Step |
Hyp |
Ref |
Expression |
1 |
|
lgamgulm.r |
|
2 |
|
lgamgulm.u |
|
3 |
|
lgamgulm.n |
|
4 |
|
lgamgulm.a |
|
5 |
|
lgamgulm.l |
|
6 |
|
1elunit |
|
7 |
|
0elunit |
|
8 |
|
0red |
|
9 |
|
1red |
|
10 |
|
eqid |
|
11 |
10
|
subcn |
|
12 |
11
|
a1i |
|
13 |
1 2
|
lgamgulmlem1 |
|
14 |
13 4
|
sseldd |
|
15 |
14
|
eldifad |
|
16 |
3
|
nnred |
|
17 |
16
|
recnd |
|
18 |
3
|
nnne0d |
|
19 |
15 17 18
|
divcld |
|
20 |
|
unitssre |
|
21 |
|
ax-resscn |
|
22 |
20 21
|
sstri |
|
23 |
22
|
a1i |
|
24 |
|
ssidd |
|
25 |
|
cncfmptc |
|
26 |
19 23 24 25
|
syl3anc |
|
27 |
|
cncfmptid |
|
28 |
22 24 27
|
sylancr |
|
29 |
26 28
|
mulcncf |
|
30 |
|
eqid |
|
31 |
30
|
logcn |
|
32 |
31
|
a1i |
|
33 |
|
cncff |
|
34 |
32 33
|
syl |
|
35 |
19
|
adantr |
|
36 |
|
simpr |
|
37 |
20 36
|
sseldi |
|
38 |
37
|
recnd |
|
39 |
35 38
|
mulcld |
|
40 |
|
1cnd |
|
41 |
39 40
|
addcld |
|
42 |
|
rere |
|
43 |
42
|
adantl |
|
44 |
41
|
recld |
|
45 |
39
|
recld |
|
46 |
45
|
recnd |
|
47 |
46
|
abscld |
|
48 |
39
|
abscld |
|
49 |
|
1red |
|
50 |
|
absrele |
|
51 |
39 50
|
syl |
|
52 |
49
|
rehalfcld |
|
53 |
1
|
nnred |
|
54 |
53
|
adantr |
|
55 |
3
|
adantr |
|
56 |
54 55
|
nndivred |
|
57 |
19
|
abscld |
|
58 |
57
|
adantr |
|
59 |
35
|
absge0d |
|
60 |
|
elicc01 |
|
61 |
60
|
simp2bi |
|
62 |
61
|
adantl |
|
63 |
15 17 18
|
absdivd |
|
64 |
3
|
nnrpd |
|
65 |
64
|
rpge0d |
|
66 |
16 65
|
absidd |
|
67 |
66
|
oveq2d |
|
68 |
63 67
|
eqtr2d |
|
69 |
15
|
abscld |
|
70 |
|
fveq2 |
|
71 |
70
|
breq1d |
|
72 |
|
fvoveq1 |
|
73 |
72
|
breq2d |
|
74 |
73
|
ralbidv |
|
75 |
71 74
|
anbi12d |
|
76 |
75 2
|
elrab2 |
|
77 |
76
|
simprbi |
|
78 |
4 77
|
syl |
|
79 |
78
|
simpld |
|
80 |
69 53 64 79
|
lediv1dd |
|
81 |
68 80
|
eqbrtrrd |
|
82 |
81
|
adantr |
|
83 |
60
|
simp3bi |
|
84 |
83
|
adantl |
|
85 |
58 56 37 49 59 62 82 84
|
lemul12ad |
|
86 |
35 38
|
absmuld |
|
87 |
37 62
|
absidd |
|
88 |
87
|
oveq2d |
|
89 |
86 88
|
eqtr2d |
|
90 |
56
|
recnd |
|
91 |
90
|
mulid1d |
|
92 |
85 89 91
|
3brtr3d |
|
93 |
|
2rp |
|
94 |
93
|
a1i |
|
95 |
53 16 94
|
lemuldiv2d |
|
96 |
5 95
|
mpbid |
|
97 |
|
2cnd |
|
98 |
|
2ne0 |
|
99 |
98
|
a1i |
|
100 |
17 97 99
|
divrecd |
|
101 |
96 100
|
breqtrd |
|
102 |
9
|
rehalfcld |
|
103 |
53 102 64
|
ledivmuld |
