Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|
2 |
|
poimirlem22.s |
|
3 |
|
poimirlem9.1 |
|
4 |
|
poimirlem9.2 |
|
5 |
|
poimirlem9.3 |
|
6 |
|
poimirlem9.4 |
|
7 |
|
resundi |
|
8 |
1
|
nncnd |
|
9 |
|
npcan1 |
|
10 |
8 9
|
syl |
|
11 |
1
|
nnzd |
|
12 |
|
peano2zm |
|
13 |
|
uzid |
|
14 |
|
peano2uz |
|
15 |
11 12 13 14
|
4syl |
|
16 |
10 15
|
eqeltrrd |
|
17 |
|
fzss2 |
|
18 |
16 17
|
syl |
|
19 |
18 4
|
sseldd |
|
20 |
|
fzp1elp1 |
|
21 |
4 20
|
syl |
|
22 |
10
|
oveq2d |
|
23 |
21 22
|
eleqtrd |
|
24 |
19 23
|
prssd |
|
25 |
|
undif |
|
26 |
24 25
|
sylib |
|
27 |
26
|
reseq2d |
|
28 |
|
elrabi |
|
29 |
28 2
|
eleq2s |
|
30 |
|
xp1st |
|
31 |
|
xp2nd |
|
32 |
5 29 30 31
|
4syl |
|
33 |
|
fvex |
|
34 |
|
f1oeq1 |
|
35 |
33 34
|
elab |
|
36 |
32 35
|
sylib |
|
37 |
|
f1ofn |
|
38 |
|
fnresdm |
|
39 |
36 37 38
|
3syl |
|
40 |
27 39
|
eqtrd |
|
41 |
7 40
|
eqtr3id |
|
42 |
|
2lt3 |
|
43 |
|
2re |
|
44 |
|
3re |
|
45 |
43 44
|
ltnlei |
|
46 |
42 45
|
mpbi |
|
47 |
|
df-pr |
|
48 |
47
|
coeq2i |
|
49 |
|
coundi |
|
50 |
48 49
|
eqtri |
|
51 |
|
elrabi |
|
52 |
51 2
|
eleq2s |
|
53 |
|
xp1st |
|
54 |
|
xp2nd |
|
55 |
3 52 53 54
|
4syl |
|
56 |
|
fvex |
|
57 |
|
f1oeq1 |
|
58 |
56 57
|
elab |
|
59 |
55 58
|
sylib |
|
60 |
|
f1of1 |
|
61 |
59 60
|
syl |
|
62 |
23
|
snssd |
|
63 |
|
f1ores |
|
64 |
61 62 63
|
syl2anc |
|
65 |
|
f1of |
|
66 |
64 65
|
syl |
|
67 |
|
f1ofn |
|
68 |
59 67
|
syl |
|
69 |
|
fnsnfv |
|
70 |
68 23 69
|
syl2anc |
|
71 |
70
|
feq3d |
|
72 |
66 71
|
mpbird |
|
73 |
|
eqid |
|
74 |
|
fvex |
|
75 |
|
ovex |
|
76 |
74 75
|
fsn |
|
77 |
73 76
|
mpbir |
|
78 |
|
fco2 |
|
79 |
72 77 78
|
sylancl |
|
80 |
|
fvex |
|
81 |
80
|
fconst2 |
|
82 |
79 81
|
sylib |
|
83 |
74 80
|
xpsn |
|
84 |
82 83
|
eqtrdi |
|
85 |
84
|
uneq1d |
|
86 |
50 85
|
syl5eq |
|
87 |
|
elfznn |
|
88 |
4 87
|
syl |
|
89 |
88
|
nnred |
|
90 |
89
|
ltp1d |
|
91 |
89 90
|
ltned |
|
92 |
91
|
necomd |
|
93 |
|
f1veqaeq |
|
94 |
61 23 19 93
|
syl12anc |
|
95 |
94
|
necon3d |
|
96 |
92 95
|
mpd |
|
97 |
96
|
neneqd |
|
98 |
74 80
|
opth |
|
99 |
98
|
simprbi |
|
100 |
97 99
|
nsyl |
|
101 |
91
|
neneqd |
|
102 |
74 80
|
opth1 |
|
103 |
