Step |
Hyp |
Ref |
Expression |
1 |
|
2sqmod.1 |
|
2 |
|
2sqmod.2 |
|
3 |
|
2sqmod.3 |
|
4 |
|
2sqmod.4 |
|
5 |
|
2sqmod.5 |
|
6 |
|
2sqmod.6 |
|
7 |
|
2sqmod.7 |
|
8 |
|
2sqmod.8 |
|
9 |
|
2sqmod.9 |
|
10 |
6
|
adantr |
|
11 |
4
|
nn0red |
|
12 |
11
|
adantr |
|
13 |
3
|
nn0red |
|
14 |
13
|
adantr |
|
15 |
4
|
nn0ge0d |
|
16 |
15
|
adantr |
|
17 |
3
|
nn0ge0d |
|
18 |
17
|
adantr |
|
19 |
4
|
nn0cnd |
|
20 |
19
|
sqcld |
|
21 |
20
|
adantr |
|
22 |
3
|
nn0cnd |
|
23 |
22
|
sqcld |
|
24 |
23
|
adantr |
|
25 |
2
|
nn0cnd |
|
26 |
25
|
sqcld |
|
27 |
5
|
nn0cnd |
|
28 |
27
|
sqcld |
|
29 |
8 9
|
eqtr4d |
|
30 |
26 23 20 28 29
|
subaddeqd |
|
31 |
30
|
adantr |
|
32 |
2
|
nn0zd |
|
33 |
4
|
nn0zd |
|
34 |
|
dvdsmul1 |
|
35 |
32 33 34
|
syl2anc |
|
36 |
35
|
adantr |
|
37 |
25 19
|
mulcld |
|
38 |
37
|
adantr |
|
39 |
22 27
|
mulcld |
|
40 |
39
|
adantr |
|
41 |
2
|
nn0red |
|
42 |
41 11
|
remulcld |
|
43 |
5
|
nn0red |
|
44 |
13 43
|
remulcld |
|
45 |
42 44
|
resubcld |
|
46 |
45
|
recnd |
|
47 |
46
|
adantr |
|
48 |
45
|
sqge0d |
|
49 |
3
|
nn0zd |
|
50 |
1 32 49 8
|
2sqn0 |
|
51 |
|
elnnne0 |
|
52 |
2 50 51
|
sylanbrc |
|
53 |
5
|
nn0zd |
|
54 |
28 20
|
addcomd |
|
55 |
54 9
|
eqtrd |
|
56 |
1 53 33 55
|
2sqn0 |
|
57 |
|
elnnne0 |
|
58 |
5 56 57
|
sylanbrc |
|
59 |
52 58
|
nnmulcld |
|
60 |
1 33 53 9
|
2sqn0 |
|
61 |
|
elnnne0 |
|
62 |
4 60 61
|
sylanbrc |
|
63 |
23 26
|
addcomd |
|
64 |
63 8
|
eqtrd |
|
65 |
1 49 32 64
|
2sqn0 |
|
66 |
|
elnnne0 |
|
67 |
3 65 66
|
sylanbrc |
|
68 |
62 67
|
nnmulcld |
|
69 |
59 68
|
nnaddcld |
|
70 |
69
|
nnsqcld |
|
71 |
70
|
nnred |
|
72 |
45
|
resqcld |
|
73 |
71 72
|
addge02d |
|
74 |
48 73
|
mpbid |
|
75 |
8 9
|
oveq12d |
|
76 |
|
bhmafibid1 |
|
77 |
41 13 11 43 76
|
syl22anc |
|
78 |
75 77
|
eqtr3d |
|
79 |
|
prmz |
|
80 |
1 79
|
syl |
|
81 |
80
|
zcnd |
|
82 |
81
|
sqvald |
|
83 |
19 22
|
mulcomd |
|
84 |
83
|
oveq2d |
|
85 |
84
|
oveq1d |
|
86 |
85
|
oveq2d |
|
87 |
78 82 86
|
3eqtr4d |
|
88 |
74 87
|
breqtrrd |
|
89 |
88
|
adantr |
|
90 |
32 53
|
zmulcld |
|
91 |
33 49
|
zmulcld |
|
92 |
90 91
|
zaddcld |
|
93 |
|
dvdssqim |
|
94 |
80 92 93
|
syl2anc |
|
95 |
|
zsqcl |
|
96 |
80 95
|
syl |
|
97 |
|
dvdsle |
|
98 |
96 70 97
|
syl2anc |
|
99 |
94 98
|
syld |
|
100 |
99
|
imp |
|
101 |
96
|
zred |
|
102 |
71 101
|
letri3d |
|
103 |
102
|
adantr |
|
104 |
89 100 103
|
mpbir2and |
|
105 |
87
|
adantr |
|
106 |
104 105
|
eqtr2d |
|
107 |
71
|
recnd |
|
108 |
72
|
recnd |
|
109 |
107 107 108
|
subadd2d |
|
110 |
109
|
adantr |
|
111 |
106 110
|
mpbird |
|
112 |
107
|
subidd |
|
113 |
112
|
adantr |
|
114 |
111 113
|
eqtr3d |
|
