Step |
Hyp |
Ref |
Expression |
1 |
|
lgsquad2.1 |
|
2 |
|
lgsquad2.2 |
|
3 |
|
lgsquad2.3 |
|
4 |
|
lgsquad2.4 |
|
5 |
|
lgsquad2.5 |
|
6 |
|
lgsquad2lem2.f |
|
7 |
|
lgsquad2lem2.s |
|
8 |
|
2nn |
|
9 |
8
|
a1i |
|
10 |
1
|
nnzd |
|
11 |
|
2z |
|
12 |
|
gcdcom |
|
13 |
10 11 12
|
sylancl |
|
14 |
|
2prm |
|
15 |
|
coprm |
|
16 |
14 10 15
|
sylancr |
|
17 |
2 16
|
mpbid |
|
18 |
13 17
|
eqtrd |
|
19 |
|
rpmulgcd |
|
20 |
1 9 3 18 19
|
syl31anc |
|
21 |
20 5
|
eqtrd |
|
22 |
|
oveq1 |
|
23 |
|
oveq2 |
|
24 |
22 23
|
oveq12d |
|
25 |
|
oveq1 |
|
26 |
|
1m1e0 |
|
27 |
25 26
|
eqtrdi |
|
28 |
27
|
oveq1d |
|
29 |
|
2cn |
|
30 |
|
2ne0 |
|
31 |
29 30
|
div0i |
|
32 |
28 31
|
eqtrdi |
|
33 |
32
|
oveq1d |
|
34 |
33
|
oveq2d |
|
35 |
24 34
|
eqeq12d |
|
36 |
35
|
imbi2d |
|
37 |
36
|
imbi2d |
|
38 |
|
oveq1 |
|
39 |
38
|
eqeq1d |
|
40 |
|
oveq1 |
|
41 |
|
oveq2 |
|
42 |
40 41
|
oveq12d |
|
43 |
|
oveq1 |
|
44 |
43
|
oveq1d |
|
45 |
44
|
oveq1d |
|
46 |
45
|
oveq2d |
|
47 |
42 46
|
eqeq12d |
|
48 |
39 47
|
imbi12d |
|
49 |
48
|
imbi2d |
|
50 |
|
oveq1 |
|
51 |
50
|
eqeq1d |
|
52 |
|
oveq1 |
|
53 |
|
oveq2 |
|
54 |
52 53
|
oveq12d |
|
55 |
|
oveq1 |
|
56 |
55
|
oveq1d |
|
57 |
56
|
oveq1d |
|
58 |
57
|
oveq2d |
|
59 |
54 58
|
eqeq12d |
|
60 |
51 59
|
imbi12d |
|
61 |
60
|
imbi2d |
|
62 |
|
oveq1 |
|
63 |
62
|
eqeq1d |
|
64 |
|
oveq1 |
|
65 |
|
oveq2 |
|
66 |
64 65
|
oveq12d |
|
67 |
|
oveq1 |
|
68 |
67
|
oveq1d |
|
69 |
68
|
oveq1d |
|
70 |
69
|
oveq2d |
|
71 |
66 70
|
eqeq12d |
|
72 |
63 71
|
imbi12d |
|
73 |
72
|
imbi2d |
|
74 |
|
oveq1 |
|
75 |
74
|
eqeq1d |
|
76 |
|
oveq1 |
|
77 |
|
oveq2 |
|
78 |
76 77
|
oveq12d |
|
79 |
|
oveq1 |
|
80 |
79
|
oveq1d |
|
81 |
80
|
oveq1d |
|
82 |
81
|
oveq2d |
|
83 |
78 82
|
eqeq12d |
|
84 |
75 83
|
imbi12d |
|
85 |
84
|
imbi2d |
|
86 |
|
1t1e1 |
|
87 |
|
neg1cn |
|
88 |
|
exp0 |
|
89 |
87 88
|
ax-mp |
|
90 |
86 89
|
eqtr4i |
|
91 |
|
sq1 |
|
92 |
91
|
oveq1i |
|
93 |
|
1z |
|
94 |
|
ax-1ne0 |
|
95 |
93 94
|
pm3.2i |
|
96 |
3
|
nnzd |
|
97 |
|
1gcd |
|
98 |
96 97
|
syl |
|
99 |
|
lgssq |
|
100 |
95 96 98 99
|
mp3an2i |
|
101 |
92 100
|
syl5eqr |
|
102 |
91
|
oveq2i |
|
103 |
|
1nn |
|
104 |
103
|
a1i |
|
105 |
|
gcd1 |
|
106 |
96 105
|
syl |
|
107 |
|
lgssq2 |
|
108 |
96 104 106 107
|
syl3anc |
|
109 |
102 108
|
syl5eqr |
|
110 |
101 109
|
oveq12d |
|
111 |
|
nnm1nn0 |
|
112 |
3 111
|
syl |
|
113 |
112
|
nn0cnd |
|
114 |
113
|
halfcld |
|
115 |
114
