Step |
Hyp |
Ref |
Expression |
1 |
|
submateq.a |
|
2 |
|
submateq.b |
|
3 |
|
submateq.n |
|
4 |
|
submateq.i |
|
5 |
|
submateq.j |
|
6 |
|
submateq.e |
|
7 |
|
submateq.f |
|
8 |
|
submateq.1 |
|
9 |
|
simprl |
|
10 |
3
|
ad2antrr |
|
11 |
4
|
ad2antrr |
|
12 |
|
simplr |
|
13 |
|
simpr |
|
14 |
10 11 12 13
|
submateqlem1 |
|
15 |
14
|
simprd |
|
16 |
9 15
|
syldanl |
|
17 |
16
|
adantr |
|
18 |
|
simprr |
|
19 |
3
|
ad2antrr |
|
20 |
5
|
ad2antrr |
|
21 |
|
simplr |
|
22 |
|
simpr |
|
23 |
19 20 21 22
|
submateqlem1 |
|
24 |
23
|
simprd |
|
25 |
18 24
|
syldanl |
|
26 |
25
|
adantlr |
|
27 |
17 26
|
jca |
|
28 |
|
ovexd |
|
29 |
|
ovexd |
|
30 |
|
simpl |
|
31 |
30
|
eleq1d |
|
32 |
|
simpr |
|
33 |
32
|
eleq1d |
|
34 |
31 33
|
anbi12d |
|
35 |
|
oveq12 |
|
36 |
|
oveq12 |
|
37 |
35 36
|
eqeq12d |
|
38 |
34 37
|
imbi12d |
|
39 |
8
|
3expib |
|
40 |
28 29 38 39
|
vtocl2d |
|
41 |
40
|
ad3antrrr |
|
42 |
27 41
|
mpd |
|
43 |
|
eqid |
|
44 |
3
|
ad3antrrr |
|
45 |
4
|
ad3antrrr |
|
46 |
5
|
ad3antrrr |
|
47 |
|
eqid |
|
48 |
1 47 2
|
matbas2i |
|
49 |
6 48
|
syl |
|
50 |
49
|
ad3antrrr |
|
51 |
14
|
simpld |
|
52 |
9 51
|
syldanl |
|
53 |
52
|
adantr |
|
54 |
23
|
simpld |
|
55 |
18 54
|
syldanl |
|
56 |
55
|
adantlr |
|
57 |
43 44 44 45 46 50 53 56
|
smatbr |
|
58 |
|
eqid |
|
59 |
1 47 2
|
matbas2i |
|
60 |
7 59
|
syl |
|
61 |
60
|
ad3antrrr |
|
62 |
58 44 44 45 46 61 53 56
|
smatbr |
|
63 |
42 57 62
|
3eqtr4d |
|
64 |
16
|
adantr |
|
65 |
3
|
ad2antrr |
|
66 |
5
|
ad2antrr |
|
67 |
|
simplr |
|
68 |
|
simpr |
|
69 |
65 66 67 68
|
submateqlem2 |
|
70 |
69
|
simprd |
|
71 |
18 70
|
syldanl |
|
72 |
71
|
adantlr |
|
73 |
64 72
|
jca |
|
74 |
|
vex |
|
75 |
74
|
a1i |
|
76 |
|
simpl |
|
77 |
76
|
eleq1d |
|
78 |
|
simpr |
|
79 |
|
eqidd |
|
80 |
78 79
|
eleq12d |
|
81 |
77 80
|
anbi12d |
|
82 |
|
oveq12 |
|
83 |
|
oveq12 |
|
84 |
82 83
|
eqeq12d |
|
85 |
81 84
|
imbi12d |
|
86 |
28 75 85 39
|
vtocl2d |
|
87 |
86
|
ad3antrrr |
|
88 |
73 87
|
mpd |
|
89 |
3
|
ad3antrrr |
|
90 |
4
|
ad3antrrr |
|
91 |
5
|
ad3antrrr |
|
92 |
49
|
ad3antrrr |
|
93 |
52
|
adantr |
|
94 |
69
|
simpld |
|
95 |
18 94
|
syldanl |
|
96 |
95
|
adantlr |
|
97 |
43 89 89 90 91 92 93 96
|
smattr |
|
98 |
60
|
ad3antrrr |
|
99 |
58 89 89 90 91 98 93 96
|
smattr |
|
100 |
88 97 99
|
3eqtr4d |
|
101 |
|
