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Theorem issibf 24601
Description: The predicate " F is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-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 )
Assertion
Ref Expression
issibf  |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S
)  /\  ran  F  e. 
Fin  /\  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) ) )
Distinct variable groups:    x, F    x, M    x, W    x,  .0.
Allowed substitution hints:    ph( x)    B( x)    S( x)    .x. ( x)    H( x)    J( x)    V( x)

Proof of Theorem issibf
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . . . . . . . 9  |-  B  =  ( Base `  W
)
2 sitgval.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
3 sitgval.s . . . . . . . . 9  |-  S  =  (sigaGen `  J )
4 sitgval.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
5 sitgval.x . . . . . . . . 9  |-  .x.  =  ( .s `  W )
6 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
7 sitgval.1 . . . . . . . . 9  |-  ( ph  ->  W  e.  V )
8 sitgval.2 . . . . . . . . 9  |-  ( ph  ->  M  e.  U. ran measures )
91, 2, 3, 4, 5, 6, 7, 8sitgval 24600 . . . . . . . 8  |-  ( 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 ) ) ) ) )
109dmeqd 5031 . . . . . . 7  |-  ( ph  ->  dom  ( Wsitg M
)  =  dom  (
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 ) ) ) ) )
11 eqid 2404 . . . . . . . 8  |-  ( 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 ) ) ) )  =  ( 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 ) ) ) )
1211dmmpt 5324 . . . . . . 7  |-  dom  (
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 ) ) ) )  =  { 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 ) ) )  e.  _V }
1310, 12syl6eq 2452 . . . . . 6  |-  ( ph  ->  dom  ( 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 ) ) )  e.  _V }
)
1413eleq2d 2471 . . . . 5  |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  F  e.  { 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 ) ) )  e.  _V }
) )
15 rneq 5054 . . . . . . . . . 10  |-  ( f  =  F  ->  ran  f  =  ran  F )
1615difeq1d 3424 . . . . . . . . 9  |-  ( f  =  F  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
17 cnveq 5005 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  `' f  =  `' F
)
1817imaeq1d 5161 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1918fveq2d 5691 . . . . . . . . . . 11  |-  ( f  =  F  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
2019fveq2d 5691 . . . . . . . . . 10  |-  ( f  =  F  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
2120oveq1d 6055 . . . . . . . . 9  |-  ( f  =  F  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
2216, 21mpteq12dv 4247 . . . . . . . 8  |-  ( f  =  F  ->  (
x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' f " {
x } ) ) )  .x.  x ) )  =  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )
2322oveq2d 6056 . . . . . . 7  |-  ( 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 ) ) ) )
2423eleq1d 2470 . . . . . 6  |-  ( f  =  F  ->  (
( W  gsumg  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' f
" { x }
) ) )  .x.  x ) ) )  e.  _V  <->  ( W  gsumg  ( x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )  e.  _V )
)
2524elrab 3052 . . . . 5  |-  ( F  e.  { 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 ) ) )  e.  _V }  <->  ( 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
) ) )  e. 
_V ) )
2614, 25syl6bb 253 . . . 4  |-  ( ph  ->  ( F  e.  dom  ( 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
) ) )  e. 
_V ) ) )
27 ovex 6065 . . . . 5  |-  ( W 
gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )  e.  _V
2827biantru 492 . . . 4  |-  ( 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.  {
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
) ) )  e. 
_V ) )
2926, 28syl6bbr 255 . . 3  |-  ( ph  ->  ( F  e.  dom  ( 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 )
) } ) )
30 rneq 5054 . . . . . 6  |-  ( g  =  F  ->  ran  g  =  ran  F )
3130eleq1d 2470 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
3230difeq1d 3424 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
33 cnveq 5005 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3433imaeq1d 5161 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3534fveq2d 5691 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3635eleq1d 2470 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) )
3732, 36raleqbidv 2876 . . . . 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 ) ) )
3831, 37anbi12d 692 . . . 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 ) ) ) )
3938elrab 3052 . . 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 ) ) ) )
4029, 39syl6bb 253 . 2  |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S
)  /\  ( ran  F  e.  Fin  /\  A. x  e.  ( ran  F 
\  {  .0.  }
) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) ) ) )
41 3anass 940 . 2  |-  ( ( F  e.  ( dom 
MMblFnM S )  /\  ran  F  e.  Fin  /\  A. x  e.  ( ran  F 
\  {  .0.  }
) ( M `  ( `' F " { 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 ) ) ) )
4240, 41syl6bbr 255 1  |-  ( ph  ->  ( F  e.  dom  ( Wsitg M )  <->  ( F  e.  ( dom  MMblFnM S
)  /\  ran  F  e. 
Fin  /\  A. x  e.  ( ran  F  \  {  .0.  } ) ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277   {csn 3774   U.cuni 3975    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946    +oocpnf 9073   [,)cico 10874   Basecbs 13424  Scalarcsca 13487   .scvsca 13488   TopOpenctopn 13604   0gc0g 13678    gsumg cgsu 13679  RRHomcrrh 24330  sigaGencsigagen 24474  measurescmeas 24502  MblFnMcmbfm 24553  sitgcsitg 24597
This theorem is referenced by:  sibf0  24602  sibfmbl  24603  sibfrn  24605  sibfima  24606  sibfof  24607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-sitg 24598
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