Step |
Hyp |
Ref |
Expression |
1 |
|
reelprrecn |
|
2 |
1
|
a1i |
|
3 |
|
elioore |
|
4 |
3
|
recnd |
|
5 |
4
|
adantl |
|
6 |
|
rpcn |
|
7 |
6
|
adantr |
|
8 |
|
rpne0 |
|
9 |
8
|
adantr |
|
10 |
5 7 9
|
divcld |
|
11 |
|
asincl |
|
12 |
10 11
|
syl |
|
13 |
|
1cnd |
|
14 |
10
|
sqcld |
|
15 |
13 14
|
subcld |
|
16 |
15
|
sqrtcld |
|
17 |
10 16
|
mulcld |
|
18 |
12 17
|
addcld |
|
19 |
|
ovexd |
|
20 |
|
rpre |
|
21 |
20
|
renegcld |
|
22 |
21
|
rexrd |
|
23 |
|
rpxr |
|
24 |
|
elioo2 |
|
25 |
22 23 24
|
syl2anc |
|
26 |
|
simpr |
|
27 |
20
|
adantr |
|
28 |
8
|
adantr |
|
29 |
26 27 28
|
redivcld |
|
30 |
29
|
a1d |
|
31 |
6
|
mulm1d |
|
32 |
31
|
adantr |
|
33 |
32
|
breq1d |
|
34 |
|
neg1rr |
|
35 |
34
|
a1i |
|
36 |
|
simpl |
|
37 |
35 26 36
|
ltmuldivd |
|
38 |
33 37
|
bitr3d |
|
39 |
38
|
biimpd |
|
40 |
39
|
adantrd |
|
41 |
|
1red |
|
42 |
26 41 36
|
ltdivmuld |
|
43 |
6
|
mulid1d |
|
44 |
43
|
adantr |
|
45 |
44
|
breq2d |
|
46 |
42 45
|
bitr2d |
|
47 |
46
|
biimpd |
|
48 |
47
|
adantld |
|
49 |
30 40 48
|
3jcad |
|
50 |
49
|
exp4b |
|
51 |
50
|
3impd |
|
52 |
25 51
|
sylbid |
|
53 |
52
|
imp |
|
54 |
34
|
rexri |
|
55 |
|
1xr |
|
56 |
|
elioo2 |
|
57 |
54 55 56
|
mp2an |
|
58 |
53 57
|
sylibr |
|
59 |
|
ovexd |
|
60 |
|
elioore |
|
61 |
60
|
recnd |
|
62 |
|
asincl |
|
63 |
|
id |
|
64 |
|
1cnd |
|
65 |
|
sqcl |
|
66 |
64 65
|
subcld |
|
67 |
66
|
sqrtcld |
|
68 |
63 67
|
mulcld |
|
69 |
62 68
|
addcld |
|
70 |
61 69
|
syl |
|
71 |
70
|
adantl |
|
72 |
|
ovexd |
|
73 |
|
recn |
|
74 |
73
|
adantl |
|
75 |
|
1cnd |
|
76 |
2
|
dvmptid |
|
77 |
|
ioossre |
|
78 |
77
|
a1i |
|
79 |
|
eqid |
|
80 |
79
|
tgioo2 |
|
81 |
|
iooretop |
|
82 |
81
|
a1i |
|
83 |
2 74 75 76 78 80 79 82
|
dvmptres |
|
84 |
2 5 13 83 6 8
|
dvmptdivc |
|
85 |
61 62
|
syl |
|
86 |
85
|
adantl |
|
87 |
|
ovexd |
|
88 |
|
asinf |
|
89 |
88
|
a1i |
|
90 |
|
ioossre |
|
91 |
|
ax-resscn |
|
92 |
90 91
|
sstri |
|
93 |
92
|
a1i |
|
94 |
89 93
|
feqresmpt |
|
95 |
94
|
oveq2d |
|
96 |
|
dvreasin |
|
97 |
95 96
|
eqtr3di |
|
98 |
61 68
|
syl |
|
99 |
98
|
adantl |
|
100 |
|
ovexd |
|
101 |
61
|
adantl |
|
102 |
|
1cnd |
|
103 |
|
recn |
|
104 |
103
|
adantl |
|
105 |
|
1cnd |
|
106 |
2
|
dvmptid |
|
107 |
90
|
a1i |
|
108 |
|
iooretop |
|
109 |
108
|
a1i |
|
110 |
2 104 105 106 107 80 79 109
|
dvmptres |
|
111 |
61 67
|
syl |
|
112 |
111
|
adantl |
|
113 |
|
ovexd |
|
114 |
|
1red |
|
115 |
60
|
resqcld |
|
116 |
114 115
|
resubcld |
|
117 |
|
elioo2 |
|
118 |
54 55 117
|
mp2an |
|
119 |
|
id |
|
120 |
|
1red |
|
121 |
119 120
|
absltd |
|
122 |
103
|
abscld |
|
123 |
103
|
absge0d |
|
124 |
|
0le1 |
|
125 |
124
|
a1i |
|
126 |
122 120 123 125
|
lt2sqd |
|
127 |
|
absresq |
|
128 |
|
sq1 |
|
129 |
128
|
a1i |
|
130 |
127 129
|
breq12d |
|
131 |
|
resqcl |
