Step |
Hyp |
Ref |
Expression |
1 |
|
4sq.1 |
|
2 |
|
mul4sq.1 |
|
3 |
|
mul4sq.2 |
|
4 |
|
mul4sq.3 |
|
5 |
|
mul4sq.4 |
|
6 |
|
mul4sq.5 |
|
7 |
|
mul4sq.6 |
|
8 |
|
mul4sq.7 |
|
9 |
|
mul4sq.8 |
|
10 |
|
mul4sq.9 |
|
11 |
|
mul4sq.10 |
|
12 |
|
gzcn |
|
13 |
2 12
|
syl |
|
14 |
|
gzcn |
|
15 |
4 14
|
syl |
|
16 |
13 15
|
mulcld |
|
17 |
16
|
absvalsqd |
|
18 |
16
|
cjcld |
|
19 |
16 18
|
mulcld |
|
20 |
17 19
|
eqeltrd |
|
21 |
|
gzcn |
|
22 |
3 21
|
syl |
|
23 |
|
gzcn |
|
24 |
5 23
|
syl |
|
25 |
22 24
|
mulcld |
|
26 |
25
|
absvalsqd |
|
27 |
25
|
cjcld |
|
28 |
25 27
|
mulcld |
|
29 |
26 28
|
eqeltrd |
|
30 |
20 29
|
addcld |
|
31 |
13
|
cjcld |
|
32 |
31 15
|
mulcld |
|
33 |
22
|
cjcld |
|
34 |
33 24
|
mulcld |
|
35 |
32 34
|
mulcld |
|
36 |
15
|
cjcld |
|
37 |
22 36
|
mulcld |
|
38 |
24
|
cjcld |
|
39 |
13 38
|
mulcld |
|
40 |
37 39
|
mulcld |
|
41 |
35 40
|
addcld |
|
42 |
13 24
|
mulcld |
|
43 |
42
|
absvalsqd |
|
44 |
42
|
cjcld |
|
45 |
42 44
|
mulcld |
|
46 |
43 45
|
eqeltrd |
|
47 |
22 15
|
mulcld |
|
48 |
47
|
absvalsqd |
|
49 |
47
|
cjcld |
|
50 |
47 49
|
mulcld |
|
51 |
48 50
|
eqeltrd |
|
52 |
46 51
|
addcld |
|
53 |
30 41 52
|
ppncand |
|
54 |
22 38
|
mulcld |
|
55 |
32 54
|
addcld |
|
56 |
55
|
absvalsqd |
|
57 |
32 54
|
cjaddd |
|
58 |
31 15
|
cjmuld |
|
59 |
13
|
cjcjd |
|
60 |
59
|
oveq1d |
|
61 |
58 60
|
eqtrd |
|
62 |
22 38
|
cjmuld |
|
63 |
24
|
cjcjd |
|
64 |
63
|
oveq2d |
|
65 |
62 64
|
eqtrd |
|
66 |
61 65
|
oveq12d |
|
67 |
57 66
|
eqtrd |
|
68 |
67
|
oveq2d |
|
69 |
13 36
|
mulcld |
|
70 |
32 69 34
|
adddid |
|
71 |
15 31 13 36
|
mul4d |
|
72 |
31 15
|
mulcomd |
|
73 |
72
|
oveq1d |
|
74 |
13 15
|
mulcomd |
|
75 |
13 15
|
cjmuld |
|
76 |
74 75
|
oveq12d |
|
77 |
71 73 76
|
3eqtr4d |
|
78 |
77 17
|
eqtr4d |
|
79 |
78
|
oveq1d |
|
80 |
70 79
|
eqtrd |
|
81 |
54 69 34
|
adddid |
|
82 |
13 36
|
mulcomd |
|
83 |
82
|
oveq2d |
|
84 |
22 38 36 13
|
mul4d |
|
85 |
38 13
|
mulcomd |
|
86 |
85
|
oveq2d |
|
87 |
83 84 86
|
3eqtrd |
|
88 |
22 38 24 33
|
mul4d |
|
89 |
33 24
|
mulcomd |
|
90 |
89
|
oveq2d |
|
91 |
22 24
|
cjmuld |
|
92 |
33 38
|
mulcomd |
|
93 |
91 92
|
eqtrd |
|
94 |
93
|
oveq2d |
|
95 |
88 90 94
|
3eqtr4d |
|
96 |
95 26
|
eqtr4d |
|
97 |
87 96
|
oveq12d |
|
98 |
81 97
|
eqtrd |
|
99 |
80 98
|
oveq12d |
|
100 |
69 34
|
addcld |
|
101 |
32 54 100
|
adddird |
|
102 |
20 29 35 40
|
add42d |
|
103 |
99 101 102
|
3eqtr4d |
|
104 |
56 68 103
|
3eqtrd |
|
105 |
31 24
|
mulcld |
|
106 |
105 37
|
subcld |
|
107 |
106
|
absvalsqd |
|
108 |
|
cjsub |
|
109 |
105 37 108
|
syl2anc |
|
110 |
31 24
|
cjmuld |
|
111 |
59
|
oveq1d |
|
112 |
110 111
|
eqtrd |
|
113 |
22 36
|
cjmuld |
|
114 |
15
|
cjcjd |
|
115 |
114
|
oveq2d |
|
116 |
113 115
|
eqtrd |
|
117 |
112 116
|
oveq12d |
|
118 |
109 117
|
eqtrd |
|
119 |
118
|
oveq2d |
