Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|
2 |
|
poimirlem22.s |
|
3 |
|
poimirlem22.1 |
|
4 |
|
poimirlem22.2 |
|
5 |
|
poimirlem18.3 |
|
6 |
|
poimirlem18.4 |
|
7 |
1 2 3 4 5 6
|
poimirlem16 |
|
8 |
|
elfznn0 |
|
9 |
8
|
nn0red |
|
10 |
9
|
adantl |
|
11 |
1
|
nnzd |
|
12 |
|
peano2zm |
|
13 |
11 12
|
syl |
|
14 |
13
|
zred |
|
15 |
14
|
adantr |
|
16 |
1
|
nnred |
|
17 |
16
|
adantr |
|
18 |
|
elfzle2 |
|
19 |
18
|
adantl |
|
20 |
16
|
ltm1d |
|
21 |
20
|
adantr |
|
22 |
10 15 17 19 21
|
lelttrd |
|
23 |
22
|
adantlr |
|
24 |
|
fveq2 |
|
25 |
|
opex |
|
26 |
|
op2ndg |
|
27 |
25 1 26
|
sylancr |
|
28 |
24 27
|
sylan9eqr |
|
29 |
28
|
adantr |
|
30 |
23 29
|
breqtrrd |
|
31 |
30
|
iftrued |
|
32 |
31
|
csbeq1d |
|
33 |
|
vex |
|
34 |
|
oveq2 |
|
35 |
34
|
imaeq2d |
|
36 |
35
|
xpeq1d |
|
37 |
|
oveq1 |
|
38 |
37
|
oveq1d |
|
39 |
38
|
imaeq2d |
|
40 |
39
|
xpeq1d |
|
41 |
36 40
|
uneq12d |
|
42 |
41
|
oveq2d |
|
43 |
33 42
|
csbie |
|
44 |
|
2fveq3 |
|
45 |
|
op1stg |
|
46 |
25 1 45
|
sylancr |
|
47 |
46
|
fveq2d |
|
48 |
|
ovex |
|
49 |
48
|
mptex |
|
50 |
|
fvex |
|
51 |
48
|
mptex |
|
52 |
50 51
|
coex |
|
53 |
49 52
|
op1st |
|
54 |
47 53
|
eqtrdi |
|
55 |
44 54
|
sylan9eqr |
|
56 |
|
fveq2 |
|
57 |
56 46
|
sylan9eqr |
|
58 |
57
|
fveq2d |
|
59 |
49 52
|
op2nd |
|
60 |
58 59
|
eqtrdi |
|
61 |
60
|
imaeq1d |
|
62 |
61
|
xpeq1d |
|
63 |
60
|
imaeq1d |
|
64 |
63
|
xpeq1d |
|
65 |
62 64
|
uneq12d |
|
66 |
55 65
|
oveq12d |
|
67 |
43 66
|
syl5eq |
|
68 |
67
|
adantr |
|
69 |
32 68
|
eqtrd |
|
70 |
69
|
mpteq2dva |
|
71 |
70
|
eqeq2d |
|
72 |
71
|
ex |
|
73 |
72
|
alrimiv |
|
74 |
|
oveq2 |
|
75 |
74
|
eleq1d |
|
76 |
|
oveq2 |
|
77 |
76
|
eleq1d |
|
78 |
|
fveq2 |
|
79 |
78
|
oveq1d |
|
80 |
79
|
adantl |
|
81 |
|
elrabi |
|
82 |
81 2
|
eleq2s |
|
83 |
|
xp1st |
|
84 |
4 82 83
|
3syl |
|
85 |
|
xp1st |
|
86 |
|
elmapi |
|
87 |
84 85 86
|
3syl |
|
88 |
4 82
|
syl |
|
89 |
|
xp2nd |
|
90 |
88 83 89
|
3syl |
|
91 |
|
f1oeq1 |
|
92 |
50 91
|
elab |
|
93 |
90 92
|
sylib |
|
94 |
|
f1of |
|
95 |
93 94
|
syl |
|
96 |
|
nnuz |
|
97 |
1 96
|
eleqtrdi |
|
98 |
|
eluzfz1 |
|
99 |
97 98
|
syl |
|
100 |
95 99
|
ffvelrnd |
|
101 |
87 100
|
ffvelrnd |
|
102 |
|
elfzonn0 |
|
103 |
|
peano2nn0 |
|
104 |
101 102 103
|
3syl |
|
105 |
|
elfzo0 |
|
106 |
101 105
|
sylib |
|
107 |
106
|
simp2d |
|
108 |
104
|
nn0red |
