Step |
Hyp |
Ref |
Expression |
1 |
|
subgntr.h |
⊢ 𝐽 = ( TopOpen ‘ 𝐺 ) |
2 |
|
nsgsubg |
⊢ ( 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
3 |
1
|
clssubg |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
4 |
2 3
|
sylan2 |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
df-ima |
⊢ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ran ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
7 |
1 6
|
tgptopon |
⊢ ( 𝐺 ∈ TopGrp → 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ) |
9 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) → 𝐽 ∈ Top ) |
10 |
8 9
|
syl |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝐽 ∈ Top ) |
11 |
2
|
ad2antlr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
12 |
6
|
subgss |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
13 |
11 12
|
syl |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
14 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) → ( Base ‘ 𝐺 ) = ∪ 𝐽 ) |
15 |
8 14
|
syl |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( Base ‘ 𝐺 ) = ∪ 𝐽 ) |
16 |
13 15
|
sseqtrd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑆 ⊆ ∪ 𝐽 ) |
17 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
18 |
17
|
clsss3 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ∪ 𝐽 ) |
19 |
10 16 18
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ∪ 𝐽 ) |
20 |
19 15
|
sseqtrrd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( Base ‘ 𝐺 ) ) |
21 |
20
|
resmptd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
22 |
21
|
rneqd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ran ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ran ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
23 |
5 22
|
eqtrid |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ran ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
24 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
25 |
|
tgptmd |
⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd ) |
26 |
25
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝐺 ∈ TopMnd ) |
27 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
28 |
8 8 27
|
cnmptc |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ 𝑥 ) ∈ ( 𝐽 Cn 𝐽 ) ) |
29 |
8
|
cnmptid |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ 𝑦 ) ∈ ( 𝐽 Cn 𝐽 ) ) |
30 |
1 24 26 8 28 29
|
cnmpt1plusg |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
31 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
32 |
1 31
|
tgpsubcn |
⊢ ( 𝐺 ∈ TopGrp → ( -g ‘ 𝐺 ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
33 |
32
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( -g ‘ 𝐺 ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
34 |
8 30 28 33
|
cnmpt12f |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) ) |
35 |
17
|
cnclsi |
⊢ ( ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ∈ ( 𝐽 Cn 𝐽 ) ∧ 𝑆 ⊆ ∪ 𝐽 ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ) ) |
36 |
34 16 35
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ) ) |
37 |
|
df-ima |
⊢ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) = ran ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ 𝑆 ) |
38 |
13
|
resmptd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ 𝑆 ) = ( 𝑦 ∈ 𝑆 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
39 |
38
|
rneqd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ran ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ↾ 𝑆 ) = ran ( 𝑦 ∈ 𝑆 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
40 |
37 39
|
eqtrid |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) = ran ( 𝑦 ∈ 𝑆 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
41 |
6 24 31
|
nsgconj |
⊢ ( ( 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ 𝑆 ) |
42 |
41
|
ad4ant234 |
⊢ ( ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ 𝑆 ) |
43 |
42
|
fmpttd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ 𝑆 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) : 𝑆 ⟶ 𝑆 ) |
44 |
43
|
frnd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ran ( 𝑦 ∈ 𝑆 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ⊆ 𝑆 ) |
45 |
40 44
|
eqsstrd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ⊆ 𝑆 ) |
46 |
17
|
clsss |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ⊆ 𝑆 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
47 |
10 16 45 46
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ 𝑆 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
48 |
36 47
|
sstrd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) “ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
49 |
23 48
|
eqsstrrd |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ran ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
50 |
|
ovex |
⊢ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ V |
51 |
|
eqid |
⊢ ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) = ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) |
52 |
50 51
|
fnmpti |
⊢ ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) Fn ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) |
53 |
|
df-f |
⊢ ( ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) : ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⟶ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ( ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) Fn ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∧ ran ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
54 |
52 53
|
mpbiran |
⊢ ( ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) : ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⟶ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ran ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
55 |
49 54
|
sylibr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) : ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⟶ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
56 |
51
|
fmpt |
⊢ ( ∀ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ( 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ) : ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⟶ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
57 |
55 56
|
sylibr |
⊢ ( ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ∀ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
58 |
57
|
ralrimiva |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
59 |
6 24 31
|
isnsg3 |
⊢ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( NrmSGrp ‘ 𝐺 ) ↔ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
60 |
4 58 59
|
sylanbrc |
⊢ ( ( 𝐺 ∈ TopGrp ∧ 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( NrmSGrp ‘ 𝐺 ) ) |