Step |
Hyp |
Ref |
Expression |
1 |
|
gausslemma2d.p |
|
2 |
|
gausslemma2d.h |
|
3 |
|
gausslemma2d.r |
|
4 |
3
|
elrnmpt |
|
5 |
4
|
elv |
|
6 |
|
iftrue |
|
7 |
6
|
eqeq2d |
|
8 |
7
|
adantr |
|
9 |
|
elfz1b |
|
10 |
|
id |
|
11 |
|
2nn |
|
12 |
11
|
a1i |
|
13 |
10 12
|
nnmulcld |
|
14 |
13
|
3ad2ant1 |
|
15 |
14
|
3ad2ant1 |
|
16 |
2
|
eleq1i |
|
17 |
16
|
biimpi |
|
18 |
17
|
3ad2ant2 |
|
19 |
18
|
3ad2ant1 |
|
20 |
|
nnoddn2prm |
|
21 |
|
nnz |
|
22 |
21
|
anim1i |
|
23 |
20 22
|
syl |
|
24 |
|
nnz |
|
25 |
|
2z |
|
26 |
25
|
a1i |
|
27 |
24 26
|
zmulcld |
|
28 |
27
|
3ad2ant1 |
|
29 |
23 28
|
anim12i |
|
30 |
|
df-3an |
|
31 |
29 30
|
sylibr |
|
32 |
31
|
ex |
|
33 |
1 32
|
syl |
|
34 |
33
|
impcom |
|
35 |
|
ltoddhalfle |
|
36 |
34 35
|
syl |
|
37 |
36
|
biimp3a |
|
38 |
15 19 37
|
3jca |
|
39 |
38
|
3exp |
|
40 |
9 39
|
sylbi |
|
41 |
40
|
impcom |
|
42 |
41
|
impcom |
|
43 |
2
|
oveq2i |
|
44 |
43
|
eleq2i |
|
45 |
|
elfz1b |
|
46 |
44 45
|
bitri |
|
47 |
42 46
|
sylibr |
|
48 |
|
eleq1 |
|
49 |
47 48
|
syl5ibrcom |
|
50 |
8 49
|
sylbid |
|
51 |
|
iffalse |
|
52 |
51
|
eqeq2d |
|
53 |
52
|
adantr |
|
54 |
|
eldifi |
|
55 |
|
prmz |
|
56 |
1 54 55
|
3syl |
|
57 |
56
|
ad2antrl |
|
58 |
|
elfzelz |
|
59 |
25
|
a1i |
|
60 |
58 59
|
zmulcld |
|
61 |
60
|
ad2antll |
|
62 |
57 61
|
zsubcld |
|
63 |
55
|
zred |
|
64 |
2
|
breq2i |
|
65 |
|
nnre |
|
66 |
65
|
adantr |
|
67 |
|
peano2rem |
|
68 |
67
|
adantl |
|
69 |
|
2re |
|
70 |
|
2pos |
|
71 |
69 70
|
pm3.2i |
|
72 |
71
|
a1i |
|
73 |
|
lemuldiv |
|
74 |
66 68 72 73
|
syl3anc |
|
75 |
64 74
|
bitr4id |
|
76 |
13
|
nnred |
|
77 |
76
|
adantr |
|
78 |
|
simpr |
|
79 |
77 68 78
|
lesub2d |
|
80 |
|
recn |
|
81 |
|
1cnd |
|
82 |
80 81
|
nncand |
|
83 |
82
|
adantl |
|
84 |
83
|
breq1d |
|
85 |
84
|
biimpd |
|
86 |
79 85
|
sylbid |
|
87 |
75 86
|
sylbid |
|
88 |
87
|
impancom |
|
89 |
88
|
3adant2 |
|
90 |
9 89
|
sylbi |
|
91 |
90
|
com12 |
|
92 |
1 54 63 91
|
4syl |
|
93 |
92
|
imp |
|
94 |
93
|
adantl |
|
95 |
|
elnnz1 |
|
96 |
62 94 95
|
sylanbrc |
|
97 |
9
|
simp2bi |
|
98 |
97
|
ad2antll |
|
99 |
|
nnre |
|
100 |
99
|
rehalfcld |
|
101 |
100
|
adantr |
|
102 |
60
|
zred |
|
103 |
|
lenlt |
|
104 |
101 102 103
|
syl2an |
|
105 |
22 60
|
anim12i |
|
106 |
105 30
|
sylibr |
|
107 |
|
