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Theorem sitgfval 29174
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 29165 . 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 463 . . . . . 6  |-  ( (
ph  /\  f  =  F )  ->  f  =  F )
1110rneqd 5062 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ran  f  =  ran  F )
1211difeq1d 3550 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
1310cnveqd 5010 . . . . . . . 8  |-  ( (
ph  /\  f  =  F )  ->  `' f  =  `' F
)
1413imaeq1d 5167 . . . . . . 7  |-  ( (
ph  /\  f  =  F )  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1514fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  f  =  F )  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
1615fveq2d 5869 . . . . 5  |-  ( (
ph  /\  f  =  F )  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
1716oveq1d 6305 . . . 4  |-  ( (
ph  /\  f  =  F )  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
1812, 17mpteq12dv 4481 . . 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 6306 . 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 29168 . . . 4  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
221, 2, 3, 4, 5, 6, 7, 8, 20sibfrn 29170 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
231, 2, 3, 4, 5, 6, 7, 8, 20sibfima 29171 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
2423ralrimiva 2802 . . . 4  |-  ( ph  ->  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `
 ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
2521, 22, 24jca32 538 . . 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 2513 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
2826difeq1d 3550 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
29 cnveq 5008 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3029imaeq1d 5167 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3130fveq2d 5869 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3231eleq1d 2513 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3328, 32raleqbidv 3001 . . . . 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 717 . . . 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 3196 . . 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 216 . 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 6318 . . 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 5954 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 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741   _Vcvv 3045    \ cdif 3401   {csn 3968   U.cuni 4198    |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290   Fincfn 7569   0cc0 9539   +oocpnf 9672   [,)cico 11637   Basecbs 15121  Scalarcsca 15193   .scvsca 15194   TopOpenctopn 15320   0gc0g 15338    gsumg cgsu 15339  RRHomcrrh 28797  sigaGencsigagen 28960  measurescmeas 29017  MblFnMcmbfm 29072  sitgcsitg 29162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-sitg 29163
This theorem is referenced by:  sitgclg  29175  sitg0  29179
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