Step |
Hyp |
Ref |
Expression |
1 |
|
tngbas.t |
|- T = ( G toNrmGrp N ) |
2 |
|
tngtset.2 |
|- D = ( dist ` T ) |
3 |
|
tngtset.3 |
|- J = ( MetOpen ` D ) |
4 |
1 2 3
|
tngtset |
|- ( ( G e. V /\ N e. W ) -> J = ( TopSet ` T ) ) |
5 |
|
df-mopn |
|- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) ) |
6 |
5
|
dmmptss |
|- dom MetOpen C_ U. ran *Met |
7 |
6
|
sseli |
|- ( D e. dom MetOpen -> D e. U. ran *Met ) |
8 |
|
eqid |
|- ( -g ` G ) = ( -g ` G ) |
9 |
1 8
|
tngds |
|- ( N e. W -> ( N o. ( -g ` G ) ) = ( dist ` T ) ) |
10 |
9 2
|
eqtr4di |
|- ( N e. W -> ( N o. ( -g ` G ) ) = D ) |
11 |
10
|
adantl |
|- ( ( G e. V /\ N e. W ) -> ( N o. ( -g ` G ) ) = D ) |
12 |
11
|
dmeqd |
|- ( ( G e. V /\ N e. W ) -> dom ( N o. ( -g ` G ) ) = dom D ) |
13 |
|
dmcoss |
|- dom ( N o. ( -g ` G ) ) C_ dom ( -g ` G ) |
14 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
15 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
16 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
17 |
14 15 16 8
|
grpsubfval |
|- ( -g ` G ) = ( x e. ( Base ` G ) , y e. ( Base ` G ) |-> ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) ) |
18 |
|
ovex |
|- ( x ( +g ` G ) ( ( invg ` G ) ` y ) ) e. _V |
19 |
17 18
|
dmmpo |
|- dom ( -g ` G ) = ( ( Base ` G ) X. ( Base ` G ) ) |
20 |
13 19
|
sseqtri |
|- dom ( N o. ( -g ` G ) ) C_ ( ( Base ` G ) X. ( Base ` G ) ) |
21 |
12 20
|
eqsstrrdi |
|- ( ( G e. V /\ N e. W ) -> dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) ) |
22 |
21
|
adantr |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) ) |
23 |
|
dmss |
|- ( dom D C_ ( ( Base ` G ) X. ( Base ` G ) ) -> dom dom D C_ dom ( ( Base ` G ) X. ( Base ` G ) ) ) |
24 |
22 23
|
syl |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D C_ dom ( ( Base ` G ) X. ( Base ` G ) ) ) |
25 |
|
dmxpid |
|- dom ( ( Base ` G ) X. ( Base ` G ) ) = ( Base ` G ) |
26 |
24 25
|
sseqtrdi |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D C_ ( Base ` G ) ) |
27 |
|
simpr |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> D e. U. ran *Met ) |
28 |
|
xmetunirn |
|- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) ) |
29 |
27 28
|
sylib |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> D e. ( *Met ` dom dom D ) ) |
30 |
|
eqid |
|- ( MetOpen ` D ) = ( MetOpen ` D ) |
31 |
30
|
mopnuni |
|- ( D e. ( *Met ` dom dom D ) -> dom dom D = U. ( MetOpen ` D ) ) |
32 |
29 31
|
syl |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> dom dom D = U. ( MetOpen ` D ) ) |
33 |
1 14
|
tngbas |
|- ( N e. W -> ( Base ` G ) = ( Base ` T ) ) |
34 |
33
|
ad2antlr |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> ( Base ` G ) = ( Base ` T ) ) |
35 |
26 32 34
|
3sstr3d |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> U. ( MetOpen ` D ) C_ ( Base ` T ) ) |
36 |
|
sspwuni |
|- ( ( MetOpen ` D ) C_ ~P ( Base ` T ) <-> U. ( MetOpen ` D ) C_ ( Base ` T ) ) |
37 |
35 36
|
sylibr |
|- ( ( ( G e. V /\ N e. W ) /\ D e. U. ran *Met ) -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) |
38 |
37
|
ex |
|- ( ( G e. V /\ N e. W ) -> ( D e. U. ran *Met -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) ) |
39 |
7 38
|
syl5 |
|- ( ( G e. V /\ N e. W ) -> ( D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) ) |
40 |
|
ndmfv |
|- ( -. D e. dom MetOpen -> ( MetOpen ` D ) = (/) ) |
41 |
|
0ss |
|- (/) C_ ~P ( Base ` T ) |
42 |
40 41
|
eqsstrdi |
|- ( -. D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) |
43 |
39 42
|
pm2.61d1 |
|- ( ( G e. V /\ N e. W ) -> ( MetOpen ` D ) C_ ~P ( Base ` T ) ) |
44 |
3 43
|
eqsstrid |
|- ( ( G e. V /\ N e. W ) -> J C_ ~P ( Base ` T ) ) |
45 |
4 44
|
eqsstrrd |
|- ( ( G e. V /\ N e. W ) -> ( TopSet ` T ) C_ ~P ( Base ` T ) ) |
46 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
47 |
|
eqid |
|- ( TopSet ` T ) = ( TopSet ` T ) |
48 |
46 47
|
topnid |
|- ( ( TopSet ` T ) C_ ~P ( Base ` T ) -> ( TopSet ` T ) = ( TopOpen ` T ) ) |
49 |
45 48
|
syl |
|- ( ( G e. V /\ N e. W ) -> ( TopSet ` T ) = ( TopOpen ` T ) ) |
50 |
4 49
|
eqtrd |
|- ( ( G e. V /\ N e. W ) -> J = ( TopOpen ` T ) ) |