halfleoddlt |
|
108 |
106 107
|
syl |
|
109 |
108
|
biimpa |
|
110 |
|
nncn |
|
111 |
|
subhalfhalf |
|
112 |
110 111
|
syl |
|
113 |
112
|
breq1d |
|
114 |
113
|
ad3antrrr |
|
115 |
109 114
|
mpbird |
|
116 |
99
|
ad2antrr |
|
117 |
100
|
ad2antrr |
|
118 |
102
|
adantl |
|
119 |
116 117 118
|
3jca |
|
120 |
119
|
adantr |
|
121 |
|
ltsub23 |
|
122 |
120 121
|
syl |
|
123 |
115 122
|
mpbid |
|
124 |
21
|
ad2antrr |
|
125 |
|
simplr |
|
126 |
60
|
adantl |
|
127 |
124 126
|
zsubcld |
|
128 |
124 125 127
|
3jca |
|
129 |
128
|
adantr |
|
130 |
|
ltoddhalfle |
|
131 |
129 130
|
syl |
|
132 |
123 131
|
mpbid |
|
133 |
132
|
ex |
|
134 |
2
|
breq2i |
|
135 |
133 134
|
imbitrrdi |
|
136 |
104 135
|
sylbird |
|
137 |
136
|
ex |
|
138 |
1 20 137
|
3syl |
|
139 |
138
|
imp |
|
140 |
139
|
impcom |
|
141 |
|
elfz1b |
|
142 |
96 98 140 141
|
syl3anbrc |
|
143 |
|
eleq1 |
|
144 |
142 143
|
syl5ibrcom |
|
145 |
53 144
|
sylbid |
|
146 |
50 145
|
pm2.61ian |
|
147 |
146
|
rexlimdva |
|
148 |
|
elfz1b |
|
149 |
|
simp1 |
|
150 |
|
simpl |
|
151 |
|
nnehalf |
|
152 |
149 150 151
|
syl2anr |
|
153 |
|
simpr2 |
|
154 |
|
nnre |
|
155 |
154
|
ad2antrr |
|
156 |
|
nnrp |
|
157 |
156
|
adantl |
|
158 |
157
|
adantr |
|
159 |
|
2rp |
|
160 |
|
1le2 |
|
161 |
159 160
|
pm3.2i |
|
162 |
161
|
a1i |
|
163 |
|
ledivge1le |
|
164 |
155 158 162 163
|
syl3anc |
|
165 |
164
|
ex |
|
166 |
165
|
com23 |
|
167 |
166
|
3impia |
|
168 |
167
|
impcom |
|
169 |
152 153 168
|
3jca |
|
170 |
169
|
ex |
|
171 |
148 170
|
biimtrid |
|
172 |
171
|
3impia |
|
173 |
|
elfz1b |
|
174 |
172 173
|
sylibr |
|
175 |
|
oveq1 |
|
176 |
175
|
breq1d |
|
177 |
175
|
oveq2d |
|
178 |
176 175 177
|
ifbieq12d |
|
179 |
178
|
eqeq2d |
|
180 |
179
|
adantl |
|
181 |
|
elfzelz |
|
182 |
181
|
zcnd |
|
183 |
182
|
3ad2ant3 |
|
184 |
|
2cnd |
|
185 |
|
2ne0 |
|
186 |
185
|
a1i |
|
187 |
183 184 186
|
divcan1d |
|
188 |
2
|
breq2i |
|
189 |
|
nnz |
|
190 |
1 20 22
|
3syl |
|
191 |
190
|
adantl |
|
192 |
189 191
|
anim12ci |
|
193 |
|
df-3an |
|
194 |
192 193
|
sylibr |
|
195 |
|
ltoddhalfle |
|
196 |
194 195
|
syl |
|
197 |
196
|
exbiri |
|
198 |
197
|
com23 |
|
199 |
188 198
|
biimtrid |
|
200 |
199
|
a1d |
|
201 |
200
|
3imp |
|
202 |
148 201
|
sylbi |
|
203 |
202
|
com12 |
|
204 |
203
|
3impia |
|
205 |