115 |
47 114
|
sqeq0d |
|
116 |
38 40 115
|
subeq0d |
|
117 |
36 116
|
breqtrd |
|
118 |
1 32 49 8
|
2sqcoprm |
|
119 |
118
|
adantr |
|
120 |
|
coprmdvds |
|
121 |
32 49 53 120
|
syl3anc |
|
122 |
121
|
adantr |
|
123 |
117 119 122
|
mp2and |
|
124 |
|
dvdsle |
|
125 |
32 58 124
|
syl2anc |
|
126 |
125
|
adantr |
|
127 |
123 126
|
mpd |
|
128 |
52
|
nnrpd |
|
129 |
128
|
rprege0d |
|
130 |
5
|
nn0ge0d |
|
131 |
|
le2sq |
|
132 |
129 43 130 131
|
syl12anc |
|
133 |
132
|
adantr |
|
134 |
127 133
|
mpbid |
|
135 |
52
|
nnsqcld |
|
136 |
135
|
nnred |
|
137 |
|
zsqcl |
|
138 |
53 137
|
syl |
|
139 |
138
|
zred |
|
140 |
136 139
|
suble0d |
|
141 |
140
|
adantr |
|
142 |
134 141
|
mpbird |
|
143 |
31 142
|
eqbrtrrd |
|
144 |
|
dvdsmul1 |
|
145 |
49 53 144
|
syl2anc |
|
146 |
145
|
adantr |
|
147 |
146 116
|
breqtrrd |
|
148 |
|
gcdcom |
|
149 |
32 49 148
|
syl2anc |
|
150 |
149 118
|
eqtr3d |
|
151 |
150
|
adantr |
|
152 |
|
coprmdvds |
|
153 |
49 32 33 152
|
syl3anc |
|
154 |
153
|
adantr |
|
155 |
147 151 154
|
mp2and |
|
156 |
|
dvdsle |
|
157 |
49 62 156
|
syl2anc |
|
158 |
157
|
adantr |
|
159 |
155 158
|
mpd |
|
160 |
13 11 17 15
|
le2sqd |
|
161 |
160
|
adantr |
|
162 |
159 161
|
mpbid |
|
163 |
11
|
resqcld |
|
164 |
|
zsqcl |
|
165 |
49 164
|
syl |
|
166 |
165
|
zred |
|
167 |
163 166
|
subge0d |
|
168 |
167
|
adantr |
|
169 |
162 168
|
mpbird |
|
170 |
136 139
|
resubcld |
|
171 |
30 170
|
eqeltrrd |
|
172 |
|
0red |
|
173 |
171 172
|
letri3d |
|
174 |
173
|
adantr |
|
175 |
143 169 174
|
mpbir2and |
|
176 |
21 24 175
|
subeq0d |
|
177 |
12 14 16 18 176
|
sq11d |
|
178 |
10 177
|
breqtrrd |
|
179 |
7
|
adantr |
|
180 |
41
|
adantr |
|
181 |
43
|
adantr |
|
182 |
2
|
nn0ge0d |
|
183 |
182
|
adantr |
|
184 |
130
|
adantr |
|
185 |
26
|
adantr |
|
186 |
28
|
adantr |
|
187 |
169 31
|
breqtrrd |
|
188 |
170 172
|
letri3d |
|
189 |
188
|
adantr |
|
190 |
142 187 189
|
mpbir2and |
|
191 |
185 186 190
|
subeq0d |
|
192 |
180 181 183 184 191
|
sq11d |
|
193 |
179 192
|
breqtrrd |
|
194 |
41 11
|
letri3d |
|
195 |
194
|
adantr |
|
196 |
178 193 195
|
mpbir2and |
|
197 |
25
|
adantr |
|
198 |
19
|
adantr |
|
199 |
22
|
adantr |
|
200 |
65
|
adantr |
|
201 |
43
|
adantr |
|
202 |
13
|
adantr |
|
203 |
130
|
adantr |
|
204 |
17
|
adantr |
|
205 |
28
|
adantr |
|
206 |
23
|
adantr |
|
207 |
|
prmnn |
|
208 |
1 207
|
syl |
|
209 |
208
|
nnne0d |
|
210 |
209
|
neneqd |
|
211 |
210
|
adantr |
|
212 |
81 28 23
|
subdid |
|
213 |
81 28
|
mulcld |
|
214 |
26 28
|
mulcld |
|
215 |
81 23
|
mulcld |
|
216 |
20 23
|
mulcld |
|
217 |
23 28
|
mulcomd |
|
218 |
8
|
oveq1d |
|
219 |