|
109 |
107
|
nnred |
|
110 |
|
elfzolt2 |
|
111 |
101 110
|
syl |
|
112 |
101 102
|
syl |
|
113 |
112
|
nn0zd |
|
114 |
107
|
nnzd |
|
115 |
|
zltp1le |
|
116 |
113 114 115
|
syl2anc |
|
117 |
111 116
|
mpbid |
|
118 |
|
fvex |
|
119 |
|
eleq1 |
|
120 |
119
|
anbi2d |
|
121 |
|
fveq2 |
|
122 |
121
|
neeq1d |
|
123 |
122
|
rexbidv |
|
124 |
120 123
|
imbi12d |
|
125 |
118 124 5
|
vtocl |
|
126 |
100 125
|
mpdan |
|
127 |
|
fveq1 |
|
128 |
87
|
ffnd |
|
129 |
128
|
adantr |
|
130 |
|
1ex |
|
131 |
|
fnconstg |
|
132 |
130 131
|
ax-mp |
|
133 |
|
c0ex |
|
134 |
|
fnconstg |
|
135 |
133 134
|
ax-mp |
|
136 |
132 135
|
pm3.2i |
|
137 |
|
dff1o3 |
|
138 |
137
|
simprbi |
|
139 |
|
imain |
|
140 |
93 138 139
|
3syl |
|
141 |
|
nn0p1nn |
|
142 |
8 141
|
syl |
|
143 |
142
|
nnred |
|
144 |
143
|
ltp1d |
|
145 |
|
fzdisj |
|
146 |
145
|
imaeq2d |
|
147 |
|
ima0 |
|
148 |
146 147
|
eqtrdi |
|
149 |
144 148
|
syl |
|
150 |
140 149
|
sylan9req |
|
151 |
|
fnun |
|
152 |
136 150 151
|
sylancr |
|
153 |
|
imaundi |
|
154 |
142
|
peano2nnd |
|
155 |
154 96
|
eleqtrdi |
|
156 |
1
|
nncnd |
|
157 |
|
npcan1 |
|
158 |
156 157
|
syl |
|
159 |
158
|
adantr |
|
160 |
|
elfzuz3 |
|
161 |
|
eluzp1p1 |
|
162 |
160 161
|
syl |
|
163 |
162
|
adantl |
|
164 |
159 163
|
eqeltrrd |
|
165 |
|
fzsplit2 |
|
166 |
155 164 165
|
syl2an2 |
|
167 |
166
|
imaeq2d |
|
168 |
|
f1ofo |
|
169 |
|
foima |
|
170 |
93 168 169
|
3syl |
|
171 |
170
|
adantr |
|
172 |
167 171
|
eqtr3d |
|
173 |
153 172
|
eqtr3id |
|
174 |
173
|
fneq2d |
|
175 |
152 174
|
mpbid |
|
176 |
48
|
a1i |
|
177 |
|
inidm |
|
178 |
|
eqidd |
|
179 |
|
f1ofn |
|
180 |
93 179
|
syl |
|
181 |
180
|
adantr |
|
182 |
|
fzss2 |
|
183 |
164 182
|
syl |
|
184 |
142 96
|
eleqtrdi |
|
185 |
|
eluzfz1 |
|
186 |
184 185
|
syl |
|
187 |
186
|
adantl |
|
188 |
|
fnfvima |
|
189 |
181 183 187 188
|
syl3anc |
|
190 |
|
fvun1 |
|
191 |
132 135 190
|
mp3an12 |
|
192 |
150 189 191
|
syl2anc |
|
193 |
130
|
fvconst2 |
|
194 |
189 193
|
syl |
|
195 |
192 194
|
eqtrd |
|
196 |
195
|
adantr |
|
197 |
129 175 176 176 177 178 196
|
ofval |
|
198 |
100 197
|
mpidan |
|
199 |
127 198
|
sylan9eqr |
|
200 |
199
|
adantllr |
|
201 |
|
fveq2 |
|
202 |
201
|
breq2d |
|
203 |
202
|
ifbid |
|
204 |
|
2fveq3 |
|
205 |
|
2fveq3 |
|
206 |
205
|
imaeq1d |
|
207 |
206
|
xpeq1d |
|
208 |
205
|
imaeq1d |
|
209 |
208
