Step | Hyp | Ref
| Expression |
1 | | subgacs.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐺) |
2 | 1 | subgss 17595 |
. . . . . . . 8
⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ⊆ 𝐵) |
3 | | selpw 4165 |
. . . . . . . 8
⊢ (𝑠 ∈ 𝒫 𝐵 ↔ 𝑠 ⊆ 𝐵) |
4 | 2, 3 | sylibr 224 |
. . . . . . 7
⊢ (𝑠 ∈ (SubGrp‘𝐺) → 𝑠 ∈ 𝒫 𝐵) |
5 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑠 → (((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧 ↔ ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
6 | 5 | raleqbi1dv 3146 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → (∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
7 | 6 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑧 = 𝑠 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
8 | 7 | elrab3 3364 |
. . . . . . 7
⊢ (𝑠 ∈ 𝒫 𝐵 → (𝑠 ∈ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧} ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
9 | 4, 8 | syl 17 |
. . . . . 6
⊢ (𝑠 ∈ (SubGrp‘𝐺) → (𝑠 ∈ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧} ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
10 | 9 | bicomd 213 |
. . . . 5
⊢ (𝑠 ∈ (SubGrp‘𝐺) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠 ↔ 𝑠 ∈ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧})) |
11 | 10 | pm5.32i 669 |
. . . 4
⊢ ((𝑠 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠) ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧})) |
12 | | eqid 2622 |
. . . . 5
⊢
(+g‘𝐺) = (+g‘𝐺) |
13 | | eqid 2622 |
. . . . 5
⊢
(-g‘𝐺) = (-g‘𝐺) |
14 | 1, 12, 13 | isnsg3 17628 |
. . . 4
⊢ (𝑠 ∈ (NrmSGrp‘𝐺) ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑠 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑠)) |
15 | | elin 3796 |
. . . 4
⊢ (𝑠 ∈ ((SubGrp‘𝐺) ∩ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧}) ↔ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝑠 ∈ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧})) |
16 | 11, 14, 15 | 3bitr4i 292 |
. . 3
⊢ (𝑠 ∈ (NrmSGrp‘𝐺) ↔ 𝑠 ∈ ((SubGrp‘𝐺) ∩ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧})) |
17 | 16 | eqriv 2619 |
. 2
⊢
(NrmSGrp‘𝐺) =
((SubGrp‘𝐺) ∩
{𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧}) |
18 | | fvex 6201 |
. . . . 5
⊢
(Base‘𝐺)
∈ V |
19 | 1, 18 | eqeltri 2697 |
. . . 4
⊢ 𝐵 ∈ V |
20 | | mreacs 16319 |
. . . 4
⊢ (𝐵 ∈ V →
(ACS‘𝐵) ∈
(Moore‘𝒫 𝐵)) |
21 | 19, 20 | mp1i 13 |
. . 3
⊢ (𝐺 ∈ Grp →
(ACS‘𝐵) ∈
(Moore‘𝒫 𝐵)) |
22 | 1 | subgacs 17629 |
. . 3
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) ∈
(ACS‘𝐵)) |
23 | | simpl 473 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp) |
24 | 1, 12 | grpcl 17430 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
25 | 24 | 3expb 1266 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
26 | | simprl 794 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) |
27 | 1, 13 | grpsubcl 17495 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ (𝑥(+g‘𝐺)𝑦) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
28 | 23, 25, 26, 27 | syl3anc 1326 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
29 | 28 | ralrimivva 2971 |
. . . 4
⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) |
30 | | acsfn1c 16323 |
. . . 4
⊢ ((𝐵 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝐵) → {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧} ∈ (ACS‘𝐵)) |
31 | 19, 29, 30 | sylancr 695 |
. . 3
⊢ (𝐺 ∈ Grp → {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧} ∈ (ACS‘𝐵)) |
32 | | mreincl 16259 |
. . 3
⊢
(((ACS‘𝐵)
∈ (Moore‘𝒫 𝐵) ∧ (SubGrp‘𝐺) ∈ (ACS‘𝐵) ∧ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧} ∈ (ACS‘𝐵)) → ((SubGrp‘𝐺) ∩ {𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧}) ∈ (ACS‘𝐵)) |
33 | 21, 22, 31, 32 | syl3anc 1326 |
. 2
⊢ (𝐺 ∈ Grp →
((SubGrp‘𝐺) ∩
{𝑧 ∈ 𝒫 𝐵 ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑧 ((𝑥(+g‘𝐺)𝑦)(-g‘𝐺)𝑥) ∈ 𝑧}) ∈ (ACS‘𝐵)) |
34 | 17, 33 | syl5eqel 2705 |
1
⊢ (𝐺 ∈ Grp →
(NrmSGrp‘𝐺) ∈
(ACS‘𝐵)) |