Step |
Hyp |
Ref |
Expression |
1 |
|
2z |
|
2 |
|
eluzelz |
|
3 |
2
|
adantr |
|
4 |
|
zmulcl |
|
5 |
1 3 4
|
sylancr |
|
6 |
|
nn0z |
|
7 |
6
|
adantl |
|
8 |
5 7
|
zmulcld |
|
9 |
|
zsqcl |
|
10 |
7 9
|
syl |
|
11 |
8 10
|
zsubcld |
|
12 |
|
peano2zm |
|
13 |
11 12
|
syl |
|
14 |
|
dvds0 |
|
15 |
13 14
|
syl |
|
16 |
|
rmx0 |
|
17 |
16
|
adantr |
|
18 |
|
rmy0 |
|
19 |
18
|
adantr |
|
20 |
19
|
oveq2d |
|
21 |
3 7
|
zsubcld |
|
22 |
21
|
zcnd |
|
23 |
22
|
mul01d |
|
24 |
20 23
|
eqtrd |
|
25 |
17 24
|
oveq12d |
|
26 |
|
1m0e1 |
|
27 |
25 26
|
eqtrdi |
|
28 |
|
nn0cn |
|
29 |
28
|
adantl |
|
30 |
29
|
exp0d |
|
31 |
27 30
|
oveq12d |
|
32 |
|
1m1e0 |
|
33 |
31 32
|
eqtrdi |
|
34 |
15 33
|
breqtrrd |
|
35 |
|
rmx1 |
|
36 |
35
|
adantr |
|
37 |
|
rmy1 |
|
38 |
37
|
adantr |
|
39 |
38
|
oveq2d |
|
40 |
22
|
mulid1d |
|
41 |
39 40
|
eqtrd |
|
42 |
36 41
|
oveq12d |
|
43 |
3
|
zcnd |
|
44 |
43 29
|
nncand |
|
45 |
42 44
|
eqtrd |
|
46 |
29
|
exp1d |
|
47 |
45 46
|
oveq12d |
|
48 |
29
|
subidd |
|
49 |
47 48
|
eqtrd |
|
50 |
15 49
|
breqtrrd |
|
51 |
|
pm3.43 |
|
52 |
13
|
adantr |
|
53 |
5
|
adantr |
|
54 |
|
simpll |
|
55 |
|
nnz |
|
56 |
55
|
adantl |
|
57 |
|
frmx |
|
58 |
57
|
fovcl |
|
59 |
54 56 58
|
syl2anc |
|
60 |
59
|
nn0zd |
|
61 |
21
|
adantr |
|
62 |
|
frmy |
|
63 |
62
|
fovcl |
|
64 |
54 56 63
|
syl2anc |
|
65 |
61 64
|
zmulcld |
|
66 |
60 65
|
zsubcld |
|
67 |
53 66
|
zmulcld |
|
68 |
|
peano2zm |
|
69 |
56 68
|
syl |
|
70 |
57
|
fovcl |
|
71 |
54 69 70
|
syl2anc |
|
72 |
71
|
nn0zd |
|
73 |
62
|
fovcl |
|
74 |
54 69 73
|
syl2anc |
|
75 |
61 74
|
zmulcld |
|
76 |
72 75
|
zsubcld |
|
77 |
67 76
|
zsubcld |
|
78 |
52 77
|
jca |
|
79 |
78
|
adantr |
|
80 |
7
|
adantr |
|
81 |
|
nnnn0 |
|
82 |
81
|
adantl |
|
83 |
|
zexpcl |
|
84 |
80 82 83
|
syl2anc |
|
85 |
53 84
|
zmulcld |
|
86 |
|
nnm1nn0 |
|
87 |
86
|
adantl |
|
88 |
|
zexpcl |
|
89 |
80 87 88
|
syl2anc |
|
90 |
85 89
|
zsubcld |
|
91 |
|
0z |
|
92 |
|
zaddcl |
|
93 |
91 10 92
|
sylancr |
|
94 |
93
|
adantr |
|
95 |
89 94
|
zmulcld |
|
96 |
90 95
|
jca |
|
97 |
96
|
adantr |
|
98 |
52 67 85
|
3jca |
|
99 |
98
|
adantr |
|
100 |
76 89
|
jca |
|
101 |
100
|
adantr |
|
102 |
13 5 5
