Step |
Hyp |
Ref |
Expression |
1 |
|
sitgval.b |
|- B = ( Base ` W ) |
2 |
|
sitgval.j |
|- J = ( TopOpen ` W ) |
3 |
|
sitgval.s |
|- S = ( sigaGen ` J ) |
4 |
|
sitgval.0 |
|- .0. = ( 0g ` W ) |
5 |
|
sitgval.x |
|- .x. = ( .s ` W ) |
6 |
|
sitgval.h |
|- H = ( RRHom ` ( Scalar ` W ) ) |
7 |
|
sitgval.1 |
|- ( ph -> W e. V ) |
8 |
|
sitgval.2 |
|- ( ph -> M e. U. ran measures ) |
9 |
|
sibfmbl.1 |
|- ( ph -> F e. dom ( W sitg M ) ) |
10 |
|
sitgclbn.1 |
|- ( ph -> W e. Ban ) |
11 |
|
sitgclbn.2 |
|- ( ph -> ( Scalar ` W ) e. RRExt ) |
12 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
13 |
|
eqid |
|- ( ( dist ` ( Scalar ` W ) ) |` ( ( Base ` ( Scalar ` W ) ) X. ( Base ` ( Scalar ` W ) ) ) ) = ( ( dist ` ( Scalar ` W ) ) |` ( ( Base ` ( Scalar ` W ) ) X. ( Base ` ( Scalar ` W ) ) ) ) |
14 |
|
bncms |
|- ( W e. Ban -> W e. CMetSp ) |
15 |
10 14
|
syl |
|- ( ph -> W e. CMetSp ) |
16 |
|
cmsms |
|- ( W e. CMetSp -> W e. MetSp ) |
17 |
|
mstps |
|- ( W e. MetSp -> W e. TopSp ) |
18 |
15 16 17
|
3syl |
|- ( ph -> W e. TopSp ) |
19 |
|
bnlmod |
|- ( W e. Ban -> W e. LMod ) |
20 |
|
lmodcmn |
|- ( W e. LMod -> W e. CMnd ) |
21 |
10 19 20
|
3syl |
|- ( ph -> W e. CMnd ) |
22 |
10 19
|
syl |
|- ( ph -> W e. LMod ) |
23 |
22
|
3ad2ant1 |
|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> W e. LMod ) |
24 |
|
imassrn |
|- ( H " ( 0 [,) +oo ) ) C_ ran H |
25 |
6
|
rneqi |
|- ran H = ran ( RRHom ` ( Scalar ` W ) ) |
26 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
27 |
26
|
rrhfe |
|- ( ( Scalar ` W ) e. RRExt -> ( RRHom ` ( Scalar ` W ) ) : RR --> ( Base ` ( Scalar ` W ) ) ) |
28 |
|
frn |
|- ( ( RRHom ` ( Scalar ` W ) ) : RR --> ( Base ` ( Scalar ` W ) ) -> ran ( RRHom ` ( Scalar ` W ) ) C_ ( Base ` ( Scalar ` W ) ) ) |
29 |
11 27 28
|
3syl |
|- ( ph -> ran ( RRHom ` ( Scalar ` W ) ) C_ ( Base ` ( Scalar ` W ) ) ) |
30 |
25 29
|
eqsstrid |
|- ( ph -> ran H C_ ( Base ` ( Scalar ` W ) ) ) |
31 |
24 30
|
sstrid |
|- ( ph -> ( H " ( 0 [,) +oo ) ) C_ ( Base ` ( Scalar ` W ) ) ) |
32 |
31
|
sselda |
|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) ) -> m e. ( Base ` ( Scalar ` W ) ) ) |
33 |
32
|
3adant3 |
|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> m e. ( Base ` ( Scalar ` W ) ) ) |
34 |
|
simp3 |
|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> x e. B ) |
35 |
1 12 5 26
|
lmodvscl |
|- ( ( W e. LMod /\ m e. ( Base ` ( Scalar ` W ) ) /\ x e. B ) -> ( m .x. x ) e. B ) |
36 |
23 33 34 35
|
syl3anc |
|- ( ( ph /\ m e. ( H " ( 0 [,) +oo ) ) /\ x e. B ) -> ( m .x. x ) e. B ) |
37 |
1 2 3 4 5 6 7 8 9 12 13 18 21 11 36
|
sitgclg |
|- ( ph -> ( ( W sitg M ) ` F ) e. B ) |