Metamath Proof Explorer


Theorem sitgclcn

Description: Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (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 ) )
sitgclcn.1
|- ( ph -> W e. Ban )
sitgclcn.2
|- ( ph -> ( Scalar ` W ) = CCfld )
Assertion sitgclcn
|- ( 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 sitgclcn.1
 |-  ( ph -> W e. Ban )
11 sitgclcn.2
 |-  ( ph -> ( Scalar ` W ) = CCfld )
12 cnrrext
 |-  CCfld e. RRExt
13 11 12 eqeltrdi
 |-  ( ph -> ( Scalar ` W ) e. RRExt )
14 1 2 3 4 5 6 7 8 9 10 13 sitgclbn
 |-  ( ph -> ( ( W sitg M ) ` F ) e. B )