Step |
Hyp |
Ref |
Expression |
1 |
|
xmspropd.1 |
|- ( ph -> B = ( Base ` K ) ) |
2 |
|
xmspropd.2 |
|- ( ph -> B = ( Base ` L ) ) |
3 |
|
xmspropd.3 |
|- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( B X. B ) ) ) |
4 |
|
xmspropd.4 |
|- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) ) |
5 |
1 2 3 4
|
xmspropd |
|- ( ph -> ( K e. *MetSp <-> L e. *MetSp ) ) |
6 |
1
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` K ) X. ( Base ` K ) ) ) |
7 |
6
|
reseq2d |
|- ( ph -> ( ( dist ` K ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) |
8 |
3 7
|
eqtr3d |
|- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) ) |
9 |
2
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` L ) X. ( Base ` L ) ) ) |
10 |
9
|
reseq2d |
|- ( ph -> ( ( dist ` L ) |` ( B X. B ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) |
11 |
8 10
|
eqtr3d |
|- ( ph -> ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) ) |
12 |
1 2
|
eqtr3d |
|- ( ph -> ( Base ` K ) = ( Base ` L ) ) |
13 |
12
|
fveq2d |
|- ( ph -> ( Met ` ( Base ` K ) ) = ( Met ` ( Base ` L ) ) ) |
14 |
11 13
|
eleq12d |
|- ( ph -> ( ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) <-> ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( Base ` L ) ) ) ) |
15 |
5 14
|
anbi12d |
|- ( ph -> ( ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) <-> ( L e. *MetSp /\ ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( Base ` L ) ) ) ) ) |
16 |
|
eqid |
|- ( TopOpen ` K ) = ( TopOpen ` K ) |
17 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
18 |
|
eqid |
|- ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) = ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) |
19 |
16 17 18
|
isms |
|- ( K e. MetSp <-> ( K e. *MetSp /\ ( ( dist ` K ) |` ( ( Base ` K ) X. ( Base ` K ) ) ) e. ( Met ` ( Base ` K ) ) ) ) |
20 |
|
eqid |
|- ( TopOpen ` L ) = ( TopOpen ` L ) |
21 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
22 |
|
eqid |
|- ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) = ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) |
23 |
20 21 22
|
isms |
|- ( L e. MetSp <-> ( L e. *MetSp /\ ( ( dist ` L ) |` ( ( Base ` L ) X. ( Base ` L ) ) ) e. ( Met ` ( Base ` L ) ) ) ) |
24 |
15 19 23
|
3bitr4g |
|- ( ph -> ( K e. MetSp <-> L e. MetSp ) ) |