|
120 |
33 15
|
mulcld |
|
121 |
39 120
|
subcld |
|
122 |
105 37 121
|
subdird |
|
123 |
105 39 120
|
subdid |
|
124 |
24 31 13 38
|
mul4d |
|
125 |
31 24
|
mulcomd |
|
126 |
125
|
oveq1d |
|
127 |
13 24
|
mulcomd |
|
128 |
13 24
|
cjmuld |
|
129 |
127 128
|
oveq12d |
|
130 |
124 126 129
|
3eqtr4d |
|
131 |
130 43
|
eqtr4d |
|
132 |
33 15
|
mulcomd |
|
133 |
132
|
oveq2d |
|
134 |
31 24 15 33
|
mul4d |
|
135 |
24 33
|
mulcomd |
|
136 |
135
|
oveq2d |
|
137 |
133 134 136
|
3eqtrd |
|
138 |
131 137
|
oveq12d |
|
139 |
123 138
|
eqtrd |
|
140 |
37 39 120
|
subdid |
|
141 |
132
|
oveq2d |
|
142 |
22 36 15 33
|
mul4d |
|
143 |
36 33
|
mulcomd |
|
144 |
22 15
|
cjmuld |
|
145 |
143 144
|
eqtr4d |
|
146 |
145
|
oveq2d |
|
147 |
141 142 146
|
3eqtrd |
|
148 |
147 48
|
eqtr4d |
|
149 |
148
|
oveq2d |
|
150 |
140 149
|
eqtrd |
|
151 |
139 150
|
oveq12d |
|
152 |
46 35 40 51
|
subadd4d |
|
153 |
122 151 152
|
3eqtrd |
|
154 |
107 119 153
|
3eqtrd |
|
155 |
104 154
|
oveq12d |
|
156 |
13 31
|
mulcld |
|
157 |
22 33
|
mulcld |
|
158 |
15 36
|
mulcld |
|
159 |
24 38
|
mulcld |
|
160 |
158 159
|
addcld |
|
161 |
156 157 160
|
adddird |
|
162 |
75
|
oveq2d |
|
163 |
13 15 31 36
|
mul4d |
|
164 |
17 162 163
|
3eqtrd |
|
165 |
128
|
oveq2d |
|
166 |
13 24 31 38
|
mul4d |
|
167 |
43 165 166
|
3eqtrd |
|
168 |
164 167
|
oveq12d |
|
169 |
156 158 159
|
adddid |
|
170 |
168 169
|
eqtr4d |
|
171 |
144
|
oveq2d |
|
172 |
22 15 33 36
|
mul4d |
|
173 |
48 171 172
|
3eqtrd |
|
174 |
91
|
oveq2d |
|
175 |
22 24 33 38
|
mul4d |
|
176 |
26 174 175
|
3eqtrd |
|
177 |
173 176
|
oveq12d |
|
178 |
157 158 159
|
adddid |
|
179 |
177 178
|
eqtr4d |
|
180 |
170 179
|
oveq12d |
|
181 |
161 180
|
eqtr4d |
|
182 |
13
|
absvalsqd |
|
183 |
22
|
absvalsqd |
|
184 |
182 183
|
oveq12d |
|
185 |
6 184
|
syl5eq |
|
186 |
15
|
absvalsqd |
|
187 |
24
|
absvalsqd |
|
188 |
186 187
|
oveq12d |
|
189 |
7 188
|
syl5eq |
|
190 |
185 189
|
oveq12d |
|
191 |
20 29 46 51
|
add42d |
|
192 |
181 190 191
|
3eqtr4d |
|
193 |
53 155 192
|
3eqtr4d |
|
194 |
193
|
oveq1d |
|
195 |
8
|
nncnd |
|
196 |
8
|
nnne0d |
|
197 |
55 195 196
|
absdivd |
|
198 |
8
|
nnred |
|
199 |
8
|
nnnn0d |
|
200 |
199
|
nn0ge0d |
|
201 |
198 200
|
absidd |
|
202 |
201
|
oveq2d |
|
203 |
197 202
|
eqtrd |
|
204 |
203
|
oveq1d |
|
205 |
55
|
abscld |
|
206 |
205
|
recnd |
|
207 |
206 195 196
|
sqdivd |
|
208 |
204 207
|
eqtrd |
|
209 |
106 195 196
|
absdivd |
|
210 |
201
|
oveq2d |
|
211 |
209 210
|
eqtrd |
|
212 |
211
|
oveq1d |
|
213 |
106
|
abscld |
|
214 |
213
|
recnd |
|
215 |
214 195 196
|
sqdivd |
|
216 |
212 215
|
eqtrd |
|
217 |
208 216
|
oveq12d |
|
218 |
30 41
|
addcld |
|
219 |
104 218
|
eqeltrd |
|
220 |
52 41
|
subcld |
|
221 |
154 220
|
eqeltrd |
|
222 |
8
|
nnsqcld |
|
223 |
222
|
nncnd |
|
224 |
222
|
nnne0d |
