Step |
Hyp |
Ref |
Expression |
1 |
|
subsubmgm.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
3 |
2
|
submgmss |
⊢ ( 𝐴 ∈ ( SubMgm ‘ 𝐻 ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
5 |
1
|
submgmbas |
⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
7 |
4 6
|
sseqtrrd |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝐴 ⊆ 𝑆 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
9 |
8
|
submgmss |
⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
11 |
7 10
|
sstrd |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
12 |
1
|
oveq1i |
⊢ ( 𝐻 ↾s 𝐴 ) = ( ( 𝐺 ↾s 𝑆 ) ↾s 𝐴 ) |
13 |
|
ressabs |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) → ( ( 𝐺 ↾s 𝑆 ) ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
14 |
12 13
|
syl5eq |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
15 |
7 14
|
syldan |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
16 |
|
eqid |
⊢ ( 𝐻 ↾s 𝐴 ) = ( 𝐻 ↾s 𝐴 ) |
17 |
16
|
submgmmgm |
⊢ ( 𝐴 ∈ ( SubMgm ‘ 𝐻 ) → ( 𝐻 ↾s 𝐴 ) ∈ Mgm ) |
18 |
17
|
adantl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → ( 𝐻 ↾s 𝐴 ) ∈ Mgm ) |
19 |
15 18
|
eqeltrrd |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → ( 𝐺 ↾s 𝐴 ) ∈ Mgm ) |
20 |
|
submgmrcl |
⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) → 𝐺 ∈ Mgm ) |
21 |
20
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝐺 ∈ Mgm ) |
22 |
|
eqid |
⊢ ( 𝐺 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) |
23 |
8 22
|
issubmgm2 |
⊢ ( 𝐺 ∈ Mgm → ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ↔ ( 𝐴 ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝐴 ) ∈ Mgm ) ) ) |
24 |
21 23
|
syl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ↔ ( 𝐴 ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝐴 ) ∈ Mgm ) ) ) |
25 |
11 19 24
|
mpbir2and |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ) |
26 |
25 7
|
jca |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) → ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) |
27 |
|
simprr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ⊆ 𝑆 ) |
28 |
5
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
29 |
27 28
|
sseqtrd |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
30 |
14
|
adantrl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
31 |
22
|
submgmmgm |
⊢ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) → ( 𝐺 ↾s 𝐴 ) ∈ Mgm ) |
32 |
31
|
ad2antrl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐺 ↾s 𝐴 ) ∈ Mgm ) |
33 |
30 32
|
eqeltrd |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐻 ↾s 𝐴 ) ∈ Mgm ) |
34 |
1
|
submgmmgm |
⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) → 𝐻 ∈ Mgm ) |
35 |
34
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐻 ∈ Mgm ) |
36 |
2 16
|
issubmgm2 |
⊢ ( 𝐻 ∈ Mgm → ( 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ↔ ( 𝐴 ⊆ ( Base ‘ 𝐻 ) ∧ ( 𝐻 ↾s 𝐴 ) ∈ Mgm ) ) ) |
37 |
35 36
|
syl |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ↔ ( 𝐴 ⊆ ( Base ‘ 𝐻 ) ∧ ( 𝐻 ↾s 𝐴 ) ∈ Mgm ) ) ) |
38 |
29 33 37
|
mpbir2and |
⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ) |
39 |
26 38
|
impbida |
⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝐺 ) → ( 𝐴 ∈ ( SubMgm ‘ 𝐻 ) ↔ ( 𝐴 ∈ ( SubMgm ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) ) |