Step |
Hyp |
Ref |
Expression |
1 |
|
issubg3.i |
⊢ 𝐼 = ( invg ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
3 |
2
|
subg0cl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) |
4 |
3
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ) |
5 |
2
|
subm0cl |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) |
6 |
5
|
adantr |
⊢ ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) |
7 |
6
|
a1i |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ) |
8 |
|
ne0i |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝑆 → 𝑆 ≠ ∅ ) |
9 |
|
id |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝑆 → ( 0g ‘ 𝐺 ) ∈ 𝑆 ) |
10 |
8 9
|
2thd |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝑆 → ( 𝑆 ≠ ∅ ↔ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( 𝑆 ≠ ∅ ↔ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ) |
12 |
|
r19.26 |
⊢ ( ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) |
13 |
12
|
a1i |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
14 |
11 13
|
3anbi23d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) ) |
15 |
|
anass |
⊢ ( ( ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ∧ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
16 |
|
df-3an |
⊢ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) |
17 |
16
|
anbi1i |
⊢ ( ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) |
18 |
|
df-3an |
⊢ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) ∧ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
19 |
15 17 18
|
3bitr4ri |
⊢ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) |
20 |
14 19
|
bitrdi |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
22 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
23 |
21 22 1
|
issubg2 |
⊢ ( 𝐺 ∈ Grp → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) ) |
25 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
26 |
21 2 22
|
issubm |
⊢ ( 𝐺 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) ) |
27 |
25 26
|
syl |
⊢ ( 𝐺 ∈ Grp → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ) ) |
28 |
27
|
anbi1d |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
30 |
20 24 29
|
3bitr4d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 0g ‘ 𝐺 ) ∈ 𝑆 ) → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |
31 |
30
|
ex |
⊢ ( 𝐺 ∈ Grp → ( ( 0g ‘ 𝐺 ) ∈ 𝑆 → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) ) |
32 |
4 7 31
|
pm5.21ndd |
⊢ ( 𝐺 ∈ Grp → ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ∈ 𝑆 ) ) ) |