Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|
2 |
|
poimirlem4.1 |
|
3 |
|
poimirlem4.2 |
|
4 |
|
poimirlem4.3 |
|
5 |
|
poimirlem3.4 |
|
6 |
|
poimirlem3.5 |
|
7 |
5
|
ffnd |
|
8 |
7
|
adantr |
|
9 |
|
1ex |
|
10 |
|
fnconstg |
|
11 |
9 10
|
ax-mp |
|
12 |
|
c0ex |
|
13 |
|
fnconstg |
|
14 |
12 13
|
ax-mp |
|
15 |
11 14
|
pm3.2i |
|
16 |
|
dff1o3 |
|
17 |
16
|
simprbi |
|
18 |
|
imain |
|
19 |
6 17 18
|
3syl |
|
20 |
|
elfznn0 |
|
21 |
20
|
nn0red |
|
22 |
21
|
ltp1d |
|
23 |
|
fzdisj |
|
24 |
22 23
|
syl |
|
25 |
24
|
imaeq2d |
|
26 |
|
ima0 |
|
27 |
25 26
|
eqtrdi |
|
28 |
19 27
|
sylan9req |
|
29 |
|
fnun |
|
30 |
15 28 29
|
sylancr |
|
31 |
|
imaundi |
|
32 |
|
nn0p1nn |
|
33 |
|
nnuz |
|
34 |
32 33
|
eleqtrdi |
|
35 |
20 34
|
syl |
|
36 |
|
elfzuz3 |
|
37 |
|
fzsplit2 |
|
38 |
35 36 37
|
syl2anc |
|
39 |
38
|
eqcomd |
|
40 |
39
|
imaeq2d |
|
41 |
|
f1ofo |
|
42 |
|
foima |
|
43 |
6 41 42
|
3syl |
|
44 |
40 43
|
sylan9eqr |
|
45 |
31 44
|
syl5eqr |
|
46 |
45
|
fneq2d |
|
47 |
30 46
|
mpbid |
|
48 |
|
ovexd |
|
49 |
|
inidm |
|
50 |
|
eqidd |
|
51 |
|
eqidd |
|
52 |
8 47 48 48 49 50 51
|
offval |
|
53 |
|
nn0p1nn |
|
54 |
3 53
|
syl |
|
55 |
54
|
nnzd |
|
56 |
|
uzid |
|
57 |
|
peano2uz |
|
58 |
55 56 57
|
3syl |
|
59 |
3
|
nn0zd |
|
60 |
1
|
nnzd |
|
61 |
|
zltp1le |
|
62 |
|
peano2z |
|
63 |
|
eluz |
|
64 |
62 63
|
sylan |
|
65 |
61 64
|
bitr4d |
|
66 |
59 60 65
|
syl2anc |
|
67 |
4 66
|
mpbid |
|
68 |
|
fzsplit2 |
|
69 |
58 67 68
|
syl2anc |
|
70 |
|
fzsn |
|
71 |
55 70
|
syl |
|
72 |
71
|
uneq1d |
|
73 |
69 72
|
eqtrd |
|
74 |
73
|
xpeq1d |
|
75 |
|
xpundir |
|
76 |
|
ovex |
|
77 |
76 12
|
xpsn |
|
78 |
77
|
uneq1i |
|
79 |
75 78
|
eqtri |
|
80 |
74 79
|
eqtrdi |
|
81 |
80
|
adantr |
|
82 |
52 81
|
uneq12d |
|
83 |
|
unass |
|
84 |
82 83
|
eqtr4di |
|
85 |
3
|
nn0red |
|
86 |
85
|
ltp1d |
|
87 |
54
|
nnred |
|
88 |
85 87
|
ltnled |
|
89 |
86 88
|
mpbid |
|
90 |
|
elfzle2 |
|
91 |
89 90
|
nsyl |
|
92 |
|
disjsn |
|
93 |
91 92
|
sylibr |
|
94 |
|
eqid |
|
95 |
76 12
|
fsn |
|
96 |
94 95
|
mpbir |
|
97 |
|
fun |
|
98 |
96 97
|
mpanl2 |
|
99 |
5 93 98
|
syl2anc |
|
100 |
|
1z |
|
101 |
|
nn0uz |
|
102 |
|
1m1e0 |
|
103 |
102
|
fveq2i |
|
104 |
101 103
|
eqtr4i |
|
105 |
3 104
|
eleqtrdi |
|
106 |
|
fzsuc2 |
|
107 |
100 105 106
|
sylancr |
|
108 |
107
|
eqcomd |
|
109 |
|
lbfzo0 |
|
110 |
2 109
|
sylibr |
|
111 |
110
|
snssd |
|
112 |
|
ssequn2 |
|
113 |
111 112
|
sylib |
|
114 |
108 113
|
feq23d |
|
115 |
99 114
|
mpbid |
|
116 |
115
|
ffnd |
|
117 |
116
|
adantr |
|
118 |
|
fnconstg |
|
119 |
9 118
|
ax-mp |
|
120 |
|
fnconstg |
|
121 |
12 120
|
ax-mp |
|
122 |
119 121
|
pm3.2i |
|
123 |
76 76
|
f1osn |
|
124 |
|
f1oun |
|
125 |
123 124
|
mpanl2 |
|
126 |
6 93 93 125
|
syl12anc |
|
127 |
|
dff1o3 |
|
128 |
127
