Step |
Hyp |
Ref |
Expression |
1 |
|
uniioombl.1 |
|
2 |
|
uniioombl.2 |
|
3 |
|
uniioombl.3 |
|
4 |
|
uniioombl.a |
|
5 |
|
uniioombl.e |
|
6 |
|
uniioombl.c |
|
7 |
|
uniioombl.g |
|
8 |
|
uniioombl.s |
|
9 |
|
uniioombl.t |
|
10 |
|
uniioombl.v |
|
11 |
|
uniioombl.m |
|
12 |
|
uniioombl.m2 |
|
13 |
|
uniioombl.k |
|
14 |
|
uniioombl.n |
|
15 |
|
uniioombl.n2 |
|
16 |
|
uniioombl.l |
|
17 |
|
inss1 |
|
18 |
|
imassrn |
|
19 |
18
|
unissi |
|
20 |
13 19
|
eqsstri |
|
21 |
7
|
uniiccdif |
|
22 |
21
|
simpld |
|
23 |
|
ovolficcss |
|
24 |
7 23
|
syl |
|
25 |
22 24
|
sstrd |
|
26 |
20 25
|
sstrid |
|
27 |
1 2 3 4 5 6 7 8 9 10
|
uniioombllem1 |
|
28 |
|
ssid |
|
29 |
9
|
ovollb |
|
30 |
7 28 29
|
sylancl |
|
31 |
|
ovollecl |
|
32 |
25 27 30 31
|
syl3anc |
|
33 |
|
ovolsscl |
|
34 |
20 25 32 33
|
mp3an2i |
|
35 |
|
ovolsscl |
|
36 |
17 26 34 35
|
mp3an2i |
|
37 |
|
inss1 |
|
38 |
|
ovolsscl |
|
39 |
37 26 34 38
|
mp3an2i |
|
40 |
|
ssun2 |
|
41 |
|
nnuz |
|
42 |
14
|
peano2nnd |
|
43 |
42 41
|
eleqtrdi |
|
44 |
|
uzsplit |
|
45 |
43 44
|
syl |
|
46 |
41 45
|
syl5eq |
|
47 |
14
|
nncnd |
|
48 |
|
ax-1cn |
|
49 |
|
pncan |
|
50 |
47 48 49
|
sylancl |
|
51 |
50
|
oveq2d |
|
52 |
51
|
uneq1d |
|
53 |
46 52
|
eqtrd |
|
54 |
53
|
iuneq1d |
|
55 |
|
iunxun |
|
56 |
54 55
|
eqtrdi |
|
57 |
|
ioof |
|
58 |
|
inss2 |
|
59 |
|
rexpssxrxp |
|
60 |
58 59
|
sstri |
|
61 |
|
fss |
|
62 |
1 60 61
|
sylancl |
|
63 |
|
fco |
|
64 |
57 62 63
|
sylancr |
|
65 |
|
ffn |
|
66 |
|
fniunfv |
|
67 |
64 65 66
|
3syl |
|
68 |
|
fvco3 |
|
69 |
1 68
|
sylan |
|
70 |
69
|
iuneq2dv |
|
71 |
67 70
|
eqtr3d |
|
72 |
4 71
|
syl5eq |
|
73 |
|
ffun |
|
74 |
|
funiunfv |
|
75 |
64 73 74
|
3syl |
|
76 |
|
elfznn |
|
77 |
1 76 68
|
syl2an |
|
78 |
77
|
iuneq2dv |
|
79 |
75 78
|
eqtr3d |
|
80 |
16 79
|
syl5eq |
|
81 |
80
|
uneq1d |
|
82 |
56 72 81
|
3eqtr4d |
|
83 |
82
|
ineq2d |
|
84 |
|
indi |
|
85 |
83 84
|
eqtrdi |
|
86 |
|
fss |
|
87 |
7 60 86
|
sylancl |
|
88 |
|
fco |
|
89 |
57 87 88
|
sylancr |
|
90 |
|
ffun |
|
91 |
|
funiunfv |
|
92 |
89 90 91
|
3syl |
|
93 |
|
elfznn |
|
94 |
|
fvco3 |
|
95 |
7 93 94
|
syl2an |
|
96 |
95
|
iuneq2dv |
|
97 |
92 96
|
eqtr3d |
|
98 |
13 97
|
syl5eq |
|
99 |
98
|
ineq2d |
|
100 |
|
incom |
|
101 |
|
iunin2 |
|
102 |
|
incom |
|
103 |
102
|
a1i |
|
104 |
103
|
iuneq2i |
|
105 |
|
incom |
|
106 |
101 104 105
|
3eqtr4ri |
|
107 |
106
|
a1i |
|
108 |
107
|
iuneq2i |
|
