Step |
Hyp |
Ref |
Expression |
1 |
|
tgpt1.j |
|- J = ( TopOpen ` G ) |
2 |
|
haust1 |
|- ( J e. Haus -> J e. Fre ) |
3 |
|
tgpgrp |
|- ( G e. TopGrp -> G e. Grp ) |
4 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
5 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
6 |
4 5
|
grpidcl |
|- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) ) |
7 |
3 6
|
syl |
|- ( G e. TopGrp -> ( 0g ` G ) e. ( Base ` G ) ) |
8 |
1 4
|
tgptopon |
|- ( G e. TopGrp -> J e. ( TopOn ` ( Base ` G ) ) ) |
9 |
|
toponuni |
|- ( J e. ( TopOn ` ( Base ` G ) ) -> ( Base ` G ) = U. J ) |
10 |
8 9
|
syl |
|- ( G e. TopGrp -> ( Base ` G ) = U. J ) |
11 |
7 10
|
eleqtrd |
|- ( G e. TopGrp -> ( 0g ` G ) e. U. J ) |
12 |
|
eqid |
|- U. J = U. J |
13 |
12
|
t1sncld |
|- ( ( J e. Fre /\ ( 0g ` G ) e. U. J ) -> { ( 0g ` G ) } e. ( Clsd ` J ) ) |
14 |
13
|
expcom |
|- ( ( 0g ` G ) e. U. J -> ( J e. Fre -> { ( 0g ` G ) } e. ( Clsd ` J ) ) ) |
15 |
11 14
|
syl |
|- ( G e. TopGrp -> ( J e. Fre -> { ( 0g ` G ) } e. ( Clsd ` J ) ) ) |
16 |
5 1
|
tgphaus |
|- ( G e. TopGrp -> ( J e. Haus <-> { ( 0g ` G ) } e. ( Clsd ` J ) ) ) |
17 |
15 16
|
sylibrd |
|- ( G e. TopGrp -> ( J e. Fre -> J e. Haus ) ) |
18 |
2 17
|
impbid2 |
|- ( G e. TopGrp -> ( J e. Haus <-> J e. Fre ) ) |