Step |
Hyp |
Ref |
Expression |
1 |
|
jensen.1 |
|
2 |
|
jensen.2 |
|
3 |
|
jensen.3 |
|
4 |
|
jensen.4 |
|
5 |
|
jensen.5 |
|
6 |
|
jensen.6 |
|
7 |
|
jensen.7 |
|
8 |
|
jensen.8 |
|
9 |
5
|
ffnd |
|
10 |
|
fnresdm |
|
11 |
9 10
|
syl |
|
12 |
11
|
oveq2d |
|
13 |
7 12
|
breqtrrd |
|
14 |
|
ssid |
|
15 |
13 14
|
jctil |
|
16 |
|
sseq1 |
|
17 |
|
reseq2 |
|
18 |
|
res0 |
|
19 |
17 18
|
syl6eq |
|
20 |
19
|
oveq2d |
|
21 |
|
cnfld0 |
|
22 |
21
|
gsum0 |
|
23 |
20 22
|
syl6eq |
|
24 |
23
|
breq2d |
|
25 |
16 24
|
anbi12d |
|
26 |
|
reseq2 |
|
27 |
26
|
oveq2d |
|
28 |
27 23
|
oveq12d |
|
29 |
|
reseq2 |
|
30 |
29
|
oveq2d |
|
31 |
30 23
|
oveq12d |
|
32 |
31
|
breq2d |
|
33 |
32
|
rabbidv |
|
34 |
28 33
|
eleq12d |
|
35 |
25 34
|
imbi12d |
|
36 |
35
|
imbi2d |
|
37 |
|
sseq1 |
|
38 |
|
reseq2 |
|
39 |
38
|
oveq2d |
|
40 |
39
|
breq2d |
|
41 |
37 40
|
anbi12d |
|
42 |
|
reseq2 |
|
43 |
42
|
oveq2d |
|
44 |
43 39
|
oveq12d |
|
45 |
|
reseq2 |
|
46 |
45
|
oveq2d |
|
47 |
46 39
|
oveq12d |
|
48 |
47
|
breq2d |
|
49 |
48
|
rabbidv |
|
50 |
44 49
|
eleq12d |
|
51 |
41 50
|
imbi12d |
|
52 |
51
|
imbi2d |
|
53 |
|
sseq1 |
|
54 |
|
reseq2 |
|
55 |
54
|
oveq2d |
|
56 |
55
|
breq2d |
|
57 |
53 56
|
anbi12d |
|
58 |
|
reseq2 |
|
59 |
58
|
oveq2d |
|
60 |
59 55
|
oveq12d |
|
61 |
|
reseq2 |
|
62 |
61
|
oveq2d |
|
63 |
62 55
|
oveq12d |
|
64 |
63
|
breq2d |
|
65 |
64
|
rabbidv |
|
66 |
60 65
|
eleq12d |
|
67 |
57 66
|
imbi12d |
|
68 |
67
|
imbi2d |
|
69 |
|
sseq1 |
|
70 |
|
reseq2 |
|
71 |
70
|
oveq2d |
|
72 |
71
|
breq2d |
|
73 |
69 72
|
anbi12d |
|
74 |
|
reseq2 |
|
75 |
74
|
oveq2d |
|
76 |
75 71
|
oveq12d |
|
77 |
|
reseq2 |
|
78 |
77
|
oveq2d |
|
79 |
78 71
|
oveq12d |
|
80 |
79
|
breq2d |
|
81 |
80
|
rabbidv |
|
82 |
76 81
|
eleq12d |
|
83 |
73 82
|
imbi12d |
|
84 |
83
|
imbi2d |
|
85 |
|
0re |
|
86 |
85
|
ltnri |
|
87 |
86
|
pm2.21i |
|
88 |
87
|
adantl |
|
89 |
88
|
a1i |
|
90 |
|
impexp |
|
91 |
|
simprl |
|
92 |
91
|
unssad |
|
93 |
|
simpr |
|
94 |
1
|
ad3antrrr |
|
95 |
2
|
ad3antrrr |
|
96 |
|
simplll |
|
97 |
96 3
|
sylan |
|
98 |
96 4
|
syl |
|
99 |
96 5
|
syl |
|
100 |
96 6
|
syl |
|
101 |
7
|
ad3antrrr |
|
102 |
96 8
|
sylan |
|
103 |
|
simpllr |
|
104 |
91
|
adantr |
|
105 |
|
eqid |
|
106 |
|
eqid |
|
107 |
|
cnring |
|
108 |
|
ringcmn |
|
109 |
107 108
|
mp1i |
|
110 |
4
|
ad2antrr |
|
111 |
110 92
|
ssfid |
|
112 |
|
rege0subm |
|
113 |
112
|
a1i |
|
114 |
5
|
ad2antrr |
|
115 |
114 92
|
fssresd |
|
116 |
|
c0ex |
|