109 |
|
iunin2 |
|
110 |
108 109
|
eqtr3i |
|
111 |
99 100 110
|
3eqtr4g |
|
112 |
111
|
uneq2d |
|
113 |
85 112
|
eqtrd |
|
114 |
40 113
|
sseqtrrid |
|
115 |
114 17
|
sstrdi |
|
116 |
|
ovolsscl |
|
117 |
115 26 34 116
|
syl3anc |
|
118 |
39 117
|
readdcld |
|
119 |
6
|
rpred |
|
120 |
39 119
|
readdcld |
|
121 |
113
|
fveq2d |
|
122 |
37 26
|
sstrid |
|
123 |
115 26
|
sstrd |
|
124 |
|
ovolun |
|
125 |
122 39 123 117 124
|
syl22anc |
|
126 |
121 125
|
eqbrtrd |
|
127 |
|
fzfid |
|
128 |
|
iunss |
|
129 |
115 128
|
sylib |
|
130 |
129
|
r19.21bi |
|
131 |
26
|
adantr |
|
132 |
34
|
adantr |
|
133 |
|
ovolsscl |
|
134 |
130 131 132 133
|
syl3anc |
|
135 |
127 134
|
fsumrecl |
|
136 |
130 131
|
sstrd |
|
137 |
136 134
|
jca |
|
138 |
137
|
ralrimiva |
|
139 |
|
ovolfiniun |
|
140 |
127 138 139
|
syl2anc |
|
141 |
119 11
|
nndivred |
|
142 |
141
|
adantr |
|
143 |
80
|
ineq2d |
|
144 |
143
|
adantr |
|
145 |
102
|
a1i |
|
146 |
145
|
iuneq2i |
|
147 |
|
iunin2 |
|
148 |
146 147
|
eqtri |
|
149 |
144 148
|
eqtr4di |
|
150 |
|
fzfid |
|
151 |
|
ffvelrn |
|
152 |
1 76 151
|
syl2an |
|
153 |
152
|
elin2d |
|
154 |
|
1st2nd2 |
|
155 |
153 154
|
syl |
|
156 |
155
|
fveq2d |
|
157 |
|
df-ov |
|
158 |
156 157
|
eqtr4di |
|
159 |
|
ioombl |
|
160 |
158 159
|
eqeltrdi |
|
161 |
160
|
adantlr |
|
162 |
|
ffvelrn |
|
163 |
7 93 162
|
syl2an |
|
164 |
163
|
elin2d |
|
165 |
|
1st2nd2 |
|
166 |
164 165
|
syl |
|
167 |
166
|
fveq2d |
|
168 |
|
df-ov |
|
169 |
167 168
|
eqtr4di |
|
170 |
|
ioombl |
|
171 |
169 170
|
eqeltrdi |
|
172 |
171
|
adantr |
|
173 |
|
inmbl |
|
174 |
161 172 173
|
syl2anc |
|
175 |
174
|
ralrimiva |
|
176 |
|
finiunmbl |
|
177 |
150 175 176
|
syl2anc |
|
178 |
149 177
|
eqeltrd |
|
179 |
|
inss2 |
|
180 |
1
|
uniiccdif |
|
181 |
180
|
simpld |
|
182 |
|
ovolficcss |
|
183 |
1 182
|
syl |
|
184 |
181 183
|
sstrd |
|
185 |
4 184
|
eqsstrid |
|
186 |
185
|
adantr |
|
187 |
179 186
|
sstrid |
|
188 |
|
inss1 |
|
189 |
|
ioossre |
|
190 |
169 189
|
eqsstrdi |
|
191 |
169
|
fveq2d |
|
192 |
|
ovolfcl |
|
193 |
7 93 192
|
syl2an |
|
194 |
|
ovolioo |
|
195 |
193 194
|
syl |
|
196 |
191 195
|
eqtrd |
|
197 |
193
|
simp2d |
|
198 |
193
|
simp1d |
|
199 |
197 198
|
resubcld |
|
200 |
196 199
|
eqeltrd |
|
201 |
|
ovolsscl |
|
202 |
188 190 200 201
|
mp3an2i |
|
203 |
|
mblsplit |
|
204 |
178 187 202 203
|
syl3anc |
|
205 |
|
imassrn |
|
206 |
205
|
unissi |
|
207 |