|
104 |
101 103
|
mpbird |
|
105 |
104
|
adantr |
|
106 |
48 56 52 92 105
|
letrd |
|
107 |
|
halflt1 |
|
108 |
107
|
a1i |
|
109 |
48 52 49 106 108
|
lelttrd |
|
110 |
47 48 49 51 109
|
lelttrd |
|
111 |
45 49
|
absltd |
|
112 |
110 111
|
mpbid |
|
113 |
112
|
simpld |
|
114 |
49
|
renegcld |
|
115 |
114 45
|
posdifd |
|
116 |
113 115
|
mpbid |
|
117 |
46 40
|
subnegd |
|
118 |
116 117
|
breqtrd |
|
119 |
39 40
|
readdd |
|
120 |
|
re1 |
|
121 |
120
|
oveq2i |
|
122 |
119 121
|
eqtrdi |
|
123 |
118 122
|
breqtrrd |
|
124 |
44 123
|
elrpd |
|
125 |
124
|
adantr |
|
126 |
43 125
|
eqeltrrd |
|
127 |
126
|
ex |
|
128 |
30
|
ellogdm |
|
129 |
41 127 128
|
sylanbrc |
|
130 |
34 129
|
cofmpt |
|
131 |
129
|
fvresd |
|
132 |
131
|
mpteq2dva |
|
133 |
130 132
|
eqtrd |
|
134 |
129
|
fmpttd |
|
135 |
|
difss |
|
136 |
10
|
addcn |
|
137 |
136
|
a1i |
|
138 |
|
1cnd |
|
139 |
|
cncfmptc |
|
140 |
138 23 24 139
|
syl3anc |
|
141 |
10 137 29 140
|
cncfmpt2f |
|
142 |
|
cncffvrn |
|
143 |
135 141 142
|
sylancr |
|
144 |
134 143
|
mpbird |
|
145 |
144 32
|
cncfco |
|
146 |
133 145
|
eqeltrrd |
|
147 |
10 12 29 146
|
cncfmpt2f |
|
148 |
21
|
a1i |
|
149 |
20
|
a1i |
|
150 |
30
|
logdmn0 |
|
151 |
129 150
|
syl |
|
152 |
41 151
|
logcld |
|
153 |
39 152
|
subcld |
|
154 |
10
|
tgioo2 |
|
155 |
|
0re |
|
156 |
|
iccntr |
|
157 |
155 9 156
|
sylancr |
|
158 |
148 149 153 154 10 157
|
dvmptntr |
|
159 |
|
reelprrecn |
|
160 |
159
|
a1i |
|
161 |
15
|
adantr |
|
162 |
17
|
adantr |
|
163 |
18
|
adantr |
|
164 |
161 162 163
|
divcld |
|
165 |
|
ioossicc |
|
166 |
165
|
sseli |
|
167 |
166 38
|
sylan2 |
|
168 |
164 167
|
mulcld |
|
169 |
15
|
adantr |
|
170 |
17
|
adantr |
|
171 |
18
|
adantr |
|
172 |
169 170 171
|
divcld |
|
173 |
148
|
sselda |
|
174 |
172 173
|
mulcld |
|
175 |
|
1cnd |
|
176 |
160
|
dvmptid |
|
177 |
160 173 175 176 19
|
dvmptcmul |
|
178 |
19
|
mulid1d |
|
179 |
178
|
mpteq2dv |
|
180 |
177 179
|
eqtrd |
|
181 |
165 149
|
sstrid |
|
182 |
|
retop |
|
183 |
|
iooretop |
|
184 |
|
isopn3i |
|
185 |
182 183 184
|
mp2an |
|
186 |
185
|
a1i |
|
187 |
160 174 172 180 181 154 10 186
|
dvmptres2 |
|
188 |
166 152
|
sylan2 |
|
189 |
|
1cnd |
|
190 |
168 189
|
addcld |
|
191 |
166 151
|
sylan2 |
|
192 |
190 191
|
