Step |
Hyp |
Ref |
Expression |
1 |
|
subsubmgm.h |
|
2 |
|
eqid |
|
3 |
2
|
submgmss |
|
4 |
3
|
adantl |
|
5 |
1
|
submgmbas |
|
6 |
5
|
adantr |
|
7 |
4 6
|
sseqtrrd |
|
8 |
|
eqid |
|
9 |
8
|
submgmss |
|
10 |
9
|
adantr |
|
11 |
7 10
|
sstrd |
|
12 |
1
|
oveq1i |
|
13 |
|
ressabs |
|
14 |
12 13
|
syl5eq |
|
15 |
7 14
|
syldan |
|
16 |
|
eqid |
|
17 |
16
|
submgmmgm |
|
18 |
17
|
adantl |
|
19 |
15 18
|
eqeltrrd |
|
20 |
|
submgmrcl |
|
21 |
20
|
adantr |
|
22 |
|
eqid |
|
23 |
8 22
|
issubmgm2 |
|
24 |
21 23
|
syl |
|
25 |
11 19 24
|
mpbir2and |
|
26 |
25 7
|
jca |
|
27 |
|
simprr |
|
28 |
5
|
adantr |
|
29 |
27 28
|
sseqtrd |
|
30 |
14
|
adantrl |
|
31 |
22
|
submgmmgm |
|
32 |
31
|
ad2antrl |
|
33 |
30 32
|
eqeltrd |
|
34 |
1
|
submgmmgm |
|
35 |
34
|
adantr |
|
36 |
2 16
|
issubmgm2 |
|
37 |
35 36
|
syl |
|
38 |
29 33 37
|
mpbir2and |
|
39 |
26 38
|
impbida |
|