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Theorem sitgfval 26679
Description: Value of the Bochner integral for a simple function  F. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
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  ( Wsitg M ) )
Assertion
Ref Expression
sitgfval  |-  ( ph  ->  ( ( Wsitg M
) `  F )  =  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) ) )
Distinct variable groups:    x, F    x, M    x, W    x,  .0.   
ph, x
Allowed substitution hints:    B( x)    S( x)    .x. ( x)    H( x)    J( x)    V( x)

Proof of Theorem sitgfval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . 3  |-  B  =  ( Base `  W
)
2 sitgval.j . . 3  |-  J  =  ( TopOpen `  W )
3 sitgval.s . . 3  |-  S  =  (sigaGen `  J )
4 sitgval.0 . . 3  |-  .0.  =  ( 0g `  W )
5 sitgval.x . . 3  |-  .x.  =  ( .s `  W )
6 sitgval.h . . 3  |-  H  =  (RRHom `  (Scalar `  W
) )
7 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
8 sitgval.2 . . 3  |-  ( ph  ->  M  e.  U. ran measures )
91, 2, 3, 4, 5, 6, 7, 8sitgval 26670 . 2  |-  ( ph  ->  ( Wsitg M )  =  ( f  e. 
{ g  e.  ( dom  MMblFnM S )  |  ( ran  g  e. 
Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " { x } ) )  e.  ( 0 [,) +oo ) ) }  |->  ( W  gsumg  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' f
" { x }
) ) )  .x.  x ) ) ) ) )
10 simpr 461 . . . . . 6  |-  ( (
ph  /\  f  =  F )  ->  f  =  F )
1110rneqd 5062 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ran  f  =  ran  F )
1211difeq1d 3468 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
1310cnveqd 5010 . . . . . . . 8  |-  ( (
ph  /\  f  =  F )  ->  `' f  =  `' F
)
1413imaeq1d 5163 . . . . . . 7  |-  ( (
ph  /\  f  =  F )  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1514fveq2d 5690 . . . . . 6  |-  ( (
ph  /\  f  =  F )  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
1615fveq2d 5690 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
1716oveq1d 6101 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
1812, 17mpteq12dv 4365 . . 3  |-  ( (
ph  /\  f  =  F )  ->  (
x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' f " {
x } ) ) )  .x.  x ) )  =  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )
1918oveq2d 6102 . 2  |-  ( (
ph  /\  f  =  F )  ->  ( W  gsumg  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' f
" { x }
) ) )  .x.  x ) ) )  =  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) ) )
20 sibfmbl.1 . . . . 5  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
211, 2, 3, 4, 5, 6, 7, 8, 20sibfmbl 26673 . . . 4  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
221, 2, 3, 4, 5, 6, 7, 8, 20sibfrn 26675 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
231, 2, 3, 4, 5, 6, 7, 8, 20sibfima 26676 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
2423ralrimiva 2794 . . . 4  |-  ( ph  ->  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
2521, 22, 24jca32 535 . . 3  |-  ( ph  ->  ( F  e.  ( dom  MMblFnM S )  /\  ( ran  F  e.  Fin  /\ 
A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) ) )
26 rneq 5060 . . . . . 6  |-  ( g  =  F  ->  ran  g  =  ran  F )
2726eleq1d 2504 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
2826difeq1d 3468 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
29 cnveq 5008 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3029imaeq1d 5163 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3130fveq2d 5690 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3231eleq1d 2504 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3328, 32raleqbidv 2926 . . . . 5  |-  ( g  =  F  ->  ( A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3427, 33anbi12d 710 . . . 4  |-  ( g  =  F  ->  (
( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " { x } ) )  e.  ( 0 [,) +oo ) )  <->  ( ran  F  e.  Fin  /\  A. x  e.  ( ran  F 
\  {  .0.  }
) ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) ) )
3534elrab 3112 . . 3  |-  ( F  e.  { g  e.  ( dom  MMblFnM S
)  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo ) ) }  <->  ( F  e.  ( dom  MMblFnM S
)  /\  ( ran  F  e.  Fin  /\  A. x  e.  ( ran  F 
\  {  .0.  }
) ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) ) )
3625, 35sylibr 212 . 2  |-  ( ph  ->  F  e.  { g  e.  ( dom  MMblFnM S )  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo ) ) } )
37 ovex 6111 . . 3  |-  ( W 
gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )  e.  _V
3837a1i 11 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e. 
_V )
399, 19, 36, 38fvmptd 5774 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  =  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   {crab 2714   _Vcvv 2967    \ cdif 3320   {csn 3872   U.cuni 4086    e. cmpt 4345   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   ` cfv 5413  (class class class)co 6086   Fincfn 7302   0cc0 9274   +oocpnf 9407   [,)cico 11294   Basecbs 14166  Scalarcsca 14233   .scvsca 14234   TopOpenctopn 14352   0gc0g 14370    gsumg cgsu 14371  RRHomcrrh 26374  sigaGencsigagen 26533  measurescmeas 26561  MblFnMcmbfm 26617  sitgcsitg 26667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-sitg 26668
This theorem is referenced by:  sitgclg  26680  sitg0  26684
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