206 16 4
|
3sstr4i |
|
208 |
|
sslin |
|
209 |
207 208
|
mp1i |
|
210 |
|
sseqin2 |
|
211 |
209 210
|
sylib |
|
212 |
211
|
fveq2d |
|
213 |
|
indifdir |
|
214 |
|
incom |
|
215 |
|
incom |
|
216 |
214 215
|
difeq12i |
|
217 |
213 216
|
eqtri |
|
218 |
82
|
eqcomd |
|
219 |
80
|
ineq1d |
|
220 |
|
2fveq3 |
|
221 |
220
|
cbvdisjv |
|
222 |
2 221
|
sylib |
|
223 |
|
fz1ssnn |
|
224 |
223
|
a1i |
|
225 |
|
uzss |
|
226 |
43 225
|
syl |
|
227 |
226 41
|
sseqtrrdi |
|
228 |
|
uzdisj |
|
229 |
51
|
ineq1d |
|
230 |
228 229
|
syl5reqr |
|
231 |
|
disjiun |
|
232 |
222 224 227 230 231
|
syl13anc |
|
233 |
219 232
|
eqtrd |
|
234 |
|
uneqdifeq |
|
235 |
207 233 234
|
sylancr |
|
236 |
218 235
|
mpbid |
|
237 |
236
|
adantr |
|
238 |
237
|
ineq2d |
|
239 |
|
incom |
|
240 |
104 101
|
eqtri |
|
241 |
238 239 240
|
3eqtr4g |
|
242 |
217 241
|
syl5eqr |
|
243 |
242
|
fveq2d |
|
244 |
212 243
|
oveq12d |
|
245 |
204 244
|
eqtrd |
|
246 |
202 142
|
resubcld |
|
247 |
|
inss2 |
|
248 |
190
|
adantr |
|
249 |
200
|
adantr |
|
250 |
|
ovolsscl |
|
251 |
247 248 249 250
|
mp3an2i |
|
252 |
150 251
|
fsumrecl |
|
253 |
15
|
r19.21bi |
|
254 |
252 202 142
|
absdifltd |
|
255 |
253 254
|
mpbid |
|
256 |
255
|
simpld |
|
257 |
246 252 256
|
ltled |
|
258 |
149
|
fveq2d |
|
259 |
|
mblvol |
|
260 |
174 259
|
syl |
|
261 |
260 251
|
eqeltrd |
|
262 |
174 261
|
jca |
|
263 |
262
|
ralrimiva |
|
264 |
|
inss1 |
|
265 |
264
|
rgenw |
|
266 |
222
|
adantr |
|
267 |
|
disjss2 |
|
268 |
265 266 267
|
mpsyl |
|
269 |
|
disjss1 |
|
270 |
223 268 269
|
mpsyl |
|
271 |
|
volfiniun |
|
272 |
150 263 270 271
|
syl3anc |
|
273 |
|
mblvol |
|
274 |
177 273
|
syl |
|
275 |
260
|
sumeq2dv |
|
276 |
272 274 275
|
3eqtr3d |
|
277 |
258 276
|
eqtrd |
|
278 |
257 277
|
breqtrrd |
|
279 |
277 252
|
eqeltrd |
|
280 |
202 142 279
|
lesubaddd |
|
281 |
278 280
|
mpbid |
|
282 |
245 281
|
eqbrtrrd |
|
283 |
134 142 279
|
leadd2d |
|
284 |
282 283
|
mpbird |
|
285 |
127 134 142 284
|
fsumle |
|
286 |
141
|
recnd |
|
287 |
|
fsumconst |
|
288 |
127 286 287
|
syl2anc |
|
289 |
|
nnnn0 |
|
290 |
|
hashfz1 |
|
291 |
11 289 290
|
3syl |
|
292 |
291
|
oveq1d |
|
293 |
119
|
recnd |
|
294 |
11
|
nncnd |
|
295 |
11
|
nnne0d |
|
296 |
293 294 295
|
divcan2d |
|
297 |
288 292 296
|
3eqtrd |
|
298 |
285 297
|
breqtrd |
|
299 |
117 135 119 140 298
|
letrd |
|
300 |
117 119 39 299
|
leadd2dd |
|
301 |
36 118 120 126 300
|
letrd |
|