Step |
Hyp |
Ref |
Expression |
1 |
|
4sq.1 |
|
2 |
|
4sq.2 |
|
3 |
|
4sq.3 |
|
4 |
|
4sq.4 |
|
5 |
|
4sqlem11.5 |
|
6 |
|
4sqlem11.6 |
|
7 |
1 2 3 4 5 6
|
4sqlem11 |
|
8 |
|
n0 |
|
9 |
7 8
|
sylib |
|
10 |
|
vex |
|
11 |
|
eqeq1 |
|
12 |
11
|
rexbidv |
|
13 |
10 12 5
|
elab2 |
|
14 |
|
abid |
|
15 |
5
|
rexeqi |
|
16 |
|
oveq1 |
|
17 |
16
|
oveq1d |
|
18 |
17
|
eqeq2d |
|
19 |
18
|
cbvrexvw |
|
20 |
|
eqeq1 |
|
21 |
20
|
rexbidv |
|
22 |
19 21
|
syl5bb |
|
23 |
22
|
rexab |
|
24 |
14 15 23
|
3bitri |
|
25 |
6
|
rnmpt |
|
26 |
25
|
eleq2i |
|
27 |
|
rexcom4 |
|
28 |
|
r19.41v |
|
29 |
28
|
exbii |
|
30 |
27 29
|
bitri |
|
31 |
24 26 30
|
3bitr4i |
|
32 |
|
ovex |
|
33 |
|
oveq2 |
|
34 |
33
|
eqeq2d |
|
35 |
32 34
|
ceqsexv |
|
36 |
35
|
rexbii |
|
37 |
31 36
|
bitri |
|
38 |
13 37
|
anbi12i |
|
39 |
|
elin |
|
40 |
|
reeanv |
|
41 |
38 39 40
|
3bitr4i |
|
42 |
|
eqtr2 |
|
43 |
4
|
3ad2ant1 |
|
44 |
|
prmnn |
|
45 |
43 44
|
syl |
|
46 |
|
nnm1nn0 |
|
47 |
45 46
|
syl |
|
48 |
47
|
nn0red |
|
49 |
45
|
nnrpd |
|
50 |
47
|
nn0ge0d |
|
51 |
45
|
nnred |
|
52 |
51
|
ltm1d |
|
53 |
|
modid |
|
54 |
48 49 50 52 53
|
syl22anc |
|
55 |
54
|
oveq1d |
|
56 |
|
simp2r |
|
57 |
|
elfzelz |
|
58 |
56 57
|
syl |
|
59 |
|
zsqcl2 |
|
60 |
58 59
|
syl |
|
61 |
60
|
nn0red |
|
62 |
|
modlt |
|
63 |
61 49 62
|
syl2anc |
|
64 |
|
zsqcl |
|
65 |
58 64
|
syl |
|
66 |
65 45
|
zmodcld |
|
67 |
66
|
nn0zd |
|
68 |
|
prmz |
|
69 |
43 68
|
syl |
|
70 |
|
zltlem1 |
|
71 |
67 69 70
|
syl2anc |
|
72 |
63 71
|
mpbid |
|
73 |
72 54
|
breqtrrd |
|
74 |
|
modsubdir |
|
75 |
48 61 49 74
|
syl3anc |
|
76 |
73 75
|
mpbid |
|
77 |
|
simp3 |
|
78 |
55 76 77
|
3eqtr4rd |
|
79 |
|
simp2l |
|
80 |
|
elfzelz |
|
81 |
79 80
|
syl |
|
82 |
|
zsqcl |
|
83 |
81 82
|
syl |
|
84 |
47
|
nn0zd |
|
85 |
84 65
|
zsubcld |
|
86 |
|
moddvds |
|
87 |
45 83 85 86
|
syl3anc |
|
88 |
78 87
|
mpbid |
|
89 |
|
zsqcl2 |
|
90 |
81 89
|
syl |
|
91 |
90
|
nn0cnd |
|
92 |
47
|
nn0cnd |
|
93 |
60
|
nn0cnd |
|
94 |
91 92 93
|
subsub3d |
|
95 |
90 60
|
nn0addcld |
|
96 |
95
|
nn0cnd |
|
97 |
45
|
nncnd |
|
98 |
|
1cnd |
|
99 |
96 97 98
|
subsub3d |
|
100 |
94 99
|
eqtrd |
|
101 |
88 100
|
breqtrd |
|
102 |
|
nn0p1nn |
|
103 |
95 102
|
syl |
|
104 |
103
|
nnzd |
|
105 |
|
dvdssubr |
|
106 |
69 104 105
|
syl2anc |
|
107 |
101 106
|
mpbird |
|
108 |
45
|
nnne0d |
|
109 |
|
dvdsval2 |
|
110 |
69 108 104 109
|
syl3anc |
|
111 |
107 110
|
mpbid |
|
112 |
|
nnrp |
|
113 |
|
nnrp |
|
114 |
|
rpdivcl |
|
115 |
112 113 114
|
syl2an |
|
116 |
103 45 115
|
syl2anc |
|
117 |
116
|
rpgt0d |
|
118 |
|
elnnz |
|
119 |
111 117 118
|
sylanbrc |
