Step |
Hyp |
Ref |
Expression |
1 |
|
subgacs.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
3 |
2
|
issubg3 |
⊢ ( 𝐺 ∈ Grp → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑠 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) ) |
4 |
1
|
submss |
⊢ ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) → 𝑠 ⊆ 𝐵 ) |
5 |
4
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ) → 𝑠 ⊆ 𝐵 ) |
6 |
|
velpw |
⊢ ( 𝑠 ∈ 𝒫 𝐵 ↔ 𝑠 ⊆ 𝐵 ) |
7 |
5 6
|
sylibr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ) → 𝑠 ∈ 𝒫 𝐵 ) |
8 |
|
eleq2w |
⊢ ( 𝑦 = 𝑠 → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 ↔ ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) |
9 |
8
|
raleqbi1dv |
⊢ ( 𝑦 = 𝑠 → ( ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 ↔ ∀ 𝑥 ∈ 𝑠 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) |
10 |
9
|
elrab3 |
⊢ ( 𝑠 ∈ 𝒫 𝐵 → ( 𝑠 ∈ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ↔ ∀ 𝑥 ∈ 𝑠 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) |
11 |
7 10
|
syl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ) → ( 𝑠 ∈ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ↔ ∀ 𝑥 ∈ 𝑠 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) |
12 |
11
|
pm5.32da |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ∧ 𝑠 ∈ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ↔ ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝑠 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑠 ) ) ) |
13 |
3 12
|
bitr4d |
⊢ ( 𝐺 ∈ Grp → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ∧ 𝑠 ∈ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ) ) |
14 |
|
elin |
⊢ ( 𝑠 ∈ ( ( SubMnd ‘ 𝐺 ) ∩ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ↔ ( 𝑠 ∈ ( SubMnd ‘ 𝐺 ) ∧ 𝑠 ∈ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ) |
15 |
13 14
|
bitr4di |
⊢ ( 𝐺 ∈ Grp → ( 𝑠 ∈ ( SubGrp ‘ 𝐺 ) ↔ 𝑠 ∈ ( ( SubMnd ‘ 𝐺 ) ∩ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ) ) |
16 |
15
|
eqrdv |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) = ( ( SubMnd ‘ 𝐺 ) ∩ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ) |
17 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
18 |
|
mreacs |
⊢ ( 𝐵 ∈ V → ( ACS ‘ 𝐵 ) ∈ ( Moore ‘ 𝒫 𝐵 ) ) |
19 |
17 18
|
mp1i |
⊢ ( 𝐺 ∈ Grp → ( ACS ‘ 𝐵 ) ∈ ( Moore ‘ 𝒫 𝐵 ) ) |
20 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
21 |
1
|
submacs |
⊢ ( 𝐺 ∈ Mnd → ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |
22 |
20 21
|
syl |
⊢ ( 𝐺 ∈ Grp → ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |
23 |
1 2
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
24 |
23
|
ralrimiva |
⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
25 |
|
acsfn1 |
⊢ ( ( 𝐵 ∈ V ∧ ∀ 𝑥 ∈ 𝐵 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) → { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ∈ ( ACS ‘ 𝐵 ) ) |
26 |
17 24 25
|
sylancr |
⊢ ( 𝐺 ∈ Grp → { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ∈ ( ACS ‘ 𝐵 ) ) |
27 |
|
mreincl |
⊢ ( ( ( ACS ‘ 𝐵 ) ∈ ( Moore ‘ 𝒫 𝐵 ) ∧ ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ∧ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ∈ ( ACS ‘ 𝐵 ) ) → ( ( SubMnd ‘ 𝐺 ) ∩ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ∈ ( ACS ‘ 𝐵 ) ) |
28 |
19 22 26 27
|
syl3anc |
⊢ ( 𝐺 ∈ Grp → ( ( SubMnd ‘ 𝐺 ) ∩ { 𝑦 ∈ 𝒫 𝐵 ∣ ∀ 𝑥 ∈ 𝑦 ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝑦 } ) ∈ ( ACS ‘ 𝐵 ) ) |
29 |
16 28
|
eqeltrd |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |