Step |
Hyp |
Ref |
Expression |
1 |
|
imassrn |
|
2 |
|
eqid |
|
3 |
|
eqid |
|
4 |
2 3
|
mgmhmf |
|
5 |
4
|
adantr |
|
6 |
5
|
frnd |
|
7 |
1 6
|
sstrid |
|
8 |
|
simpll |
|
9 |
2
|
submgmss |
|
10 |
9
|
adantl |
|
11 |
10
|
adantr |
|
12 |
|
simprl |
|
13 |
11 12
|
sseldd |
|
14 |
|
simprr |
|
15 |
11 14
|
sseldd |
|
16 |
|
eqid |
|
17 |
|
eqid |
|
18 |
2 16 17
|
mgmhmlin |
|
19 |
8 13 15 18
|
syl3anc |
|
20 |
5
|
ffnd |
|
21 |
20
|
adantr |
|
22 |
16
|
submgmcl |
|
23 |
22
|
3expb |
|
24 |
23
|
adantll |
|
25 |
|
fnfvima |
|
26 |
21 11 24 25
|
syl3anc |
|
27 |
19 26
|
eqeltrrd |
|
28 |
27
|
anassrs |
|
29 |
28
|
ralrimiva |
|
30 |
|
oveq2 |
|
31 |
30
|
eleq1d |
|
32 |
31
|
ralima |
|
33 |
20 10 32
|
syl2anc |
|
34 |
33
|
adantr |
|
35 |
29 34
|
mpbird |
|
36 |
35
|
ralrimiva |
|
37 |
|
oveq1 |
|
38 |
37
|
eleq1d |
|
39 |
38
|
ralbidv |
|
40 |
39
|
ralima |
|
41 |
20 10 40
|
syl2anc |
|
42 |
36 41
|
mpbird |
|
43 |
|
mgmhmrcl |
|
44 |
43
|
simprd |
|
45 |
44
|
adantr |
|
46 |
3 17
|
issubmgm |
|
47 |
45 46
|
syl |
|
48 |
7 42 47
|
mpbir2and |
|