|
120 |
119
|
nnge1d |
|
121 |
95
|
nn0red |
|
122 |
|
2nn |
|
123 |
2
|
3ad2ant1 |
|
124 |
|
nnmulcl |
|
125 |
122 123 124
|
sylancr |
|
126 |
125
|
nnred |
|
127 |
126
|
resqcld |
|
128 |
|
nnmulcl |
|
129 |
122 125 128
|
sylancr |
|
130 |
129
|
nnred |
|
131 |
127 130
|
readdcld |
|
132 |
|
1red |
|
133 |
123
|
nnsqcld |
|
134 |
|
nnmulcl |
|
135 |
122 133 134
|
sylancr |
|
136 |
135
|
nnred |
|
137 |
90
|
nn0red |
|
138 |
133
|
nnred |
|
139 |
81
|
zred |
|
140 |
|
elfzle1 |
|
141 |
79 140
|
syl |
|
142 |
123
|
nnred |
|
143 |
|
elfzle2 |
|
144 |
79 143
|
syl |
|
145 |
|
le2sq2 |
|
146 |
139 141 142 144 145
|
syl22anc |
|
147 |
58
|
zred |
|
148 |
|
elfzle1 |
|
149 |
56 148
|
syl |
|
150 |
|
elfzle2 |
|
151 |
56 150
|
syl |
|
152 |
|
le2sq2 |
|
153 |
147 149 142 151 152
|
syl22anc |
|
154 |
137 61 138 138 146 153
|
le2addd |
|
155 |
133
|
nncnd |
|
156 |
155
|
2timesd |
|
157 |
154 156
|
breqtrrd |
|
158 |
|
2lt4 |
|
159 |
|
2re |
|
160 |
159
|
a1i |
|
161 |
|
4re |
|
162 |
161
|
a1i |
|
163 |
133
|
nngt0d |
|
164 |
|
ltmul1 |
|
165 |
160 162 138 163 164
|
syl112anc |
|
166 |
158 165
|
mpbii |
|
167 |
|
2cn |
|
168 |
123
|
nncnd |
|
169 |
|
sqmul |
|
170 |
167 168 169
|
sylancr |
|
171 |
|
sq2 |
|
172 |
171
|
oveq1i |
|
173 |
170 172
|
eqtrdi |
|
174 |
166 173
|
breqtrrd |
|
175 |
121 136 127 157 174
|
lelttrd |
|
176 |
129
|
nnrpd |
|
177 |
127 176
|
ltaddrpd |
|
178 |
121 127 131 175 177
|
lttrd |
|
179 |
121 131 132 178
|
ltadd1dd |
|
180 |
3
|
3ad2ant1 |
|
181 |
180
|
oveq1d |
|
182 |
97
|
sqvald |
|
183 |
125
|
nncnd |
|
184 |
|
binom21 |
|
185 |
183 184
|
syl |
|
186 |
181 182 185
|
3eqtr3d |
|
187 |
179 186
|
breqtrrd |
|
188 |
103
|
nnred |
|
189 |
45
|
nngt0d |
|
190 |
|
ltdivmul |
|
191 |
188 51 51 189 190
|
syl112anc |
|
192 |
187 191
|
mpbird |
|
193 |
|
1z |
|
194 |
|
elfzm11 |
|
195 |
193 69 194
|
sylancr |
|
196 |
111 120 192 195
|
mpbir3and |
|
197 |
|
gzreim |
|
198 |
81 58 197
|
syl2anc |
|
199 |
|
gzcn |
|
200 |
198 199
|
syl |
|
201 |
200
|
absvalsq2d |
|
202 |
139 147
|
crred |
|
203 |
202
|
oveq1d |
|
204 |
139 147
|
crimd |
|
205 |
204
|
oveq1d |
|
206 |
203 205
|
oveq12d |
|
207 |
201 206
|
eqtrd |
|
208 |
207
|
oveq1d |
|
209 |
103
|
nncnd |
|
210 |
209 97 108
|
divcan1d |
|
211 |
208 210
|
eqtr4d |
|
212 |
|
oveq1 |
|
213 |
212
|
eqeq2d |
|
214 |
|
fveq2 |
|
215 |
214
|
oveq1d |
|
216 |
215
|
oveq1d |
|
217 |
216
|
eqeq1d |
|
218 |
213 217
|
rspc2ev |
|
219 |
196 198 211 218
|
syl3anc |
|
220 |
219
|
3expia |
|
221 |
42 220
|
syl5 |
|
222 |
221
|
rexlimdvva |
|
223 |
41 222
|
syl5bi |
|
224 |
223
|
exlimdv |
|
225 |
9 224
|
mpd |
|