Step |
Hyp |
Ref |
Expression |
1 |
|
symgtrf.t |
|
2 |
|
symgtrf.g |
|
3 |
|
symgtrf.b |
|
4 |
|
symggen.k |
|
5 |
|
elex |
|
6 |
2
|
symggrp |
|
7 |
|
grpmnd |
|
8 |
6 7
|
syl |
|
9 |
3
|
submacs |
|
10 |
|
acsmre |
|
11 |
8 9 10
|
3syl |
|
12 |
5 11
|
syl |
|
13 |
1 2 3
|
symgtrf |
|
14 |
13
|
a1i |
|
15 |
|
2onn |
|
16 |
|
nnfi |
|
17 |
15 16
|
ax-mp |
|
18 |
|
eqid |
|
19 |
18 1
|
pmtrfb |
|
20 |
19
|
simp3bi |
|
21 |
|
enfi |
|
22 |
20 21
|
syl |
|
23 |
22
|
adantl |
|
24 |
17 23
|
mpbiri |
|
25 |
14 24
|
ssrabdv |
|
26 |
2 3
|
symgfisg |
|
27 |
|
subgsubm |
|
28 |
26 27
|
syl |
|
29 |
4
|
mrcsscl |
|
30 |
12 25 28 29
|
syl3anc |
|
31 |
|
vex |
|
32 |
31
|
a1i |
|
33 |
|
finnum |
|
34 |
|
domfi |
|
35 |
2 3
|
symgbasf1o |
|
36 |
35
|
adantl |
|
37 |
|
f1ofn |
|
38 |
|
fnnfpeq0 |
|
39 |
36 37 38
|
3syl |
|
40 |
2 3
|
elbasfv |
|
41 |
40
|
adantl |
|
42 |
2
|
symgid |
|
43 |
41 42
|
syl |
|
44 |
41 11
|
syl |
|
45 |
4
|
mrccl |
|
46 |
44 13 45
|
sylancl |
|
47 |
|
eqid |
|
48 |
47
|
subm0cl |
|
49 |
46 48
|
syl |
|
50 |
43 49
|
eqeltrd |
|
51 |
|
eleq1a |
|
52 |
50 51
|
syl |
|
53 |
39 52
|
sylbid |
|
54 |
53
|
adantr |
|
55 |
|
n0 |
|
56 |
41
|
adantr |
|
57 |
|
simpr |
|
58 |
|
f1omvdmvd |
|
59 |
36 58
|
sylan |
|
60 |
59
|
eldifad |
|
61 |
57 60
|
prssd |
|
62 |
|
difss |
|
63 |
|
dmss |
|
64 |
62 63
|
ax-mp |
|
65 |
|
f1odm |
|
66 |
36 65
|
syl |
|
67 |
64 66
|
sseqtrid |
|
68 |
67
|
adantr |
|
69 |
61 68
|
sstrd |
|
70 |
|
vex |
|
71 |
|
fvex |
|
72 |
36
|
adantr |
|
73 |
72 37
|
syl |
|
74 |
67
|
sselda |
|
75 |
|
fnelnfp |
|
76 |
73 74 75
|
syl2anc |
|
77 |
57 76
|
mpbid |
|
78 |
77
|
necomd |
|
79 |
|
pr2nelem |
|
80 |
70 71 78 79
|
mp3an12i |
|
81 |
18 1
|
pmtrrn |
|
82 |
56 69 80 81
|
syl3anc |
|
83 |
13 82
|
sseldi |
|
84 |
|
simplr |
|
85 |
|
eqid |
|
86 |
2 3 85
|
symgov |
|
87 |
83 84 86
|
syl2anc |
|
88 |
87
|
oveq2d |
|
89 |
41 6
|
syl |
|
90 |
89
|
adantr |
|
91 |
3 85
|
grpcl |
|
92 |
90 83 84 91
|
syl3anc |
|
93 |
87 92
|
eqeltrrd |
|
94 |
2 3 85
|
symgov |
|
95 |
83 93 94
|
syl2anc |
|
96 |
|
coass |
|
97 |
18 1
|