|
3jca |
|
103 |
102
|
ad2antrr |
|
104 |
66 84
|
jca |
|
105 |
104
|
adantr |
|
106 |
|
congid |
|
107 |
13 5 106
|
syl2anc |
|
108 |
107
|
ad2antrr |
|
109 |
|
simpr |
|
110 |
|
congmul |
|
111 |
103 105 108 109 110
|
syl112anc |
|
112 |
111
|
adantrl |
|
113 |
|
simprl |
|
114 |
|
congsub |
|
115 |
99 101 112 113 114
|
syl112anc |
|
116 |
13 10
|
zaddcld |
|
117 |
116
|
adantr |
|
118 |
|
congid |
|
119 |
52 89 118
|
syl2anc |
|
120 |
|
0zd |
|
121 |
|
iddvds |
|
122 |
13 121
|
syl |
|
123 |
13
|
zcnd |
|
124 |
123
|
subid1d |
|
125 |
122 124
|
breqtrrd |
|
126 |
|
congid |
|
127 |
13 10 126
|
syl2anc |
|
128 |
|
congadd |
|
129 |
13 13 120 10 10 125 127 128
|
syl322anc |
|
130 |
129
|
adantr |
|
131 |
|
congmul |
|
132 |
52 89 89 117 94 119 130 131
|
syl322anc |
|
133 |
11
|
zcnd |
|
134 |
29
|
sqcld |
|
135 |
|
1cnd |
|
136 |
133 134 135
|
addsubd |
|
137 |
8
|
zcnd |
|
138 |
137 134
|
npcand |
|
139 |
138
|
oveq1d |
|
140 |
136 139
|
eqtr3d |
|
141 |
140
|
adantr |
|
142 |
141
|
oveq2d |
|
143 |
28
|
ad2antlr |
|
144 |
143 87
|
expcld |
|
145 |
137
|
adantr |
|
146 |
|
1cnd |
|
147 |
144 145 146
|
subdid |
|
148 |
5
|
zcnd |
|
149 |
148
|
adantr |
|
150 |
144 149 143
|
mul12d |
|
151 |
|
simpr |
|
152 |
|
expm1t |
|
153 |
143 151 152
|
syl2anc |
|
154 |
153
|
oveq2d |
|
155 |
150 154
|
eqtr4d |
|
156 |
144
|
mulid1d |
|
157 |
155 156
|
oveq12d |
|
158 |
142 147 157
|
3eqtrrd |
|
159 |
158
|
oveq1d |
|
160 |
132 159
|
breqtrrd |
|
161 |
160
|
adantr |
|
162 |
|
congtr |
|
163 |
79 97 115 161 162
|
syl112anc |
|
164 |
|
rmxluc |
|
165 |
54 56 164
|
syl2anc |
|
166 |
|
rmyluc |
|
167 |
54 56 166
|
syl2anc |
|
168 |
167
|
oveq2d |
|
169 |
2
|
zcnd |
|
170 |
169
|
ad2antrr |
|
171 |
170 143
|
subcld |
|
172 |
|
2cn |
|
173 |
63
|
zcnd |
|
174 |
54 56 173
|
syl2anc |
|
175 |
174 170
|
mulcld |
|
176 |
|
mulcl |
|
177 |
172 175 176
|
sylancr |
|
178 |
73
|
zcnd |
|
179 |
54 69 178
|
syl2anc |
|
180 |
171 177 179
|
subdid |
|
181 |
|
2cnd |
|
182 |
181 174 170
|
mul12d |
|
183 |
174 149
|
mulcomd |
|
184 |
182 183
|
eqtrd |
|
185 |
184
|
oveq2d |
|
186 |
171 149 174
|
mul12d |
|
187 |
185 186
|
eqtrd |
|
188 |
187
|
oveq1d |
|
189 |
168 180 188