|
132 |
131 120
|
posdifd |
|
133 |
126 130 132
|
3bitrd |
|
134 |
121 133
|
bitr3d |
|
135 |
134
|
biimpd |
|
136 |
135
|
3impib |
|
137 |
118 136
|
sylbi |
|
138 |
116 137
|
elrpd |
|
139 |
138
|
adantl |
|
140 |
|
negex |
|
141 |
140
|
a1i |
|
142 |
|
rpcn |
|
143 |
142
|
sqrtcld |
|
144 |
143
|
adantl |
|
145 |
|
ovexd |
|
146 |
|
1cnd |
|
147 |
103
|
sqcld |
|
148 |
146 147
|
subcld |
|
149 |
148
|
adantl |
|
150 |
140
|
a1i |
|
151 |
|
0red |
|
152 |
|
1cnd |
|
153 |
2 152
|
dvmptc |
|
154 |
147
|
adantl |
|
155 |
|
ovexd |
|
156 |
79
|
cnfldtopon |
|
157 |
|
toponmax |
|
158 |
156 157
|
mp1i |
|
159 |
|
df-ss |
|
160 |
91 159
|
mpbi |
|
161 |
160
|
a1i |
|
162 |
65
|
adantl |
|
163 |
|
ovexd |
|
164 |
|
2nn |
|
165 |
|
dvexp |
|
166 |
164 165
|
ax-mp |
|
167 |
|
2m1e1 |
|
168 |
167
|
oveq2i |
|
169 |
|
exp1 |
|
170 |
168 169
|
eqtrid |
|
171 |
170
|
oveq2d |
|
172 |
171
|
mpteq2ia |
|
173 |
166 172
|
eqtri |
|
174 |
173
|
a1i |
|
175 |
79 2 158 161 162 163 174
|
dvmptres3 |
|
176 |
2 105 151 153 154 155 175
|
dvmptsub |
|
177 |
|
df-neg |
|
178 |
177
|
mpteq2i |
|
179 |
176 178
|
eqtr4di |
|
180 |
2 149 150 179 107 80 79 109
|
dvmptres |
|
181 |
|
dvsqrt |
|
182 |
181
|
a1i |
|
183 |
|
fveq2 |
|
184 |
183
|
oveq2d |
|
185 |
184
|
oveq2d |
|
186 |
2 2 139 141 144 145 180 182 183 185
|
dvmptco |
|
187 |
|
2cnd |
|
188 |
187 61
|
mulneg2d |
|
189 |
188
|
oveq1d |
|
190 |
61
|
negcld |
|
191 |
137
|
gt0ne0d |
|
192 |
61 66
|
syl |
|
193 |
192
|
adantr |
|
194 |
|
simpr |
|
195 |
193 194
|
sqr00d |
|
196 |
195
|
ex |
|
197 |
196
|
necon3d |
|
198 |
191 197
|
mpd |
|
199 |
|
2ne0 |
|
200 |
199
|
a1i |
|
201 |
190 111 187 198 200
|
divcan5d |
|
202 |
187 61
|
mulcld |
|
203 |
202
|
negcld |
|
204 |
187 111
|
mulcld |
|
205 |
187 111 200 198
|
mulne0d |
|
206 |
203 204 205
|
divrec2d |
|
207 |
189 201 206
|
3eqtr3rd |
|
208 |
207
|
mpteq2ia |
|
209 |
186 208
|
eqtrdi |
|
210 |
2 101 102 110 112 113 209
|
dvmptmul |
|
211 |
2 86 87 97 99 100 210
|
dvmptadd |
|
212 |
111
|
mulid2d |
|
213 |
190 111 198
|
divcld |
|
214 |
213 61
|
mulcomd |
|
215 |
61 190 111 198
|
divassd |
|
216 |
61 61
|
mulneg2d |
|
217 |
61
|
sqvald |
|
218 |
217
|
negeqd |
|
219 |
216 218
|
eqtr4d |
|
220 |
219
|
oveq1d |
|
221 |
214 215 220
|
3eqtr2d |
|
222 |
212 221
|
oveq12d |
|
223 |
61
|
sqcld |
|
224 |
223
|
negcld |
|
225 |
224 111 198
|
divcld |
|
226 |
111 225
|
addcomd |
|
227 |
222 226
|
eqtrd |
|
228 |
227
|
oveq2d |
|
229 |
111
|
2timesd |
|
230 |
64 65
|
negsubd |
|
231 |
66
|
sqsqrtd |
|
232 |
67
|
sqvald |
|
233 |
230 231 232
|
3eqtr2d |
|
234 |
61 233
|
syl |
|
235 |
234
|
oveq1d |
|
236 |
|
1cnd |
|
237 |
236 224 111 198
|
divdird |
|
238 |
111 111 198
|
divcan3d |
|
239 |
235 237 238
|
3eqtr3rd |
|
240 |
239
|
oveq1d |
|
241 |
111 198
