Step |
Hyp |
Ref |
Expression |
1 |
|
uspreg.1 |
|- J = ( TopOpen ` W ) |
2 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
3 |
|
eqid |
|- ( UnifSt ` W ) = ( UnifSt ` W ) |
4 |
2 3 1
|
isusp |
|- ( W e. UnifSp <-> ( ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) /\ J = ( unifTop ` ( UnifSt ` W ) ) ) ) |
5 |
4
|
simprbi |
|- ( W e. UnifSp -> J = ( unifTop ` ( UnifSt ` W ) ) ) |
6 |
5
|
adantr |
|- ( ( W e. UnifSp /\ J e. Haus ) -> J = ( unifTop ` ( UnifSt ` W ) ) ) |
7 |
4
|
simplbi |
|- ( W e. UnifSp -> ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) ) |
8 |
|
simpr |
|- ( ( W e. UnifSp /\ J e. Haus ) -> J e. Haus ) |
9 |
6 8
|
eqeltrrd |
|- ( ( W e. UnifSp /\ J e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Haus ) |
10 |
|
eqid |
|- ( unifTop ` ( UnifSt ` W ) ) = ( unifTop ` ( UnifSt ` W ) ) |
11 |
10
|
utopreg |
|- ( ( ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) /\ ( unifTop ` ( UnifSt ` W ) ) e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Reg ) |
12 |
7 9 11
|
syl2an2r |
|- ( ( W e. UnifSp /\ J e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Reg ) |
13 |
6 12
|
eqeltrd |
|- ( ( W e. UnifSp /\ J e. Haus ) -> J e. Reg ) |