|
3eqtrd |
|
190 |
165 189
|
oveq12d |
|
191 |
58
|
nn0cnd |
|
192 |
54 56 191
|
syl2anc |
|
193 |
149 192
|
mulcld |
|
194 |
70
|
nn0cnd |
|
195 |
54 69 194
|
syl2anc |
|
196 |
171 174
|
mulcld |
|
197 |
149 196
|
mulcld |
|
198 |
171 179
|
mulcld |
|
199 |
193 195 197 198
|
sub4d |
|
200 |
149 192 196
|
subdid |
|
201 |
200
|
eqcomd |
|
202 |
201
|
oveq1d |
|
203 |
190 199 202
|
3eqtrd |
|
204 |
143 82
|
expp1d |
|
205 |
|
nncn |
|
206 |
205
|
adantl |
|
207 |
|
ax-1cn |
|
208 |
|
npcan |
|
209 |
206 207 208
|
sylancl |
|
210 |
209
|
oveq2d |
|
211 |
143 87
|
expp1d |
|
212 |
210 211
|
eqtr3d |
|
213 |
212
|
oveq1d |
|
214 |
144 143 143
|
mulassd |
|
215 |
134
|
addid2d |
|
216 |
29
|
sqvald |
|
217 |
215 216
|
eqtr2d |
|
218 |
217
|
adantr |
|
219 |
218
|
oveq2d |
|
220 |
214 219
|
eqtrd |
|
221 |
204 213 220
|
3eqtrd |
|
222 |
203 221
|
oveq12d |
|
223 |
222
|
adantr |
|
224 |
163 223
|
breqtrrd |
|
225 |
224
|
ex |
|
226 |
225
|
expcom |
|
227 |
226
|
a2d |
|
228 |
51 227
|
syl5 |
|
229 |
|
oveq2 |
|
230 |
|
oveq2 |
|
231 |
230
|
oveq2d |
|
232 |
229 231
|
oveq12d |
|
233 |
|
oveq2 |
|
234 |
232 233
|
oveq12d |
|
235 |
234
|
breq2d |
|
236 |
235
|
imbi2d |
|
237 |
|
oveq2 |
|
238 |
|
oveq2 |
|
239 |
238
|
oveq2d |
|
240 |
237 239
|
oveq12d |
|
241 |
|
oveq2 |
|
242 |
240 241
|
oveq12d |
|
243 |
242
|
breq2d |
|
244 |
243
|
imbi2d |
|
245 |
|
oveq2 |
|
246 |
|
oveq2 |
|
247 |
246
|
oveq2d |
|
248 |
245 247
|
oveq12d |
|
249 |
|
oveq2 |
|
250 |
248 249
|
oveq12d |
|
251 |
250
|
breq2d |
|
252 |
251
|
imbi2d |
|
253 |
|
oveq2 |
|
254 |
|
oveq2 |
|
255 |
254
|
oveq2d |
|
256 |
253 255
|
oveq12d |
|
257 |
|
oveq2 |
|
258 |
256 257
|
oveq12d |
|
259 |
258
|
breq2d |
|
260 |
259
|
imbi2d |
|
261 |
|
oveq2 |
|
262 |
|
oveq2 |
|
263 |
262
|
oveq2d |
|
264 |
261 263
|
oveq12d |
|
265 |
|
oveq2 |
|
266 |
264 265
|
oveq12d |
|
267 |
266
|
breq2d |
|
268 |
267
|
imbi2d |
|
269 |
|
oveq2 |
|
270 |
|
oveq2 |
|
271 |
270
|
oveq2d |
|
272 |
269 271
|
oveq12d |
|
273 |
|
oveq2 |
|
274 |
272 273
|
oveq12d |
|
275 |
274
|
breq2d |
|
276 |
275
|
imbi2d |
|
277 |
34 50 228 236 244 252 260 268 276
|
2nn0ind |
|
278 |
277
|
impcom |
|
279 |
278
|
3impa |
|