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Theorem sitgfval 26870
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 26861 . 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 5174 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ran  f  =  ran  F )
1211difeq1d 3580 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
1310cnveqd 5122 . . . . . . . 8  |-  ( (
ph  /\  f  =  F )  ->  `' f  =  `' F
)
1413imaeq1d 5275 . . . . . . 7  |-  ( (
ph  /\  f  =  F )  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1514fveq2d 5802 . . . . . 6  |-  ( (
ph  /\  f  =  F )  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
1615fveq2d 5802 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
1716oveq1d 6214 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
1812, 17mpteq12dv 4477 . . 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 6215 . 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 26864 . . . 4  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
221, 2, 3, 4, 5, 6, 7, 8, 20sibfrn 26866 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
231, 2, 3, 4, 5, 6, 7, 8, 20sibfima 26867 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
2423ralrimiva 2829 . . . 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 5172 . . . . . 6  |-  ( g  =  F  ->  ran  g  =  ran  F )
2726eleq1d 2523 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
2826difeq1d 3580 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
29 cnveq 5120 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3029imaeq1d 5275 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3130fveq2d 5802 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3231eleq1d 2523 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3328, 32raleqbidv 3035 . . . . 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 3222 . . 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 6224 . . 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 5887 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 1370    e. wcel 1758   A.wral 2798   {crab 2802   _Vcvv 3076    \ cdif 3432   {csn 3984   U.cuni 4198    |-> cmpt 4457   `'ccnv 4946   dom cdm 4947   ran crn 4948   "cima 4950   ` cfv 5525  (class class class)co 6199   Fincfn 7419   0cc0 9392   +oocpnf 9525   [,)cico 11412   Basecbs 14291  Scalarcsca 14359   .scvsca 14360   TopOpenctopn 14478   0gc0g 14496    gsumg cgsu 14497  RRHomcrrh 26566  sigaGencsigagen 26725  measurescmeas 26753  MblFnMcmbfm 26808  sitgcsitg 26858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-sitg 26859
This theorem is referenced by:  sitgclg  26871  sitg0  26875
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