Step |
Hyp |
Ref |
Expression |
1 |
|
subsubg.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑆 ) |
2 |
|
subgrcl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
3 |
2
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐺 ∈ Grp ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
5 |
4
|
subgss |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐻 ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
7 |
1
|
subgbas |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
9 |
6 8
|
sseqtrrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐴 ⊆ 𝑆 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
11 |
10
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
13 |
9 12
|
sstrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
14 |
1
|
oveq1i |
⊢ ( 𝐻 ↾s 𝐴 ) = ( ( 𝐺 ↾s 𝑆 ) ↾s 𝐴 ) |
15 |
|
ressabs |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) → ( ( 𝐺 ↾s 𝑆 ) ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
16 |
14 15
|
eqtrid |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
17 |
9 16
|
syldan |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
18 |
|
eqid |
⊢ ( 𝐻 ↾s 𝐴 ) = ( 𝐻 ↾s 𝐴 ) |
19 |
18
|
subggrp |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐻 ) → ( 𝐻 ↾s 𝐴 ) ∈ Grp ) |
20 |
19
|
adantl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐻 ↾s 𝐴 ) ∈ Grp ) |
21 |
17 20
|
eqeltrrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐺 ↾s 𝐴 ) ∈ Grp ) |
22 |
10
|
issubg |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝐺 ∈ Grp ∧ 𝐴 ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝐴 ) ∈ Grp ) ) |
23 |
3 13 21 22
|
syl3anbrc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ) |
24 |
23 9
|
jca |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) |
25 |
1
|
subggrp |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp ) |
26 |
25
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐻 ∈ Grp ) |
27 |
|
simprr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ⊆ 𝑆 ) |
28 |
7
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝑆 = ( Base ‘ 𝐻 ) ) |
29 |
27 28
|
sseqtrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ⊆ ( Base ‘ 𝐻 ) ) |
30 |
16
|
adantrl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐻 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) ) |
31 |
|
eqid |
⊢ ( 𝐺 ↾s 𝐴 ) = ( 𝐺 ↾s 𝐴 ) |
32 |
31
|
subggrp |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s 𝐴 ) ∈ Grp ) |
33 |
32
|
ad2antrl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐺 ↾s 𝐴 ) ∈ Grp ) |
34 |
30 33
|
eqeltrd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → ( 𝐻 ↾s 𝐴 ) ∈ Grp ) |
35 |
4
|
issubg |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐻 ∈ Grp ∧ 𝐴 ⊆ ( Base ‘ 𝐻 ) ∧ ( 𝐻 ↾s 𝐴 ) ∈ Grp ) ) |
36 |
26 29 34 35
|
syl3anbrc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) → 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ) |
37 |
24 36
|
impbida |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐴 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ⊆ 𝑆 ) ) ) |