pmtrfinv |
|
98 |
82 97
|
syl |
|
99 |
98
|
coeq1d |
|
100 |
|
f1of |
|
101 |
|
fcoi2 |
|
102 |
72 100 101
|
3syl |
|
103 |
99 102
|
eqtrd |
|
104 |
96 103
|
syl5eqr |
|
105 |
88 95 104
|
3eqtrd |
|
106 |
105
|
adantlr |
|
107 |
46
|
ad2antrr |
|
108 |
4
|
mrcssid |
|
109 |
44 13 108
|
sylancl |
|
110 |
109
|
adantr |
|
111 |
110 82
|
sseldd |
|
112 |
111
|
adantlr |
|
113 |
87
|
difeq1d |
|
114 |
113
|
dmeqd |
|
115 |
|
simpll |
|
116 |
|
mvdco |
|
117 |
18
|
pmtrmvd |
|
118 |
56 69 80 117
|
syl3anc |
|
119 |
118 61
|
eqsstrd |
|
120 |
|
ssidd |
|
121 |
119 120
|
unssd |
|
122 |
116 121
|
sstrid |
|
123 |
|
fvco2 |
|
124 |
73 74 123
|
syl2anc |
|
125 |
|
prcom |
|
126 |
125
|
fveq2i |
|
127 |
126
|
fveq1i |
|
128 |
68 60
|
sseldd |
|
129 |
18
|
pmtrprfv |
|
130 |
56 128 74 77 129
|
syl13anc |
|
131 |
127 130
|
syl5eq |
|
132 |
124 131
|
eqtrd |
|
133 |
2 3
|
symgbasf1o |
|
134 |
|
f1ofn |
|
135 |
93 133 134
|
3syl |
|
136 |
|
fnelnfp |
|
137 |
136
|
necon2bbid |
|
138 |
135 74 137
|
syl2anc |
|
139 |
132 138
|
mpbid |
|
140 |
122 57 139
|
ssnelpssd |
|
141 |
|
php3 |
|
142 |
115 140 141
|
syl2anc |
|
143 |
114 142
|
eqbrtrd |
|
144 |
143
|
adantlr |
|
145 |
92
|
adantlr |
|
146 |
|
ovex |
|
147 |
|
difeq1 |
|
148 |
147
|
dmeqd |
|
149 |
148
|
breq1d |
|
150 |
|
eleq1 |
|
151 |
|
eleq1 |
|
152 |
150 151
|
imbi12d |
|
153 |
149 152
|
imbi12d |
|
154 |
146 153
|
spcv |
|
155 |
154
|
ad2antlr |
|
156 |
144 145 155
|
mp2d |
|
157 |
85
|
submcl |
|
158 |
107 112 156 157
|
syl3anc |
|
159 |
106 158
|
eqeltrrd |
|
160 |
159
|
ex |
|
161 |
160
|
exlimdv |
|
162 |
55 161
|
syl5bi |
|
163 |
54 162
|
pm2.61dne |
|
164 |
163
|
exp31 |
|
165 |
164
|
com23 |
|
166 |
34 165
|
syl |
|
167 |
166
|
3impia |
|
168 |
|
eleq1w |
|
169 |
|
eleq1w |
|
170 |
168 169
|
imbi12d |
|
171 |
|
eleq1w |
|
172 |
|
eleq1w |
|
173 |
171 172
|
imbi12d |
|
174 |
|
difeq1 |
|
175 |
174
|
dmeqd |
|
176 |
|
difeq1 |
|
177 |
176
|
dmeqd |
|
178 |
32 33 167 170 173 175 177
|
indcardi |
|
179 |
178
|
impcom |
|
180 |
179
|
3adant1 |
|
181 |
180
|
rabssdv |
|
182 |
30 181
|
eqssd |
|