|
reccld |
|
242 |
241 225 111
|
addassd |
|
243 |
229 240 242
|
3eqtrrd |
|
244 |
228 243
|
eqtrd |
|
245 |
244
|
mpteq2ia |
|
246 |
211 245
|
eqtrdi |
|
247 |
|
fveq2 |
|
248 |
|
id |
|
249 |
|
oveq1 |
|
250 |
249
|
oveq2d |
|
251 |
250
|
fveq2d |
|
252 |
248 251
|
oveq12d |
|
253 |
247 252
|
oveq12d |
|
254 |
251
|
oveq2d |
|
255 |
2 2 58 59 71 72 84 246 253 254
|
dvmptco |
|
256 |
6
|
sqcld |
|
257 |
2 18 19 255 256
|
dvmptcmul |
|
258 |
|
2cnd |
|
259 |
258 16
|
mulcld |
|
260 |
6 8
|
reccld |
|
261 |
260
|
adantr |
|
262 |
259 261
|
mulcomd |
|
263 |
262
|
oveq2d |
|
264 |
256
|
adantr |
|
265 |
264 261 259
|
mulassd |
|
266 |
6
|
sqvald |
|
267 |
266
|
oveq1d |
|
268 |
256 6 8
|
divrecd |
|
269 |
6 6 8
|
divcan3d |
|
270 |
267 268 269
|
3eqtr3d |
|
271 |
270
|
adantr |
|
272 |
271
|
oveq1d |
|
273 |
7 258 16
|
mul12d |
|
274 |
20
|
resqcld |
|
275 |
274
|
adantr |
|
276 |
20
|
sqge0d |
|
277 |
276
|
adantr |
|
278 |
|
1red |
|
279 |
3
|
adantl |
|
280 |
20
|
adantr |
|
281 |
279 280 9
|
redivcld |
|
282 |
281
|
resqcld |
|
283 |
278 282
|
resubcld |
|
284 |
|
0red |
|
285 |
26 27
|
absltd |
|
286 |
73
|
abscld |
|
287 |
286
|
adantl |
|
288 |
73
|
absge0d |
|
289 |
288
|
adantl |
|
290 |
|
rpge0 |
|
291 |
290
|
adantr |
|
292 |
287 27 289 291
|
lt2sqd |
|
293 |
|
absresq |
|
294 |
293
|
adantl |
|
295 |
256
|
adantr |
|
296 |
295
|
mulid1d |
|
297 |
296
|
eqcomd |
|
298 |
294 297
|
breq12d |
|
299 |
6
|
adantr |
|
300 |
74 299 28
|
sqdivd |
|
301 |
300
|
breq1d |
|
302 |
29
|
resqcld |
|
303 |
302 41
|
posdifd |
|
304 |
|
resqcl |
|
305 |
304
|
adantl |
|
306 |
|
rpgt0 |
|
307 |
|
0red |
|
308 |
|
0le0 |
|
309 |
308
|
a1i |
|
310 |
307 20 309 290
|
lt2sqd |
|
311 |
|
sq0 |
|
312 |
311
|
a1i |
|
313 |
312
|
breq1d |
|
314 |
310 313
|
bitrd |
|
315 |
306 314
|
mpbid |
|
316 |
274 315
|
elrpd |
|
317 |
316
|
adantr |
|
318 |
305 41 317
|
ltdivmuld |
|
319 |
301 303 318
|
3bitr3rd |
|
320 |
292 298 319
|
3bitrd |
|
321 |
285 320
|
bitr3d |
|
322 |
321
|
biimpd |
|
323 |
322
|
exp4b |
|
324 |
323
|
3impd |
|
325 |
25 324
|
sylbid |
|
326 |
325
|
imp |
|
327 |
284 283 326
|
ltled |
|
328 |
275 277 283 327
|
sqrtmuld |
|
329 |
264 13 14
|
subdid |
|
330 |
264
|
mulid1d |
|
331 |
5 7 9
|
sqdivd |
|
332 |
331
|
oveq2d |
|
333 |
4
|
sqcld |
|
334 |
333
|
adantl |
|
335 |
|
sqne0 |
|
336 |
6 335
|
syl |
|
337 |
8 336
|
mpbird |
|
338 |
337
|
adantr |
|
339 |
334 264 338
|
divcan2d |
|
340 |
332 339
|
eqtrd |
|
341 |
330 340
|
oveq12d |
|
342 |
329 341
|
eqtrd |
|
343 |
342
|
fveq2d |
|
344 |
20 290
|
sqrtsqd |
|
345 |
344
|
adantr |
|
346 |
345
|
oveq1d |
|
347 |
328 343 346
|
3eqtr3rd |
|
348 |
347
|
oveq2d |
|
349 |
272 273 348
|
3eqtrd |
|
350 |
263 265 349
|
3eqtr2d |
|
351 |
350
|
mpteq2dva |
|
352 |
257 351
|
eqtrd |
|