reccld |
|
193 |
192 164
|
mulcld |
|
194 |
|
cnelprrecn |
|
195 |
194
|
a1i |
|
196 |
166 129
|
sylan2 |
|
197 |
|
eldifi |
|
198 |
197
|
adantl |
|
199 |
30
|
logdmn0 |
|
200 |
199
|
adantl |
|
201 |
198 200
|
logcld |
|
202 |
198 200
|
reccld |
|
203 |
174 175
|
addcld |
|
204 |
|
0cnd |
|
205 |
160 138
|
dvmptc |
|
206 |
160 174 172 180 175 204 205
|
dvmptadd |
|
207 |
19
|
addid1d |
|
208 |
207
|
mpteq2dv |
|
209 |
206 208
|
eqtrd |
|
210 |
160 203 172 209 181 154 10 186
|
dvmptres2 |
|
211 |
34
|
feqmptd |
|
212 |
|
fvres |
|
213 |
212
|
mpteq2ia |
|
214 |
211 213
|
eqtr2di |
|
215 |
214
|
oveq2d |
|
216 |
30
|
dvlog |
|
217 |
215 216
|
eqtrdi |
|
218 |
|
fveq2 |
|
219 |
|
oveq2 |
|
220 |
160 195 196 164 201 202 210 217 218 219
|
dvmptco |
|
221 |
160 168 164 187 188 193 220
|
dvmptsub |
|
222 |
158 221
|
eqtrd |
|
223 |
222
|
dmeqd |
|
224 |
|
ovex |
|
225 |
|
eqid |
|
226 |
224 225
|
dmmpti |
|
227 |
223 226
|
eqtrdi |
|
228 |
|
2re |
|
229 |
228
|
a1i |
|
230 |
229 53
|
remulcld |
|
231 |
1
|
nnrpd |
|
232 |
53 231
|
ltaddrpd |
|
233 |
53
|
recnd |
|
234 |
233
|
2timesd |
|
235 |
232 234
|
breqtrrd |
|
236 |
53 230 16 235 5
|
ltletrd |
|
237 |
|
difrp |
|
238 |
53 16 237
|
syl2anc |
|
239 |
236 238
|
mpbid |
|
240 |
239
|
rprecred |
|
241 |
3
|
nnrecred |
|
242 |
240 241
|
resubcld |
|
243 |
53 242
|
remulcld |
|
244 |
222
|
fveq1d |
|
245 |
244
|
fveq2d |
|
246 |
245
|
adantr |
|
247 |
|
nfv |
|
248 |
|
nfcv |
|
249 |
|
nffvmpt1 |
|
250 |
248 249
|
nffv |
|
251 |
|
nfcv |
|
252 |
|
nfcv |
|
253 |
250 251 252
|
nfbr |
|
254 |
247 253
|
nfim |
|
255 |
|
eleq1w |
|
256 |
255
|
anbi2d |
|
257 |
|
2fveq3 |
|
258 |
257
|
breq1d |
|
259 |
256 258
|
imbi12d |
|
260 |
|
simpr |
|
261 |
225
|
fvmpt2 |
|
262 |
260 224 261
|
sylancl |
|
263 |
262
|
fveq2d |
|
264 |
164 189 192
|
subdid |
|
265 |
164
|
mulid1d |
|
266 |
164 192
|
mulcomd |
|
267 |
265 266
|
oveq12d |
|
268 |
264 267
|
eqtr2d |
|
269 |
268
|
fveq2d |
|
270 |
161 162 163
|
absdivd |
|
271 |
16
|
adantr |
|
272 |
65
|
adantr |
|
273 |
271 272
|
absidd |
|
274 |
273
|
oveq2d |
|
275 |
270 274
|
eqtrd |
|
276 |
275
|
oveq1d |
|
277 |
189 192
|
subcld |
|
278 |
164 277
|
absmuld |
|
279 |
69
|
adantr |
|
280 |
279
|
recnd |
|
281 |
277
|
abscld |
|
282 |
281
