Step |
Hyp |
Ref |
Expression |
1 |
|
tgpt1.j |
⊢ 𝐽 = ( TopOpen ‘ 𝐺 ) |
2 |
|
haust1 |
⊢ ( 𝐽 ∈ Haus → 𝐽 ∈ Fre ) |
3 |
|
tgpgrp |
⊢ ( 𝐺 ∈ TopGrp → 𝐺 ∈ Grp ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
5 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
6 |
4 5
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ) |
7 |
3 6
|
syl |
⊢ ( 𝐺 ∈ TopGrp → ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ) |
8 |
1 4
|
tgptopon |
⊢ ( 𝐺 ∈ TopGrp → 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) ) |
9 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ ( Base ‘ 𝐺 ) ) → ( Base ‘ 𝐺 ) = ∪ 𝐽 ) |
10 |
8 9
|
syl |
⊢ ( 𝐺 ∈ TopGrp → ( Base ‘ 𝐺 ) = ∪ 𝐽 ) |
11 |
7 10
|
eleqtrd |
⊢ ( 𝐺 ∈ TopGrp → ( 0g ‘ 𝐺 ) ∈ ∪ 𝐽 ) |
12 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
13 |
12
|
t1sncld |
⊢ ( ( 𝐽 ∈ Fre ∧ ( 0g ‘ 𝐺 ) ∈ ∪ 𝐽 ) → { ( 0g ‘ 𝐺 ) } ∈ ( Clsd ‘ 𝐽 ) ) |
14 |
13
|
expcom |
⊢ ( ( 0g ‘ 𝐺 ) ∈ ∪ 𝐽 → ( 𝐽 ∈ Fre → { ( 0g ‘ 𝐺 ) } ∈ ( Clsd ‘ 𝐽 ) ) ) |
15 |
11 14
|
syl |
⊢ ( 𝐺 ∈ TopGrp → ( 𝐽 ∈ Fre → { ( 0g ‘ 𝐺 ) } ∈ ( Clsd ‘ 𝐽 ) ) ) |
16 |
5 1
|
tgphaus |
⊢ ( 𝐺 ∈ TopGrp → ( 𝐽 ∈ Haus ↔ { ( 0g ‘ 𝐺 ) } ∈ ( Clsd ‘ 𝐽 ) ) ) |
17 |
15 16
|
sylibrd |
⊢ ( 𝐺 ∈ TopGrp → ( 𝐽 ∈ Fre → 𝐽 ∈ Haus ) ) |
18 |
2 17
|
impbid2 |
⊢ ( 𝐺 ∈ TopGrp → ( 𝐽 ∈ Haus ↔ 𝐽 ∈ Fre ) ) |