26 23
|
pncan2d |
|
220 |
218 219
|
eqtr3d |
|
221 |
220
|
oveq1d |
|
222 |
9
|
oveq1d |
|
223 |
20 28
|
pncan2d |
|
224 |
222 223
|
eqtr3d |
|
225 |
224
|
oveq1d |
|
226 |
217 221 225
|
3eqtr4d |
|
227 |
81 26 28
|
subdird |
|
228 |
81 20 23
|
subdird |
|
229 |
226 227 228
|
3eqtr3d |
|
230 |
213 214 215 216 229
|
subeqxfrd |
|
231 |
212 230
|
eqtrd |
|
232 |
25 27
|
sqmuld |
|
233 |
19 22
|
sqmuld |
|
234 |
232 233
|
oveq12d |
|
235 |
25 27
|
mulcld |
|
236 |
19 22
|
mulcld |
|
237 |
|
subsq |
|
238 |
235 236 237
|
syl2anc |
|
239 |
231 234 238
|
3eqtr2d |
|
240 |
239
|
adantr |
|
241 |
235
|
adantr |
|
242 |
|
simpll |
|
243 |
|
simpr |
|
244 |
243
|
neqned |
|
245 |
90 91
|
zsubcld |
|
246 |
|
dvdssqim |
|
247 |
80 245 246
|
syl2anc |
|
248 |
247
|
imp |
|
249 |
248
|
adantr |
|
250 |
96
|
adantr |
|
251 |
245
|
adantr |
|
252 |
235
|
adantr |
|
253 |
236
|
adantr |
|
254 |
|
simpr |
|
255 |
252 253 254
|
subne0d |
|
256 |
251 255
|
znsqcld |
|
257 |
|
dvdsle |
|
258 |
250 256 257
|
syl2anc |
|
259 |
258
|
imp |
|
260 |
242 244 249 259
|
syl21anc |
|
261 |
41 43
|
remulcld |
|
262 |
11 13
|
remulcld |
|
263 |
261 262
|
resubcld |
|
264 |
263
|
resqcld |
|
265 |
62
|
nnrpd |
|
266 |
128 265
|
rpmulcld |
|
267 |
67
|
nnrpd |
|
268 |
58
|
nnrpd |
|
269 |
267 268
|
rpmulcld |
|
270 |
266 269
|
rpaddcld |
|
271 |
|
2z |
|
272 |
271
|
a1i |
|
273 |
270 272
|
rpexpcld |
|
274 |
264 273
|
ltaddrp2d |
|
275 |
|
bhmafibid2 |
|
276 |
41 13 11 43 275
|
syl22anc |
|
277 |
75 276
|
eqtr3d |
|
278 |
83
|
oveq2d |
|
279 |
278
|
oveq1d |
|
280 |
279
|
oveq2d |
|
281 |
277 280
|
eqtr4d |
|
282 |
274 281
|
breqtrrd |
|
283 |
282 82
|
breqtrrd |
|
284 |
242 283
|
syl |
|
285 |
264 101
|
ltnled |
|
286 |
242 285
|
syl |
|
287 |
284 286
|
mpbid |
|
288 |
260 287
|
condan |
|
289 |
241 288
|
subeq0bd |
|
290 |
289
|
oveq2d |
|
291 |
235 236
|
addcld |
|
292 |
291
|
mul01d |
|
293 |
292
|
adantr |
|
294 |
240 290 293
|
3eqtrd |
|
295 |
28 23
|
subcld |
|
296 |
81 295
|
mul0ord |
|
297 |
296
|
adantr |
|
298 |
294 297
|
mpbid |
|
299 |
298
|
ord |
|
300 |
211 299
|
mpd |
|
301 |
205 206 300
|
subeq0d |
|
302 |
201 202 203 204 301
|
sq11d |
|
303 |
302
|
oveq2d |
|
304 |
303 288
|
eqtr3d |
|
305 |
197 198 199 200 304
|
mulcan2ad |
|
306 |
138 165
|
zsubcld |
|
307 |
|
dvdsmul1 |
|
308 |
80 306 307
|
syl2anc |
|
309 |
308 239
|
breqtrd |
|
310 |
|
euclemma |
|
311 |
1 92 245 310
|
syl3anc |
|
312 |
309 311
|
mpbid |
|
313 |
196 305 312
|
mpjaodan |
|
314 |
313
|
oveq1d |
|
315 |
314
|
oveq2d |
|
316 |
315 220 224
|
3eqtr3d |
|
317 |
13 43 17 130 316
|
sq11d |
|
318 |
313 317
|
jca |
|