|
recnd |
|
283 |
280 282 162 163
|
div23d |
|
284 |
276 278 283
|
3eqtr4d |
|
285 |
263 269 284
|
3eqtrd |
|
286 |
53
|
adantr |
|
287 |
240
|
adantr |
|
288 |
241
|
adantr |
|
289 |
287 288
|
resubcld |
|
290 |
271 289
|
remulcld |
|
291 |
15
|
absge0d |
|
292 |
291
|
adantr |
|
293 |
277
|
absge0d |
|
294 |
79
|
adantr |
|
295 |
239
|
adantr |
|
296 |
231
|
adantr |
|
297 |
295 296
|
rpdivcld |
|
298 |
14
|
dmgmn0 |
|
299 |
298
|
adantr |
|
300 |
161 162 299 163
|
divne0d |
|
301 |
|
eliooord |
|
302 |
301
|
adantl |
|
303 |
302
|
simpld |
|
304 |
303
|
gt0ne0d |
|
305 |
164 167 300 304
|
mulne0d |
|
306 |
168 305
|
reccld |
|
307 |
189 306
|
addcld |
|
308 |
168 189 168 305
|
divdird |
|
309 |
168 305
|
dividd |
|
310 |
309
|
oveq1d |
|
311 |
308 310
|
eqtrd |
|
312 |
190 168 191 305
|
divne0d |
|
313 |
311 312
|
eqnetrrd |
|
314 |
307 313
|
absrpcld |
|
315 |
|
1red |
|
316 |
|
0le1 |
|
317 |
316
|
a1i |
|
318 |
297
|
rpred |
|
319 |
306
|
negcld |
|
320 |
319
|
abscld |
|
321 |
320 315
|
resubcld |
|
322 |
307
|
abscld |
|
323 |
233
|
adantr |
|
324 |
296
|
rpne0d |
|
325 |
162 323 323 324
|
divsubdird |
|
326 |
323 324
|
dividd |
|
327 |
326
|
oveq2d |
|
328 |
325 327
|
eqtrd |
|
329 |
271 296
|
rerpdivcld |
|
330 |
323 162 324 163
|
recdivd |
|
331 |
166 92
|
sylan2 |
|
332 |
168 305
|
absrpcld |
|
333 |
64
|
adantr |
|
334 |
296 333
|
rpdivcld |
|
335 |
332 334
|
lerecd |
|
336 |
331 335
|
mpbid |
|
337 |
330 336
|
eqbrtrrd |
|
338 |
306
|
absnegd |
|
339 |
189 168 305
|
absdivd |
|
340 |
|
abs1 |
|
341 |
340
|
oveq1i |
|
342 |
339 341
|
eqtrdi |
|
343 |
338 342
|
eqtrd |
|
344 |
337 343
|
breqtrrd |
|
345 |
329 320 315 344
|
lesub1dd |
|
346 |
328 345
|
eqbrtrd |
|
347 |
340
|
oveq2i |
|
348 |
319 189
|
abs2difd |
|
349 |
347 348
|
eqbrtrrid |
|
350 |
189 306
|
addcomd |
|
351 |
350
|
negeqd |
|
352 |
306 189
|
negdi2d |
|
353 |
351 352
|
eqtrd |
|
354 |
353
|
fveq2d |
|
355 |
307
|
absnegd |
|
356 |
354 355
|
eqtr3d |
|
357 |
349 356
|
breqtrd |
|
358 |
318 321 322 346 357
|
letrd |
|
359 |
297 314 315 317 358
|
lediv2ad |
|
360 |
17 233
|
subcld |
|
361 |
360
|
adantr |
|
362 |
53 236
|
gtned |
|
363 |
17 233 362
|
subne0d |
|
364 |
363
|
adantr |
|
365 |
361 323 364 324
|
recdivd |
|
366 |
162 323
|