101 102
|
nsyl |
|
104 |
|
opex |
|
105 |
104
|
snid |
|
106 |
|
elun1 |
|
107 |
105 106
|
ax-mp |
|
108 |
|
eleq2 |
|
109 |
107 108
|
mpbii |
|
110 |
104
|
elpr |
|
111 |
|
oran |
|
112 |
110 111
|
bitri |
|
113 |
109 112
|
sylib |
|
114 |
113
|
necon2ai |
|
115 |
100 103 114
|
syl2anc |
|
116 |
86 115
|
eqnetrd |
|
117 |
|
fnressn |
|
118 |
68 19 117
|
syl2anc |
|
119 |
|
fnressn |
|
120 |
68 23 119
|
syl2anc |
|
121 |
118 120
|
uneq12d |
|
122 |
|
df-pr |
|
123 |
122
|
reseq2i |
|
124 |
|
resundi |
|
125 |
123 124
|
eqtri |
|
126 |
|
df-pr |
|
127 |
121 125 126
|
3eqtr4g |
|
128 |
1 2 3 4 5
|
poimirlem8 |
|
129 |
|
uneq12 |
|
130 |
|
resundi |
|
131 |
26
|
reseq2d |
|
132 |
|
fnresdm |
|
133 |
59 67 132
|
3syl |
|
134 |
131 133
|
eqtrd |
|
135 |
130 134
|
eqtr3id |
|
136 |
41 135
|
eqeq12d |
|
137 |
129 136
|
syl5ib |
|
138 |
128 137
|
mpan2d |
|
139 |
138
|
necon3d |
|
140 |
6 139
|
mpd |
|
141 |
140
|
necomd |
|
142 |
127 141
|
eqnetrrd |
|
143 |
|
prex |
|
144 |
56 143
|
coex |
|
145 |
|
prex |
|
146 |
33
|
resex |
|
147 |
|
hashtpg |
|
148 |
144 145 146 147
|
mp3an |
|
149 |
148
|
biimpi |
|
150 |
149
|
3expia |
|
151 |
116 142 150
|
syl2anc |
|
152 |
|
prex |
|
153 |
|
prex |
|
154 |
152 153
|
mapval |
|
155 |
|
prfi |
|
156 |
|
prfi |
|
157 |
|
mapfi |
|
158 |
155 156 157
|
mp2an |
|
159 |
154 158
|
eqeltrri |
|
160 |
|
f1of |
|
161 |
160
|
ss2abi |
|
162 |
|
ssfi |
|
163 |
159 161 162
|
mp2an |
|
164 |
23 19
|
prssd |
|
165 |
|
f1ores |
|
166 |
61 164 165
|
syl2anc |
|
167 |
|
fnimapr |
|
168 |
68 23 19 167
|
syl3anc |
|
169 |
168
|
f1oeq3d |
|
170 |
166 169
|
mpbid |
|
171 |
|
f1oprg |
|
172 |
74 75 75 74 171
|
mp4an |
|
173 |
91 92 172
|
syl2anc |
|
174 |
|
f1oco |
|
175 |
170 173 174
|
syl2anc |
|
176 |
|
rnpropg |
|
177 |
74 75 176
|
mp2an |
|
178 |
177
|
eqimssi |
|
179 |
|
cores |
|
180 |
|
f1oeq1 |
|
181 |
178 179 180
|
mp2b |
|
182 |
175 181
|
sylib |
|
183 |
96
|
necomd |
|
184 |
|
fvex |
|
185 |
|
f1oprg |
|
186 |
74 184 75 80 185
|
mp4an |
|
187 |
91 183 186
|
syl2anc |
|
188 |
|
prcom |
|
189 |
|
f1oeq3 |
|
190 |
188 189
|
ax-mp |
|
191 |
187 190
|
sylib |
|
192 |
|
f1of1 |
|
193 |
36 192
|
syl |