|
mul02d |
|
116 |
115
|
oveq2d |
|
117 |
90 110 116
|
3eqtr4a |
|
118 |
117
|
a1d |
|
119 |
|
simprl |
|
120 |
|
prmz |
|
121 |
120
|
ad2antrl |
|
122 |
11
|
a1i |
|
123 |
3
|
adantr |
|
124 |
123
|
nnzd |
|
125 |
|
zmulcl |
|
126 |
11 124 125
|
sylancr |
|
127 |
|
simprr |
|
128 |
|
dvdsmul1 |
|
129 |
11 124 128
|
sylancr |
|
130 |
|
rpdvds |
|
131 |
121 122 126 127 129 130
|
syl32anc |
|
132 |
|
prmrp |
|
133 |
119 14 132
|
sylancl |
|
134 |
131 133
|
mpbid |
|
135 |
|
eldifsn |
|
136 |
119 134 135
|
sylanbrc |
|
137 |
|
prmnn |
|
138 |
137
|
ad2antrl |
|
139 |
8
|
a1i |
|
140 |
|
rpmulgcd |
|
141 |
138 139 123 131 140
|
syl31anc |
|
142 |
141 127
|
eqtr3d |
|
143 |
136 142
|
jca |
|
144 |
143 6
|
syldan |
|
145 |
144
|
exp32 |
|
146 |
145
|
com12 |
|
147 |
|
jcab |
|
148 |
|
simplrl |
|
149 |
|
eluz2nn |
|
150 |
148 149
|
syl |
|
151 |
|
simplrr |
|
152 |
|
eluz2nn |
|
153 |
151 152
|
syl |
|
154 |
150 153
|
nnmulcld |
|
155 |
|
n2dvds1 |
|
156 |
96
|
ad2antrr |
|
157 |
11 156 128
|
sylancr |
|
158 |
|
eluzelz |
|
159 |
|
eluzelz |
|
160 |
158 159
|
anim12i |
|
161 |
160
|
ad2antlr |
|
162 |
|
zmulcl |
|
163 |
161 162
|
syl |
|
164 |
11 156 125
|
sylancr |
|
165 |
|
dvdsgcd |
|
166 |
11 163 164 165
|
mp3an2i |
|
167 |
157 166
|
mpan2d |
|
168 |
|
simpr |
|
169 |
168
|
breq2d |
|
170 |
167 169
|
sylibd |
|
171 |
155 170
|
mtoi |
|
172 |
171
|
adantrr |
|
173 |
3
|
ad2antrr |
|
174 |
4
|
ad2antrr |
|
175 |
|
dvdsmul2 |
|
176 |
11 156 175
|
sylancr |
|
177 |
|
rpdvds |
|
178 |
163 156 164 168 176 177
|
syl32anc |
|
179 |
178
|
adantrr |
|
180 |
|
eqidd |
|
181 |
161
|
simpld |
|
182 |
181 164
|
gcdcomd |
|
183 |
164 163
|
gcdcomd |
|
184 |
183 168
|
eqtrd |
|
185 |
|
dvdsmul1 |
|
186 |
161 185
|
syl |
|
187 |
|
rpdvds |
|
188 |
164 181 163 184 186 187
|
syl32anc |
|
189 |
182 188
|
eqtrd |
|
190 |
189
|
adantrr |
|
191 |
|
simprrl |
|
192 |
190 191
|
mpd |
|
193 |
161
|
simprd |
|
194 |
193 164
|
gcdcomd |
|
195 |
|
dvdsmul2 |
|
196 |
161 195
|
syl |
|
197 |
|
rpdvds |
|
198 |
164 193 163 184 196 197
|
syl32anc |
|
199 |
194 198
|
eqtrd |
|
200 |
199
|
adantrr |
|
201 |
|
simprrr |
|
202 |
200 201
|
mpd |
|
203 |
154 172 173 174 179 150 153 180 192 202
|
lgsquad2lem1 |
|
204 |
203
|
exp32 |
|
205 |
204
|
com23 |
|
206 |
205
|
expcom |
|
207 |
206
|
a2d |
|
208 |
147 207
|
syl5bir |
|
209 |
37 49 61 73 85 118 146 208
|
prmind |
|
210 |
1 209
|
mpcom |
|
211 |
21 210
|
mpd |
|