nncand |
|
367 |
366
|
oveq1d |
|
368 |
162 361 361 364
|
divsubdird |
|
369 |
367 368
|
eqtr3d |
|
370 |
361 364
|
dividd |
|
371 |
370
|
oveq2d |
|
372 |
365 369 371
|
3eqtrd |
|
373 |
359 372
|
breqtrd |
|
374 |
190 189 190 191
|
divsubdird |
|
375 |
168 189
|
pncand |
|
376 |
375
|
oveq1d |
|
377 |
190 191
|
dividd |
|
378 |
377
|
oveq1d |
|
379 |
374 376 378
|
3eqtr3rd |
|
380 |
190 168 191 305
|
recdivd |
|
381 |
311
|
oveq2d |
|
382 |
379 380 381
|
3eqtr2d |
|
383 |
382
|
fveq2d |
|
384 |
189 307 313
|
absdivd |
|
385 |
340
|
oveq1i |
|
386 |
384 385
|
eqtrdi |
|
387 |
383 386
|
eqtrd |
|
388 |
360 363
|
reccld |
|
389 |
388
|
adantr |
|
390 |
241
|
recnd |
|
391 |
390
|
adantr |
|
392 |
162 389 391
|
subdid |
|
393 |
162 361 364
|
divrecd |
|
394 |
393
|
eqcomd |
|
395 |
162 163
|
recidd |
|
396 |
394 395
|
oveq12d |
|
397 |
392 396
|
eqtrd |
|
398 |
373 387 397
|
3brtr4d |
|
399 |
279 286 281 290 292 293 294 398
|
lemul12ad |
|
400 |
242
|
recnd |
|
401 |
400
|
adantr |
|
402 |
323 162 401
|
mul12d |
|
403 |
399 402
|
breqtrd |
|
404 |
279 281
|
remulcld |
|
405 |
243
|
adantr |
|
406 |
404 405 333
|
ledivmuld |
|
407 |
403 406
|
mpbird |
|
408 |
285 407
|
eqbrtrd |
|
409 |
254 259 408
|
chvarfv |
|
410 |
246 409
|
eqbrtrd |
|
411 |
8 9 147 227 243 410
|
dvlip |
|
412 |
6 7 411
|
mpanr12 |
|
413 |
|
eqidd |
|
414 |
|
oveq2 |
|
415 |
414 178
|
sylan9eqr |
|
416 |
415
|
fvoveq1d |
|
417 |
415 416
|
oveq12d |
|
418 |
6
|
a1i |
|
419 |
|
ovexd |
|
420 |
413 417 418 419
|
fvmptd |
|
421 |
|
oveq2 |
|
422 |
19
|
mul01d |
|
423 |
421 422
|
sylan9eqr |
|
424 |
423
|
oveq1d |
|
425 |
|
0p1e1 |
|
426 |
424 425
|
eqtrdi |
|
427 |
426
|
fveq2d |
|
428 |
|
log1 |
|
429 |
427 428
|
eqtrdi |
|
430 |
423 429
|
oveq12d |
|
431 |
|
0m0e0 |
|
432 |
430 431
|
eqtrdi |
|
433 |
7
|
a1i |
|
434 |
413 432 433 433
|
fvmptd |
|
435 |
420 434
|
oveq12d |
|
436 |
19 138
|
addcld |
|
437 |
14 3
|
dmgmdivn0 |
|
438 |
436 437
|
logcld |
|
439 |
19 438
|
subcld |
|
440 |
439
|
subid1d |
|
441 |
435 440
|
eqtr2d |
|
442 |
441
|
fveq2d |
|
443 |
|
1m0e1 |
|
444 |
443
|
fveq2i |
|
445 |
444 340
|
eqtri |
|
446 |
445
|
oveq2i |
|
447 |
233 400
|
mulcld |
|
448 |
447
|
mulid1d |
|
449 |
446 448
|
syl5req |
|
450 |
412 442 449
|
3brtr4d |
|