|
194 |
|
f1ores |
|
195 |
193 24 194
|
syl2anc |
|
196 |
|
dff1o3 |
|
197 |
196
|
simprbi |
|
198 |
|
imadif |
|
199 |
36 197 198
|
3syl |
|
200 |
|
f1ofo |
|
201 |
|
foima |
|
202 |
36 200 201
|
3syl |
|
203 |
|
f1ofo |
|
204 |
|
foima |
|
205 |
59 203 204
|
3syl |
|
206 |
202 205
|
eqtr4d |
|
207 |
128
|
rneqd |
|
208 |
|
df-ima |
|
209 |
|
df-ima |
|
210 |
207 208 209
|
3eqtr4g |
|
211 |
206 210
|
difeq12d |
|
212 |
|
dff1o3 |
|
213 |
212
|
simprbi |
|
214 |
|
imadif |
|
215 |
59 213 214
|
3syl |
|
216 |
|
dfin4 |
|
217 |
|
sseqin2 |
|
218 |
24 217
|
sylib |
|
219 |
216 218
|
eqtr3id |
|
220 |
219
|
imaeq2d |
|
221 |
215 220
|
eqtr3d |
|
222 |
199 211 221
|
3eqtrd |
|
223 |
219
|
imaeq2d |
|
224 |
|
fnimapr |
|
225 |
68 19 23 224
|
syl3anc |
|
226 |
225 188
|
eqtrdi |
|
227 |
222 223 226
|
3eqtr3d |
|
228 |
227
|
f1oeq3d |
|
229 |
195 228
|
mpbid |
|
230 |
|
ssabral |
|
231 |
|
f1oeq1 |
|
232 |
|
f1oeq1 |
|
233 |
|
f1oeq1 |
|
234 |
144 145 146 231 232 233
|
raltp |
|
235 |
230 234
|
bitri |
|
236 |
182 191 229 235
|
syl3anbrc |
|
237 |
|
hashss |
|
238 |
163 236 237
|
sylancr |
|
239 |
153
|
enref |
|
240 |
|
hashprg |
|
241 |
80 184 240
|
mp2an |
|
242 |
96 241
|
sylib |
|
243 |
|
hashprg |
|
244 |
74 75 243
|
mp2an |
|
245 |
91 244
|
sylib |
|
246 |
242 245
|
eqtr4d |
|
247 |
|
hashen |
|
248 |
155 156 247
|
mp2an |
|
249 |
246 248
|
sylib |
|
250 |
|
hashfacen |
|
251 |
239 249 250
|
sylancr |
|
252 |
153 153
|
mapval |
|
253 |
|
mapfi |
|
254 |
156 156 253
|
mp2an |
|
255 |
252 254
|
eqeltrri |
|
256 |
|
f1of |
|
257 |
256
|
ss2abi |
|
258 |
|
ssfi |
|
259 |
255 257 258
|
mp2an |
|
260 |
|
hashen |
|
261 |
163 259 260
|
mp2an |
|
262 |
251 261
|
sylibr |
|
263 |
|
hashfac |
|
264 |
156 263
|
ax-mp |
|
265 |
245
|
fveq2d |
|
266 |
|
fac2 |
|
267 |
265 266
|
eqtrdi |
|
268 |
264 267
|
syl5eq |
|
269 |
262 268
|
eqtrd |
|
270 |
238 269
|
breqtrd |
|
271 |
|
breq1 |
|
272 |
270 271
|
syl5ibcom |
|
273 |
151 272
|
syld |
|
274 |
273
|
necon1bd |
|
275 |
46 274
|
mpi |
|
276 |
|
coires1 |
|
277 |
128 276
|
eqtr4di |
|
278 |
275 277
|
uneq12d |
|
279 |
41 278
|
eqtr3d |
|
280 |
|
coundi |
|
281 |
279 280
|
eqtr4di |
|