117 |
116
|
a1i |
|
118 |
115 111 117
|
fdmfifsupp |
|
119 |
21 109 111 113 115 118
|
gsumsubmcl |
|
120 |
|
elrege0 |
|
121 |
120
|
simplbi |
|
122 |
119 121
|
syl |
|
123 |
122
|
adantr |
|
124 |
|
simprl |
|
125 |
123 124
|
elrpd |
|
126 |
|
simprr |
|
127 |
|
fveq2 |
|
128 |
127
|
breq1d |
|
129 |
128
|
elrab |
|
130 |
126 129
|
sylib |
|
131 |
130
|
simpld |
|
132 |
130
|
simprd |
|
133 |
94 95 97 98 99 100 101 102 103 104 105 106 125 131 132
|
jensenlem2 |
|
134 |
|
fveq2 |
|
135 |
134
|
breq1d |
|
136 |
135
|
elrab |
|
137 |
133 136
|
sylibr |
|
138 |
137
|
expr |
|
139 |
93 138
|
embantd |
|
140 |
|
cnfldbas |
|
141 |
|
ringmnd |
|
142 |
107 141
|
mp1i |
|
143 |
110 91
|
ssfid |
|
144 |
143
|
adantr |
|
145 |
|
ssun2 |
|
146 |
|
vsnid |
|
147 |
145 146
|
sselii |
|
148 |
147
|
a1i |
|
149 |
|
remulcl |
|
150 |
149
|
adantl |
|
151 |
|
rge0ssre |
|
152 |
|
fss |
|
153 |
5 151 152
|
sylancl |
|
154 |
6 1
|
fssd |
|
155 |
|
inidm |
|
156 |
150 153 154 4 4 155
|
off |
|
157 |
|
ax-resscn |
|
158 |
|
fss |
|
159 |
156 157 158
|
sylancl |
|
160 |
159
|
ad3antrrr |
|
161 |
91
|
adantr |
|
162 |
160 161
|
fssresd |
|
163 |
5
|
ad3antrrr |
|
164 |
110
|
adantr |
|
165 |
|
fex |
|
166 |
163 164 165
|
syl2anc |
|
167 |
6
|
ad3antrrr |
|
168 |
|
fex |
|
169 |
167 164 168
|
syl2anc |
|
170 |
|
offres |
|
171 |
166 169 170
|
syl2anc |
|
172 |
171
|
oveq1d |
|
173 |
151 157
|
sstri |
|
174 |
|
fss |
|
175 |
163 173 174
|
sylancl |
|
176 |
175 161
|
fssresd |
|
177 |
|
eldifi |
|
178 |
177
|
adantl |
|
179 |
178
|
fvresd |
|
180 |
|
difun2 |
|
181 |
|
difss |
|
182 |
180 181
|
eqsstri |
|
183 |
182
|
sseli |
|
184 |
|
simpr |
|
185 |
92
|
adantr |
|
186 |
163 185
|
feqresmpt |
|
187 |
186
|
oveq2d |
|
188 |
111
|
adantr |
|
189 |
185
|
sselda |
|
190 |
163
|
ffvelrnda |
|
191 |
189 190
|
syldan |
|
192 |
173 191
|
sseldi |
|
193 |
188 192
|
gsumfsum |
|
194 |
184 187 193
|
3eqtrrd |
|
195 |
|
elrege0 |
|
196 |
191 195
|
sylib |
|
197 |
196
|
simpld |
|
198 |
196
|
simprd |
|
199 |
188 197 198
|
fsum00 |
|
200 |
194 199
|
mpbid |
|
201 |
200
|
r19.21bi |
|
202 |
183 201
|
sylan2 |
|
203 |
179 202
|
eqtrd |
|
204 |
176 203
|
suppss |
|
205 |
|
mul02 |
|
206 |
205
|
adantl |
|
207 |
1
|
ad3antrrr |
|
208 |
207 157
|
sstrdi |
|
209 |
167 208
|
fssd |
|
210 |
209 161
|
fssresd |
|
211 |
116
|
a1i |
|
212 |
204 206 176 210 144 211
|
suppssof1 |
|
213 |
172 212
|
eqsstrd |
|
214 |
140 21 142 144 148 162 213
|
gsumpt |
|
215 |
148
|
fvresd |