|
simprbi |
|
129 |
|
imain |
|
130 |
126 128 129
|
3syl |
|
131 |
|
fzdisj |
|
132 |
22 131
|
syl |
|
133 |
132
|
imaeq2d |
|
134 |
|
ima0 |
|
135 |
133 134
|
eqtrdi |
|
136 |
130 135
|
sylan9req |
|
137 |
|
fnun |
|
138 |
122 136 137
|
sylancr |
|
139 |
|
f1ofo |
|
140 |
|
foima |
|
141 |
126 139 140
|
3syl |
|
142 |
107
|
imaeq2d |
|
143 |
141 142 107
|
3eqtr4d |
|
144 |
|
peano2uz |
|
145 |
36 144
|
syl |
|
146 |
|
fzsplit2 |
|
147 |
35 145 146
|
syl2anc |
|
148 |
147
|
imaeq2d |
|
149 |
143 148
|
sylan9req |
|
150 |
|
imaundi |
|
151 |
149 150
|
eqtrdi |
|
152 |
151
|
fneq2d |
|
153 |
138 152
|
mpbird |
|
154 |
|
ovexd |
|
155 |
|
inidm |
|
156 |
|
eqidd |
|
157 |
|
eqidd |
|
158 |
117 153 154 154 155 156 157
|
offval |
|
159 |
|
imadmrn |
|
160 |
76 76
|
xpsn |
|
161 |
160
|
imaeq1i |
|
162 |
|
dmxpid |
|
163 |
162
|
imaeq2i |
|
164 |
161 163
|
eqtri |
|
165 |
|
rnxpid |
|
166 |
159 164 165
|
3eqtr3ri |
|
167 |
|
eluzp1p1 |
|
168 |
|
eluzfz2 |
|
169 |
36 167 168
|
3syl |
|
170 |
169
|
snssd |
|
171 |
|
imass2 |
|
172 |
170 171
|
syl |
|
173 |
166 172
|
eqsstrid |
|
174 |
76
|
snid |
|
175 |
|
ssel |
|
176 |
173 174 175
|
mpisyl |
|
177 |
|
elun2 |
|
178 |
176 177
|
syl |
|
179 |
|
imaundir |
|
180 |
178 179
|
eleqtrrdi |
|
181 |
180
|
adantl |
|
182 |
12
|
a1i |
|
183 |
108
|
adantr |
|
184 |
|
fveq2 |
|
185 |
76 12
|
fnsn |
|
186 |
|
fvun2 |
|
187 |
185 186
|
mp3an2 |
|
188 |
174 187
|
mpanr2 |
|
189 |
7 93 188
|
syl2anc |
|
190 |
76 12
|
fvsn |
|
191 |
189 190
|
eqtrdi |
|
192 |
184 191
|
sylan9eqr |
|
193 |
192
|
adantlr |
|
194 |
|
fveq2 |
|
195 |
|
fvun2 |
|
196 |
119 121 195
|
mp3an12 |
|
197 |
136 181 196
|
syl2anc |
|
198 |
12
|
fvconst2 |
|
199 |
180 198
|
syl |
|
200 |
199
|
adantl |
|
201 |
197 200
|
eqtrd |
|
202 |
194 201
|
sylan9eqr |
|
203 |
193 202
|
oveq12d |
|
204 |
|
00id |
|
205 |
203 204
|
eqtrdi |
|
206 |
181 182 183 205
|
fmptapd |
|
207 |
7 93
|
jca |
|
208 |
|
fvun1 |
|
209 |
185 208
|
mp3an2 |
|
210 |
209
|
anassrs |
|
211 |
207 210
|
sylan |
|
212 |
211
|
adantlr |
|
213 |
|
fvres |
|
214 |
213
|
eqcomd |
|
215 |
|
resundir |
|
216 |
|
relxp |
|
217 |
|
dmxpss |
|
218 |
|
imassrn |
|
219 |
217 218
|
sstri |
|
220 |
|
f1of |
|
221 |
|
frn |
|
222 |
6 220 221
|
3syl |
|
223 |
219 222
|
sstrid |
|
224 |
|
relssres |
|
225 |
216 223 224
|
sylancr |
|
226 |
225
|
adantr |
|
227 |
|
imassrn |
|
228 |
76
|
rnsnop |
|
229 |
227 228
|
sseqtri |
|
230 |
|
ssrin |
|
231 |
229 230
|
ax-mp |
|
232 |
|
incom |
|
233 |
232 93
|
syl5eq |
|
234 |
231 233
|
sseqtrid |
|
235 |
|
ss0 |
|
236 |
234 235
|
syl |
|
237 |
|
fnconstg |
|
238 |
|
fnresdisj |
|
239 |
9 237 238
|
mp2b |
|
240 |
236 239
|
sylib |
|
241 |
240
|
adantr |
|
242 |
226 241
|
uneq12d |
|
243 |
|
imaundir |
|
244 |
243
|
xpeq1i |
|
245 |
|
xpundir |