|
xpeq1d |
|
210 |
207 209
|
uneq12d |
|
211 |
204 210
|
oveq12d |
|
212 |
203 211
|
csbeq12dv |
|
213 |
212
|
mpteq2dv |
|
214 |
213
|
eqeq2d |
|
215 |
214 2
|
elrab2 |
|
216 |
215
|
simprbi |
|
217 |
4 216
|
syl |
|
218 |
217
|
rneqd |
|
219 |
218
|
eleq2d |
|
220 |
|
eqid |
|
221 |
|
ovex |
|
222 |
221
|
csbex |
|
223 |
220 222
|
elrnmpti |
|
224 |
219 223
|
bitrdi |
|
225 |
6
|
adantr |
|
226 |
|
elfzle1 |
|
227 |
226
|
adantl |
|
228 |
225 227
|
eqbrtrd |
|
229 |
|
0re |
|
230 |
6 229
|
eqeltrdi |
|
231 |
|
lenlt |
|
232 |
230 9 231
|
syl2an |
|
233 |
228 232
|
mpbid |
|
234 |
233
|
iffalsed |
|
235 |
234
|
csbeq1d |
|
236 |
|
ovex |
|
237 |
|
oveq2 |
|
238 |
237
|
imaeq2d |
|
239 |
238
|
xpeq1d |
|
240 |
|
oveq1 |
|
241 |
240
|
oveq1d |
|
242 |
241
|
imaeq2d |
|
243 |
242
|
xpeq1d |
|
244 |
239 243
|
uneq12d |
|
245 |
244
|
oveq2d |
|
246 |
236 245
|
csbie |
|
247 |
235 246
|
eqtrdi |
|
248 |
247
|
eqeq2d |
|
249 |
248
|
rexbidva |
|
250 |
224 249
|
bitrd |
|
251 |
250
|
biimpa |
|
252 |
200 251
|
r19.29a |
|
253 |
|
eqtr3 |
|
254 |
253
|
ex |
|
255 |
252 254
|
syl |
|
256 |
255
|
necon3d |
|
257 |
256
|
rexlimdva |
|
258 |
126 257
|
mpd |
|
259 |
108 109 117 258
|
leneltd |
|
260 |
|
elfzo0 |
|
261 |
104 107 259 260
|
syl3anbrc |
|
262 |
261
|
adantr |
|
263 |
80 262
|
eqeltrd |
|
264 |
263
|
adantlr |
|
265 |
87
|
ffvelrnda |
|
266 |
|
elfzonn0 |
|
267 |
265 266
|
syl |
|
268 |
267
|
nn0cnd |
|
269 |
268
|
addid1d |
|
270 |
269 265
|
eqeltrd |
|
271 |
270
|
adantr |
|
272 |
75 77 264 271
|
ifbothda |
|
273 |
272
|
fmpttd |
|
274 |
|
ovex |
|
275 |
274 48
|
elmap |
|
276 |
273 275
|
sylibr |
|
277 |
|
simpr |
|
278 |
|
1z |
|
279 |
13 278
|
jctil |
|
280 |
|
elfzelz |
|
281 |
280 278
|
jctir |
|
282 |
|
fzaddel |
|
283 |
279 281 282
|
syl2an |
|
284 |
277 283
|
mpbid |
|
285 |
158
|
oveq2d |
|
286 |
285
|
adantr |
|
287 |
284 286
|
eleqtrd |
|
288 |
287
|
ralrimiva |
|
289 |
|
simpr |
|
290 |
|
peano2z |
|
291 |
278 290
|
ax-mp |
|
292 |
11 291
|
jctil |
|
293 |
|
elfzelz |
|
294 |
293 278
|
jctir |
|
295 |
|
fzsubel |
|
296 |
292 294 295
|
syl2an |
|
297 |
289 296
|
mpbid |
|
298 |
|
ax-1cn |
|
299 |
298 298
|
pncan3oi |
|
300 |
299
|
oveq1i |
|
301 |
297 300
|
eleqtrdi |
|
302 |
293
|
zcnd |
|
303 |
|
elfznn |
|
304 |
303
|
nncnd |
|
305 |
|
subadd2 |
|
306 |
298 305
|
mp3an2 |
|
307 |
306
|
bicomd |