|
216 |
163
|
ffnd |
|
217 |
167
|
ffnd |
|
218 |
161 148
|
sseldd |
|
219 |
|
fnfvof |
|
220 |
216 217 164 218 219
|
syl22anc |
|
221 |
214 215 220
|
3eqtrd |
|
222 |
140 21 142 144 148 176 204
|
gsumpt |
|
223 |
148
|
fvresd |
|
224 |
222 223
|
eqtrd |
|
225 |
221 224
|
oveq12d |
|
226 |
209 218
|
ffvelrnd |
|
227 |
175 218
|
ffvelrnd |
|
228 |
|
simplrr |
|
229 |
228 224
|
breqtrd |
|
230 |
229
|
gt0ne0d |
|
231 |
226 227 230
|
divcan3d |
|
232 |
225 231
|
eqtrd |
|
233 |
167 218
|
ffvelrnd |
|
234 |
232 233
|
eqeltrd |
|
235 |
2
|
ad3antrrr |
|
236 |
235 233
|
ffvelrnd |
|
237 |
236
|
leidd |
|
238 |
232
|
fveq2d |
|
239 |
|
fco |
|
240 |
2 6 239
|
syl2anc |
|
241 |
150 153 240 4 4 155
|
off |
|
242 |
|
fss |
|
243 |
241 157 242
|
sylancl |
|
244 |
243
|
ad3antrrr |
|
245 |
244 161
|
fssresd |
|
246 |
240
|
ad3antrrr |
|
247 |
|
fex |
|
248 |
246 164 247
|
syl2anc |
|
249 |
|
offres |
|
250 |
166 248 249
|
syl2anc |
|
251 |
250
|
oveq1d |
|
252 |
|
fss |
|
253 |
246 157 252
|
sylancl |
|
254 |
253 161
|
fssresd |
|
255 |
204 206 176 254 144 211
|
suppssof1 |
|
256 |
251 255
|
eqsstrd |
|
257 |
140 21 142 144 148 245 256
|
gsumpt |
|
258 |
148
|
fvresd |
|
259 |
2
|
ffnd |
|
260 |
|
fnfco |
|
261 |
259 6 260
|
syl2anc |
|
262 |
261
|
ad3antrrr |
|
263 |
|
fnfvof |
|
264 |
216 262 164 218 263
|
syl22anc |
|
265 |
|
fvco3 |
|
266 |
167 218 265
|
syl2anc |
|
267 |
266
|
oveq2d |
|
268 |
264 267
|
eqtrd |
|
269 |
257 258 268
|
3eqtrd |
|
270 |
269 224
|
oveq12d |
|
271 |
236
|
recnd |
|
272 |
271 227 230
|
divcan3d |
|
273 |
270 272
|
eqtrd |
|
274 |
237 238 273
|
3brtr4d |
|
275 |
135 234 274
|
elrabd |
|
276 |
275
|
a1d |
|
277 |
120
|
simprbi |
|
278 |
119 277
|
syl |
|
279 |
|
leloe |
|
280 |
85 122 279
|
sylancr |
|
281 |
278 280
|
mpbid |
|
282 |
139 276 281
|
mpjaodan |
|
283 |
92 282
|
embantd |
|
284 |
90 283
|
syl5bi |
|
285 |
284
|
ex |
|
286 |
285
|
com23 |
|
287 |
286
|
expcom |
|
288 |
287
|
adantl |
|
289 |
288
|
a2d |
|
290 |
36 52 68 84 89 289
|
findcard2s |
|
291 |
4 290
|
mpcom |
|
292 |
15 291
|
mpd |
|
293 |
156
|
ffnd |
|
294 |
|
fnresdm |
|
295 |
293 294
|
syl |
|
296 |
295
|
oveq2d |
|
297 |
296 12
|
oveq12d |
|
298 |
9 261 4 4 155
|
offn |
|
299 |
|
fnresdm |
|
300 |
298 299
|
syl |
|
301 |
300
|
oveq2d |
|
302 |
301 12
|
oveq12d |
|
303 |
302
|
breq2d |
|
304 |
303
|
rabbidv |
|
305 |
292 297 304
|
3eltr3d |
|
306 |
|
fveq2 |
|
307 |
306
|
breq1d |
|
308 |
307
|
elrab |
|
309 |
305 308
|
sylib |
|