fz1ssnn |
|
102 |
101 5
|
sseldi |
|
103 |
102
|
nnred |
|
104 |
103
|
adantr |
|
105 |
|
fz1ssnn |
|
106 |
105 18
|
sseldi |
|
107 |
106
|
nnred |
|
108 |
|
lelttric |
|
109 |
104 107 108
|
syl2anc |
|
110 |
109
|
adantr |
|
111 |
63 100 110
|
mpjaodan |
|
112 |
3
|
ad2antrr |
|
113 |
4
|
ad2antrr |
|
114 |
|
simplr |
|
115 |
|
simpr |
|
116 |
112 113 114 115
|
submateqlem2 |
|
117 |
116
|
simprd |
|
118 |
9 117
|
syldanl |
|
119 |
118
|
adantr |
|
120 |
25
|
adantlr |
|
121 |
119 120
|
jca |
|
122 |
|
vex |
|
123 |
122
|
a1i |
|
124 |
|
simpl |
|
125 |
124
|
eleq1d |
|
126 |
|
simpr |
|
127 |
126
|
eleq1d |
|
128 |
125 127
|
anbi12d |
|
129 |
|
oveq12 |
|
130 |
|
oveq12 |
|
131 |
129 130
|
eqeq12d |
|
132 |
128 131
|
imbi12d |
|
133 |
123 29 132 39
|
vtocl2d |
|
134 |
133
|
ad3antrrr |
|
135 |
121 134
|
mpd |
|
136 |
3
|
ad3antrrr |
|
137 |
4
|
ad3antrrr |
|
138 |
5
|
ad3antrrr |
|
139 |
49
|
ad3antrrr |
|
140 |
116
|
simpld |
|
141 |
9 140
|
syldanl |
|
142 |
141
|
adantr |
|
143 |
55
|
adantlr |
|
144 |
43 136 136 137 138 139 142 143
|
smatbl |
|
145 |
60
|
ad3antrrr |
|
146 |
58 136 136 137 138 145 142 143
|
smatbl |
|
147 |
135 144 146
|
3eqtr4d |
|
148 |
118
|
adantr |
|
149 |
71
|
adantlr |
|
150 |
148 149
|
jca |
|
151 |
|
simpl |
|
152 |
151
|
eleq1d |
|
153 |
|
simpr |
|
154 |
153
|
eleq1d |
|
155 |
152 154
|
anbi12d |
|
156 |
|
oveq12 |
|
157 |
|
oveq12 |
|
158 |
156 157
|
eqeq12d |
|
159 |
155 158
|
imbi12d |
|
160 |
123 75 159 39
|
vtocl2d |
|
161 |
160
|
ad3antrrr |
|
162 |
150 161
|
mpd |
|
163 |
3
|
ad3antrrr |
|
164 |
4
|
ad3antrrr |
|
165 |
5
|
ad3antrrr |
|
166 |
49
|
ad3antrrr |
|
167 |
141
|
adantr |
|
168 |
95
|
adantlr |
|
169 |
43 163 163 164 165 166 167 168
|
smattl |
|
170 |
60
|
ad3antrrr |
|
171 |
58 163 163 164 165 170 167 168
|
smattl |
|
172 |
162 169 171
|
3eqtr4d |
|
173 |
109
|
adantr |
|
174 |
147 172 173
|
mpjaodan |
|
175 |
101 4
|
sseldi |
|
176 |
175
|
nnred |
|
177 |
176
|
adantr |
|
178 |
105 9
|
sseldi |
|
179 |
178
|
nnred |
|
180 |
|
lelttric |
|
181 |
177 179 180
|
syl2anc |
|
182 |
111 174 181
|
mpjaodan |
|
183 |
182
|
ralrimivva |
|
184 |
|
eqid |
|
185 |
1 2 184 43 3 4 5 6
|
smatcl |
|
186 |
1 2 184 58 3 4 5 7
|
smatcl |
|
187 |
|
eqid |
|
188 |
187 184
|
eqmat |
|
189 |
185 186 188
|
syl2anc |
|
190 |
183 189
|
mpbird |
|