Metamath Proof Explorer


Theorem sitgclbn

Description: Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018)

Ref Expression
Hypotheses sitgval.b
|- B = ( Base ` W )
sitgval.j
|- J = ( TopOpen ` W )
sitgval.s
|- S = ( sigaGen ` J )
sitgval.0
|- .0. = ( 0g ` W )
sitgval.x
|- .x. = ( .s ` W )
sitgval.h
|- H = ( RRHom ` ( Scalar ` W ) )
sitgval.1
|- ( ph -> W e. V )
sitgval.2
|- ( ph -> M e. U. ran measures )
sibfmbl.1
|- ( ph -> F e. dom ( W sitg M ) )
sitgclbn.1
|- ( ph -> W e. Ban )
sitgclbn.2
|- ( ph -> ( Scalar ` W ) e. RRExt )
Assertion sitgclbn
|- ( ph -> ( ( W sitg M ) ` F ) e. B )

Proof

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 )