|
308 |
|
eqcom |
|
309 |
|
eqcom |
|
310 |
307 308 309
|
3bitr4g |
|
311 |
302 304 310
|
syl2an |
|
312 |
311
|
ralrimiva |
|
313 |
312
|
adantl |
|
314 |
|
reu6i |
|
315 |
301 313 314
|
syl2anc |
|
316 |
315
|
ralrimiva |
|
317 |
|
eqid |
|
318 |
317
|
f1ompt |
|
319 |
288 316 318
|
sylanbrc |
|
320 |
|
f1osng |
|
321 |
1 130 320
|
sylancl |
|
322 |
14 16
|
ltnled |
|
323 |
20 322
|
mpbid |
|
324 |
|
elfzle2 |
|
325 |
323 324
|
nsyl |
|
326 |
|
disjsn |
|
327 |
325 326
|
sylibr |
|
328 |
|
1re |
|
329 |
328
|
ltp1i |
|
330 |
291
|
zrei |
|
331 |
328 330
|
ltnlei |
|
332 |
329 331
|
mpbi |
|
333 |
|
elfzle1 |
|
334 |
332 333
|
mto |
|
335 |
|
disjsn |
|
336 |
334 335
|
mpbir |
|
337 |
336
|
a1i |
|
338 |
|
f1oun |
|
339 |
319 321 327 337 338
|
syl22anc |
|
340 |
130
|
a1i |
|
341 |
158 97
|
eqeltrd |
|
342 |
|
uzid |
|
343 |
|
peano2uz |
|
344 |
13 342 343
|
3syl |
|
345 |
158 344
|
eqeltrrd |
|
346 |
|
fzsplit2 |
|
347 |
341 345 346
|
syl2anc |
|
348 |
158
|
oveq1d |
|
349 |
|
fzsn |
|
350 |
11 349
|
syl |
|
351 |
348 350
|
eqtrd |
|
352 |
351
|
uneq2d |
|
353 |
347 352
|
eqtr2d |
|
354 |
|
iftrue |
|
355 |
354
|
adantl |
|
356 |
1 340 353 355
|
fmptapd |
|
357 |
|
eleq1 |
|
358 |
357
|
notbid |
|
359 |
325 358
|
syl5ibrcom |
|
360 |
359
|
necon2ad |
|
361 |
360
|
imp |
|
362 |
|
ifnefalse |
|
363 |
361 362
|
syl |
|
364 |
363
|
mpteq2dva |
|
365 |
364
|
uneq1d |
|
366 |
356 365
|
eqtr3d |
|
367 |
347 352
|
eqtrd |
|
368 |
|
uzid |
|
369 |
|
peano2uz |
|
370 |
278 368 369
|
mp2b |
|
371 |
|
fzsplit2 |
|
372 |
370 97 371
|
sylancr |
|
373 |
|
fzsn |
|
374 |
278 373
|
ax-mp |
|
375 |
374
|
uneq1i |
|
376 |
375
|
equncomi |
|
377 |
372 376
|
eqtrdi |
|
378 |
366 367 377
|
f1oeq123d |
|
379 |
339 378
|
mpbird |
|
380 |
|
f1oco |
|
381 |
93 379 380
|
syl2anc |
|
382 |
|
f1oeq1 |
|
383 |
52 382
|
elab |
|
384 |
381 383
|
sylibr |
|
385 |
276 384
|
opelxpd |
|
386 |
1
|
nnnn0d |
|
387 |
|
nn0fz0 |
|
388 |
386 387
|
sylib |
|
389 |
385 388
|
opelxpd |
|
390 |
|
elrab3t |
|
391 |
73 389 390
|
syl2anc |
|
392 |
7 391
|
mpbird |
|
393 |
392 2
|
eleqtrrdi |
|
394 |
|
fveqeq2 |
|
395 |
27 394
|
syl5ibcom |
|
396 |
1
|
nnne0d |
|
397 |
|
neeq1 |
|
398 |
396 397
|
syl5ibrcom |
|
399 |
395 398
|
syld |
|
400 |
399
|
necon2d |
|
401 |
6 400
|
mpd |
|
402 |
|
neeq1 |
|
403 |
402
|
rspcev |
|
404 |
393 401 403
|
syl2anc |
|