187 204
|
eqbrtrd |
|
206 |
205
|
iftrued |
|
207 |
206 187
|
eqtr2d |
|
208 |
174 180 207
|
rspcedvd |
|
209 |
208
|
3exp |
|
210 |
54 55
|
syl |
|
211 |
210
|
ad2antrr |
|
212 |
189
|
3ad2ant1 |
|
213 |
212
|
adantl |
|
214 |
211 213
|
zsubcld |
|
215 |
154
|
ad2antrl |
|
216 |
67
|
rehalfcld |
|
217 |
216
|
adantr |
|
218 |
|
simpl |
|
219 |
215 217 218
|
3jca |
|
220 |
219
|
ex |
|
221 |
54 63 220
|
3syl |
|
222 |
221
|
adantr |
|
223 |
222
|
impcom |
|
224 |
|
lesub2 |
|
225 |
223 224
|
syl |
|
226 |
55
|
zcnd |
|
227 |
|
1cnd |
|
228 |
|
2cnne0 |
|
229 |
228
|
a1i |
|
230 |
|
divsubdir |
|
231 |
227 229 230
|
mpd3an23 |
|
232 |
231
|
oveq2d |
|
233 |
|
id |
|
234 |
|
halfcl |
|
235 |
|
halfcn |
|
236 |
235
|
a1i |
|
237 |
233 234 236
|
subsubd |
|
238 |
111
|
oveq1d |
|
239 |
232 237 238
|
3eqtrd |
|
240 |
54 226 239
|
3syl |
|
241 |
240
|
ad2antrl |
|
242 |
241
|
breq1d |
|
243 |
|
prmnn |
|
244 |
|
halfre |
|
245 |
244
|
a1i |
|
246 |
|
nngt0 |
|
247 |
71
|
a1i |
|
248 |
|
divgt0 |
|
249 |
99 246 247 248
|
syl21anc |
|
250 |
|
halfgt0 |
|
251 |
250
|
a1i |
|
252 |
100 245 249 251
|
addgt0d |
|
253 |
54 243 252
|
3syl |
|
254 |
253
|
ad2antrl |
|
255 |
|
0red |
|
256 |
|
simpr |
|
257 |
256
|
rehalfcld |
|
258 |
244
|
a1i |
|
259 |
257 258
|
readdcld |
|
260 |
|
resubcl |
|
261 |
260
|
ancoms |
|
262 |
255 259 261
|
3jca |
|
263 |
262
|
ex |
|
264 |
154 263
|
syl |
|
265 |
264
|
adantr |
|
266 |
265
|
com12 |
|
267 |
54 63 266
|
3syl |
|
268 |
267
|
adantr |
|
269 |
268
|
impcom |
|
270 |
|
ltletr |
|
271 |
269 270
|
syl |
|
272 |
254 271
|
mpand |
|
273 |
242 272
|
sylbid |
|
274 |
225 273
|
sylbid |
|
275 |
274
|
ex |
|
276 |
275
|
com23 |
|
277 |
188 276
|
biimtrid |
|
278 |
277
|
3impia |
|
279 |
278
|
impcom |
|
280 |
|
elnnz |
|
281 |
214 279 280
|
sylanbrc |
|
282 |
23
|
adantr |
|
283 |
|
simpr |
|
284 |
283 212
|
anim12ci |
|
285 |
|
omoe |
|
286 |
282 284 285
|
syl2an2r |
|
287 |
|
nnehalf |
|
288 |
281 286 287
|
syl2anc |
|
289 |
|
simpr2 |
|
290 |
|
1red |
|
291 |
154
|
3ad2ant1 |
|
292 |
291
|
adantl |
|
293 |
54 63
|
syl |
|
294 |
293
|
ad2antrr |
|
295 |
|
nnge1 |
|
296 |
295
|
3ad2ant1 |
|
297 |
296
|
adantl |
|
298 |
290 292 294 297
|
lesub2dd |
|
299 |
294 292
|
resubcld |
|
300 |
54 63 67
|
3syl |
|
301 |
300
|
