Step |
Hyp |
Ref |
Expression |
1 |
|
tlmlmod |
|- ( W e. TopMod -> W e. LMod ) |
2 |
|
lmodgrp |
|- ( W e. LMod -> W e. Grp ) |
3 |
1 2
|
syl |
|- ( W e. TopMod -> W e. Grp ) |
4 |
|
tlmtmd |
|- ( W e. TopMod -> W e. TopMnd ) |
5 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
6 |
|
eqid |
|- ( invg ` W ) = ( invg ` W ) |
7 |
5 6
|
grpinvf |
|- ( W e. Grp -> ( invg ` W ) : ( Base ` W ) --> ( Base ` W ) ) |
8 |
3 7
|
syl |
|- ( W e. TopMod -> ( invg ` W ) : ( Base ` W ) --> ( Base ` W ) ) |
9 |
8
|
feqmptd |
|- ( W e. TopMod -> ( invg ` W ) = ( x e. ( Base ` W ) |-> ( ( invg ` W ) ` x ) ) ) |
10 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
11 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
12 |
|
eqid |
|- ( 1r ` ( Scalar ` W ) ) = ( 1r ` ( Scalar ` W ) ) |
13 |
|
eqid |
|- ( invg ` ( Scalar ` W ) ) = ( invg ` ( Scalar ` W ) ) |
14 |
5 6 10 11 12 13
|
lmodvneg1 |
|- ( ( W e. LMod /\ x e. ( Base ` W ) ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) x ) = ( ( invg ` W ) ` x ) ) |
15 |
1 14
|
sylan |
|- ( ( W e. TopMod /\ x e. ( Base ` W ) ) -> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) x ) = ( ( invg ` W ) ` x ) ) |
16 |
15
|
mpteq2dva |
|- ( W e. TopMod -> ( x e. ( Base ` W ) |-> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) x ) ) = ( x e. ( Base ` W ) |-> ( ( invg ` W ) ` x ) ) ) |
17 |
9 16
|
eqtr4d |
|- ( W e. TopMod -> ( invg ` W ) = ( x e. ( Base ` W ) |-> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) x ) ) ) |
18 |
|
eqid |
|- ( TopOpen ` W ) = ( TopOpen ` W ) |
19 |
|
eqid |
|- ( TopOpen ` ( Scalar ` W ) ) = ( TopOpen ` ( Scalar ` W ) ) |
20 |
|
id |
|- ( W e. TopMod -> W e. TopMod ) |
21 |
|
tlmtps |
|- ( W e. TopMod -> W e. TopSp ) |
22 |
5 18
|
istps |
|- ( W e. TopSp <-> ( TopOpen ` W ) e. ( TopOn ` ( Base ` W ) ) ) |
23 |
21 22
|
sylib |
|- ( W e. TopMod -> ( TopOpen ` W ) e. ( TopOn ` ( Base ` W ) ) ) |
24 |
10
|
tlmscatps |
|- ( W e. TopMod -> ( Scalar ` W ) e. TopSp ) |
25 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
26 |
25 19
|
istps |
|- ( ( Scalar ` W ) e. TopSp <-> ( TopOpen ` ( Scalar ` W ) ) e. ( TopOn ` ( Base ` ( Scalar ` W ) ) ) ) |
27 |
24 26
|
sylib |
|- ( W e. TopMod -> ( TopOpen ` ( Scalar ` W ) ) e. ( TopOn ` ( Base ` ( Scalar ` W ) ) ) ) |
28 |
10
|
lmodring |
|- ( W e. LMod -> ( Scalar ` W ) e. Ring ) |
29 |
1 28
|
syl |
|- ( W e. TopMod -> ( Scalar ` W ) e. Ring ) |
30 |
|
ringgrp |
|- ( ( Scalar ` W ) e. Ring -> ( Scalar ` W ) e. Grp ) |
31 |
29 30
|
syl |
|- ( W e. TopMod -> ( Scalar ` W ) e. Grp ) |
32 |
25 12
|
ringidcl |
|- ( ( Scalar ` W ) e. Ring -> ( 1r ` ( Scalar ` W ) ) e. ( Base ` ( Scalar ` W ) ) ) |
33 |
29 32
|
syl |
|- ( W e. TopMod -> ( 1r ` ( Scalar ` W ) ) e. ( Base ` ( Scalar ` W ) ) ) |
34 |
25 13
|
grpinvcl |
|- ( ( ( Scalar ` W ) e. Grp /\ ( 1r ` ( Scalar ` W ) ) e. ( Base ` ( Scalar ` W ) ) ) -> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) ) |
35 |
31 33 34
|
syl2anc |
|- ( W e. TopMod -> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) e. ( Base ` ( Scalar ` W ) ) ) |
36 |
23 27 35
|
cnmptc |
|- ( W e. TopMod -> ( x e. ( Base ` W ) |-> ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ) e. ( ( TopOpen ` W ) Cn ( TopOpen ` ( Scalar ` W ) ) ) ) |
37 |
23
|
cnmptid |
|- ( W e. TopMod -> ( x e. ( Base ` W ) |-> x ) e. ( ( TopOpen ` W ) Cn ( TopOpen ` W ) ) ) |
38 |
10 11 18 19 20 23 36 37
|
cnmpt1vsca |
|- ( W e. TopMod -> ( x e. ( Base ` W ) |-> ( ( ( invg ` ( Scalar ` W ) ) ` ( 1r ` ( Scalar ` W ) ) ) ( .s ` W ) x ) ) e. ( ( TopOpen ` W ) Cn ( TopOpen ` W ) ) ) |
39 |
17 38
|
eqeltrd |
|- ( W e. TopMod -> ( invg ` W ) e. ( ( TopOpen ` W ) Cn ( TopOpen ` W ) ) ) |
40 |
18 6
|
istgp |
|- ( W e. TopGrp <-> ( W e. Grp /\ W e. TopMnd /\ ( invg ` W ) e. ( ( TopOpen ` W ) Cn ( TopOpen ` W ) ) ) ) |
41 |
3 4 39 40
|
syl3anbrc |
|- ( W e. TopMod -> W e. TopGrp ) |