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Theorem issibf 28450
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 28449 . . . . . . . 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 5215 . . . . . . 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 2457 . . . . . . . 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 5508 . . . . . . 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 2514 . . . . . 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 2527 . . . . 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 5238 . . . . . . . . . 10  |-  ( f  =  F  ->  ran  f  =  ran  F )
1615difeq1d 3617 . . . . . . . . 9  |-  ( f  =  F  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
17 cnveq 5186 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  `' f  =  `' F
)
1817imaeq1d 5346 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1918fveq2d 5876 . . . . . . . . . . 11  |-  ( f  =  F  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
2019fveq2d 5876 . . . . . . . . . 10  |-  ( f  =  F  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
2120oveq1d 6311 . . . . . . . . 9  |-  ( f  =  F  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
2216, 21mpteq12dv 4535 . . . . . . . 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 6312 . . . . . . 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 2526 . . . . . 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 3257 . . . . 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 261 . . . 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 6324 . . . . 5  |-  ( W 
gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )  e.  _V
2827biantru 505 . . . 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 263 . . 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 5238 . . . . . 6  |-  ( g  =  F  ->  ran  g  =  ran  F )
3130eleq1d 2526 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
3230difeq1d 3617 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
33 cnveq 5186 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3433imaeq1d 5346 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3534fveq2d 5876 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3635eleq1d 2526 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3732, 36raleqbidv 3068 . . . . 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 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 ) ) ) )
3938elrab 3257 . . 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 261 . 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 977 . 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 263 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   U.cuni 4251    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   Fincfn 7535   0cc0 9509   +oocpnf 9642   [,)cico 11556   Basecbs 14643  Scalarcsca 14714   .scvsca 14715   TopOpenctopn 14838   0gc0g 14856    gsumg cgsu 14857  RRHomcrrh 28127  sigaGencsigagen 28299  measurescmeas 28327  MblFnMcmbfm 28382  sitgcsitg 28446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-sitg 28447
This theorem is referenced by:  sibf0  28451  sibfmbl  28452  sibfrn  28454  sibfima  28455  sibfof  28457
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