|
246 |
244 245
|
eqtri |
|
247 |
246
|
reseq1i |
|
248 |
|
resundir |
|
249 |
247 248
|
eqtr2i |
|
250 |
|
un0 |
|
251 |
242 249 250
|
3eqtr3g |
|
252 |
|
f1odm |
|
253 |
6 252
|
syl |
|
254 |
253
|
ineq2d |
|
255 |
254
|
reseq2d |
|
256 |
|
f1orel |
|
257 |
|
resindm |
|
258 |
6 256 257
|
3syl |
|
259 |
255 258
|
eqtr3d |
|
260 |
38
|
ineq2d |
|
261 |
|
fzssp1 |
|
262 |
|
sseqin2 |
|
263 |
261 262
|
mpbi |
|
264 |
263
|
a1i |
|
265 |
|
incom |
|
266 |
265 132
|
syl5eq |
|
267 |
264 266
|
uneq12d |
|
268 |
|
uncom |
|
269 |
|
indi |
|
270 |
268 269
|
eqtr4i |
|
271 |
|
un0 |
|
272 |
267 270 271
|
3eqtr3g |
|
273 |
260 272
|
eqtrd |
|
274 |
273
|
reseq2d |
|
275 |
259 274
|
sylan9req |
|
276 |
275
|
rneqd |
|
277 |
|
df-ima |
|
278 |
|
df-ima |
|
279 |
276 277 278
|
3eqtr4g |
|
280 |
279
|
xpeq1d |
|
281 |
280
|
reseq1d |
|
282 |
|
relxp |
|
283 |
|
dmxpss |
|
284 |
|
imassrn |
|
285 |
283 284
|
sstri |
|
286 |
285 222
|
sstrid |
|
287 |
|
relssres |
|
288 |
282 286 287
|
sylancr |
|
289 |
288
|
adantr |
|
290 |
281 289
|
eqtrd |
|
291 |
|
imassrn |
|
292 |
291 228
|
sseqtri |
|
293 |
|
ssrin |
|
294 |
292 293
|
ax-mp |
|
295 |
294 233
|
sseqtrid |
|
296 |
|
ss0 |
|
297 |
295 296
|
syl |
|
298 |
|
fnconstg |
|
299 |
|
fnresdisj |
|
300 |
12 298 299
|
mp2b |
|
301 |
297 300
|
sylib |
|
302 |
301
|
adantr |
|
303 |
290 302
|
uneq12d |
|
304 |
179
|
xpeq1i |
|
305 |
|
xpundir |
|
306 |
304 305
|
eqtri |
|
307 |
306
|
reseq1i |
|
308 |
|
resundir |
|
309 |
307 308
|
eqtr2i |
|
310 |
|
un0 |
|
311 |
303 309 310
|
3eqtr3g |
|
312 |
251 311
|
uneq12d |
|
313 |
215 312
|
syl5eq |
|
314 |
313
|
fveq1d |
|
315 |
214 314
|
sylan9eqr |
|
316 |
212 315
|
oveq12d |
|
317 |
316
|
mpteq2dva |
|
318 |
317
|
uneq1d |
|
319 |
158 206 318
|
3eqtr2d |
|
320 |
319
|
uneq1d |
|
321 |
84 320
|
eqtr4d |
|
322 |
321
|
csbeq1d |
|
323 |
322
|
eqeq2d |
|
324 |
323
|
rexbidva |
|
325 |
324
|
ralbidv |
|
326 |
325
|
biimpd |
|
327 |
|
f1ofn |
|
328 |
6 327
|
syl |
|
329 |
76 76
|
fnsn |
|
330 |
|
fvun2 |
|
331 |
329 330
|
mp3an2 |
|
332 |
174 331
|
mpanr2 |
|
333 |
328 93 332
|
syl2anc |
|
334 |
76 76
|
fvsn |
|
335 |
333 334
|
eqtrdi |
|
336 |
191 335
|
jca |
|
337 |
326 336
|
jctird |
|
338 |
|
3anass |
|
339 |
337 338
|
syl6ibr |
|
340 |
5 96
|
jctir |
|
341 |
340 93 97
|
syl2anc |
|
342 |
341 114
|
mpbid |
|
343 |
|
ovex |
|
344 |
|
ovex |
|
345 |
343 344
|
elmap |
|
346 |
342 345
|
sylibr |
|
347 |
|
ovex |
|
348 |
|
f1oexrnex |
|
349 |
6 347 348
|
sylancl |
|
350 |
|
snex |
|
351 |
|
unexg |
|
352 |
349 350 351
|
sylancl |
|
353 |
|
f1oeq1 |
|
354 |
353
|
elabg |
|
355 |
352 354
|
syl |
|
356 |
|
f1oeq23 |
|
357 |
107 107 356
|
syl2anc |
|
358 |
355 357
|
bitrd |
|
359 |
126 358
|
mpbird |
|
360 |
346 359
|
opelxpd |
|
361 |
339 360
|
jctild |
|