ad2antrr |
|
302 |
71
|
a1i |
|
303 |
|
lediv1 |
|
304 |
299 301 302 303
|
syl3anc |
|
305 |
298 304
|
mpbid |
|
306 |
2
|
breq2i |
|
307 |
305 306
|
sylibr |
|
308 |
288 289 307
|
3jca |
|
309 |
308
|
ex |
|
310 |
|
elfz1b |
|
311 |
309 148 310
|
3imtr4g |
|
312 |
311
|
ex |
|
313 |
1 312
|
syl |
|
314 |
313
|
3imp21 |
|
315 |
|
oveq1 |
|
316 |
315
|
breq1d |
|
317 |
315
|
oveq2d |
|
318 |
316 315 317
|
ifbieq12d |
|
319 |
318
|
eqeq2d |
|
320 |
319
|
adantl |
|
321 |
1 54 226
|
3syl |
|
322 |
321
|
3ad2ant2 |
|
323 |
182
|
3ad2ant3 |
|
324 |
322 323
|
subcld |
|
325 |
|
2cnd |
|
326 |
185
|
a1i |
|
327 |
324 325 326
|
divcan1d |
|
328 |
|
zre |
|
329 |
|
halfge0 |
|
330 |
|
rehalfcl |
|
331 |
330
|
adantl |
|
332 |
331 258
|
subge02d |
|
333 |
329 332
|
mpbii |
|
334 |
|
simpl |
|
335 |
244
|
a1i |
|
336 |
330 335
|
resubcld |
|
337 |
336
|
adantl |
|
338 |
|
letr |
|
339 |
334 337 331 338
|
syl3anc |
|
340 |
333 339
|
mpan2d |
|
341 |
80
|
adantl |
|
342 |
|
1cnd |
|
343 |
228
|
a1i |
|
344 |
341 342 343 230
|
syl3anc |
|
345 |
344
|
breq2d |
|
346 |
|
lesub |
|
347 |
331 256 334 346
|
syl3anc |
|
348 |
257 261
|
lenltd |
|
349 |
|
2cnd |
|
350 |
185
|
a1i |
|
351 |
80 349 350
|
divcan1d |
|
352 |
351
|
eqcomd |
|
353 |
352
|
oveq1d |
|
354 |
330
|
recnd |
|
355 |
354 349
|
mulcomd |
|
356 |
355
|
oveq1d |
|
357 |
349 354
|
mulsubfacd |
|
358 |
|
2m1e1 |
|
359 |
358
|
a1i |
|
360 |
359
|
oveq1d |
|
361 |
354
|
mullidd |
|
362 |
357 360 361
|
3eqtrd |
|
363 |
353 356 362
|
3eqtrd |
|
364 |
363
|
adantl |
|
365 |
364
|
breq2d |
|
366 |
347 348 365
|
3bitr3d |
|
367 |
340 345 366
|
3imtr4d |
|
368 |
367
|
ex |
|
369 |
154 368
|
syl |
|
370 |
369
|
com3l |
|
371 |
328 370
|
syl |
|
372 |
1 54 55 371
|
4syl |
|
373 |
372
|
adantl |
|
374 |
373
|
com13 |
|
375 |
188 374
|
biimtrid |
|
376 |
375
|
a1d |
|
377 |
376
|
3imp |
|
378 |
377
|
com12 |
|
379 |
148 378
|
biimtrid |
|
380 |
379
|
3impia |
|
381 |
327 380
|
eqnbrtrd |
|
382 |
381
|
iffalsed |
|
383 |
327
|
oveq2d |
|
384 |
321 182
|
anim12i |
|
385 |
384
|
3adant1 |
|
386 |
|
nncan |
|
387 |
385 386
|
syl |
|
388 |
382 383 387
|
3eqtrrd |
|
389 |
314 320 388
|
rspcedvd |
|
390 |
389
|
3exp |
|
391 |
209 390
|
pm2.61i |
|
392 |
147 391
|
impbid |
|
393 |
5 392
|
bitrid |
|
394 |
393
|
eqrdv |
|