308 317 319 320
|
syl22anc |
|
322 |
314 321
|
eqeq12d |
|
323 |
|
moddvds |
|
324 |
265 271 277 323
|
syl3anc |
|
325 |
|
f1of1 |
|
326 |
4 325
|
syl |
|
327 |
|
f1fveq |
|
328 |
326 327
|
sylan |
|
329 |
322 324 328
|
3bitr3d |
|
330 |
306 329
|
sylibd |
|
331 |
302 330
|
mpan2d |
|
332 |
298 331
|
sylbid |
|
333 |
332
|
ralrimivva |
|
334 |
|
dff13 |
|
335 |
160 333 334
|
sylanbrc |
|
336 |
3
|
ovexi |
|
337 |
336
|
f1oen |
|
338 |
4 337
|
syl |
|
339 |
|
fzofi |
|
340 |
|
ssfi |
|
341 |
339 159 340
|
mp2an |
|
342 |
|
f1finf1o |
|
343 |
338 341 342
|
sylancl |
|
344 |
335 343
|
mpbid |
|
345 |
|
f1oco |
|
346 |
264 344 345
|
syl2anc |
|
347 |
|
f1oeq23 |
|
348 |
3 3 347
|
mp2an |
|
349 |
346 348
|
sylib |
|
350 |
252
|
zcnd |
|
351 |
3
|
eleq2i |
|
352 |
|
fvco3 |
|
353 |
160 352
|
sylan |
|
354 |
353
|
fveq2d |
|
355 |
4
|
adantr |
|
356 |
160
|
ffvelrnda |
|
357 |
|
f1ocnvfv2 |
|
358 |
355 356 357
|
syl2anc |
|
359 |
354 358
|
eqtr2d |
|
360 |
351 359
|
sylan2br |
|
361 |
257 259 261 67 262 349 350 360
|
seqf1o |
|
362 |
361 254
|
eqeltrd |
|
363 |
|
moddvds |
|
364 |
6 255 362 363
|
syl3anc |
|
365 |
247 364
|
mpbid |
|
366 |
254
|
zcnd |
|
367 |
366
|
mulid2d |
|
368 |
361 367
|
eqtr4d |
|
369 |
368
|
oveq2d |
|
370 |
250
|
zcnd |
|
371 |
|
ax-1cn |
|
372 |
|
subdir |
|
373 |
371 372
|
mp3an2 |
|
374 |
370 366 373
|
syl2anc |
|
375 |
|
zsubcl |
|
376 |
250 84 375
|
sylancl |
|
377 |
376
|
zcnd |
|
378 |
377 366
|
mulcomd |
|
379 |
369 374 378
|
3eqtr2d |
|
380 |
365 379
|
breqtrd |
|
381 |
246
|
simprd |
|
382 |
|
coprmdvds |
|
383 |
102 254 376 382
|
syl3anc |
|
384 |
380 381 383
|
mp2and |
|
385 |
|
moddvds |
|
386 |
84 385
|
mp3an3 |
|
387 |
6 250 386
|
syl2anc |
|
388 |
384 387
|
mpbird |
|