|
225 |
219 221 223 224
|
divdird |
|
226 |
217 225
|
eqtr4d |
|
227 |
182 156
|
eqeltrd |
|
228 |
183 157
|
eqeltrd |
|
229 |
227 228
|
addcld |
|
230 |
6 229
|
eqeltrid |
|
231 |
189 160
|
eqeltrd |
|
232 |
230 195 231 195 196 196
|
divmuldivd |
|
233 |
195
|
sqvald |
|
234 |
233
|
oveq2d |
|
235 |
232 234
|
eqtr4d |
|
236 |
194 226 235
|
3eqtr4d |
|
237 |
230 55
|
nncand |
|
238 |
156 157 32 54
|
addsub4d |
|
239 |
185
|
oveq1d |
|
240 |
31 13 15
|
subdid |
|
241 |
31 13
|
mulcomd |
|
242 |
241
|
oveq1d |
|
243 |
240 242
|
eqtrd |
|
244 |
|
cjsub |
|
245 |
22 24 244
|
syl2anc |
|
246 |
245
|
oveq2d |
|
247 |
22 33 38
|
subdid |
|
248 |
246 247
|
eqtrd |
|
249 |
243 248
|
oveq12d |
|
250 |
238 239 249
|
3eqtr4d |
|
251 |
250
|
oveq2d |
|
252 |
237 251
|
eqtr3d |
|
253 |
252
|
oveq1d |
|
254 |
13 15
|
subcld |
|
255 |
31 254
|
mulcld |
|
256 |
22 24
|
subcld |
|
257 |
256
|
cjcld |
|
258 |
22 257
|
mulcld |
|
259 |
255 258
|
addcld |
|
260 |
230 259 195 196
|
divsubdird |
|
261 |
255 258 195 196
|
divdird |
|
262 |
31 254 195 196
|
divassd |
|
263 |
22 257 195 196
|
divassd |
|
264 |
256 195 196
|
cjdivd |
|
265 |
198
|
cjred |
|
266 |
265
|
oveq2d |
|
267 |
264 266
|
eqtrd |
|
268 |
267
|
oveq2d |
|
269 |
263 268
|
eqtr4d |
|
270 |
262 269
|
oveq12d |
|
271 |
261 270
|
eqtrd |
|
272 |
271
|
oveq2d |
|
273 |
253 260 272
|
3eqtrd |
|
274 |
11
|
nn0zd |
|
275 |
|
zgz |
|
276 |
274 275
|
syl |
|
277 |
|
gzcjcl |
|
278 |
2 277
|
syl |
|
279 |
|
gzmulcl |
|
280 |
278 9 279
|
syl2anc |
|
281 |
|
gzcjcl |
|
282 |
10 281
|
syl |
|
283 |
|
gzmulcl |
|
284 |
3 282 283
|
syl2anc |
|
285 |
|
gzaddcl |
|
286 |
280 284 285
|
syl2anc |
|
287 |
|
gzsubcl |
|
288 |
276 286 287
|
syl2anc |
|
289 |
273 288
|
eqeltrd |
|
290 |
254
|
cjcld |
|
291 |
22 290
|
mulcld |
|
292 |
31 256
|
mulcld |
|
293 |
291 292 195 196
|
divsubdird |
|
294 |
|
cjsub |
|
295 |
13 15 294
|
syl2anc |
|
296 |
295
|
oveq2d |
|
297 |
22 31 36
|
subdid |
|
298 |
296 297
|
eqtrd |
|
299 |
31 22 24
|
subdid |
|
300 |
31 22
|
mulcomd |
|
301 |
300
|
oveq1d |
|
302 |
299 301
|
eqtrd |
|
303 |
298 302
|
oveq12d |
|
304 |
22 31
|
mulcld |
|
305 |
304 37 105
|
nnncan1d |
|
306 |
303 305
|
eqtrd |
|
307 |
306
|
oveq1d |
|
308 |
293 307
|
eqtr3d |
|
309 |
22 290 195 196
|
divassd |
|
310 |
254 195 196
|
cjdivd |
|
311 |
265
|
oveq2d |
|
312 |
310 311
|
eqtrd |
|
313 |
312
|
oveq2d |
|
314 |
309 313
|
eqtr4d |
|
315 |
31 256 195 196
|
divassd |
|
316 |
314 315
|
oveq12d |
|
317 |
308 316
|
eqtr3d |
|
318 |
|
gzcjcl |
|
319 |
9 318
|
syl |
|
320 |
|
gzmulcl |
|
321 |
3 319 320
|
syl2anc |
|
322 |
|
gzmulcl |
|
323 |
278 10 322
|
syl2anc |
|
324 |
|
gzsubcl |
|
325 |
321 323 324
|
syl2anc |
|
326 |
317 325
|
eqeltrd |
|
327 |
1
|
4sqlem4a |
|
328 |
289 326 327
|
syl2anc |
|
329 |
236 328
|
eqeltrrd |
|