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Theorem issibf 26567
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 26566 . . . . . . . 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 5029 . . . . . . 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 2433 . . . . . . . 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 5321 . . . . . . 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 2481 . . . . . 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 2500 . . . . 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 5052 . . . . . . . . . 10  |-  ( f  =  F  ->  ran  f  =  ran  F )
1615difeq1d 3461 . . . . . . . . 9  |-  ( f  =  F  ->  ( ran  f  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
17 cnveq 5000 . . . . . . . . . . . . 13  |-  ( f  =  F  ->  `' f  =  `' F
)
1817imaeq1d 5156 . . . . . . . . . . . 12  |-  ( f  =  F  ->  ( `' f " {
x } )  =  ( `' F " { x } ) )
1918fveq2d 5683 . . . . . . . . . . 11  |-  ( f  =  F  ->  ( M `  ( `' f " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
2019fveq2d 5683 . . . . . . . . . 10  |-  ( f  =  F  ->  ( H `  ( M `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' F " { x } ) ) ) )
2120oveq1d 6095 . . . . . . . . 9  |-  ( f  =  F  ->  (
( H `  ( M `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
2216, 21mpteq12dv 4358 . . . . . . . 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 6096 . . . . . . 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 2499 . . . . . 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 3106 . . . . 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 6105 . . . . 5  |-  ( W 
gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) )  e.  _V
2827biantru 502 . . . 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 5052 . . . . . 6  |-  ( g  =  F  ->  ran  g  =  ran  F )
3130eleq1d 2499 . . . . 5  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
3230difeq1d 3461 . . . . . 6  |-  ( g  =  F  ->  ( ran  g  \  {  .0.  } )  =  ( ran 
F  \  {  .0.  } ) )
33 cnveq 5000 . . . . . . . . 9  |-  ( g  =  F  ->  `' g  =  `' F
)
3433imaeq1d 5156 . . . . . . . 8  |-  ( g  =  F  ->  ( `' g " {
x } )  =  ( `' F " { x } ) )
3534fveq2d 5683 . . . . . . 7  |-  ( g  =  F  ->  ( M `  ( `' g " { x }
) )  =  ( M `  ( `' F " { x } ) ) )
3635eleq1d 2499 . . . . . 6  |-  ( g  =  F  ->  (
( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) ) )
3732, 36raleqbidv 2921 . . . . 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 703 . . . 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 3106 . . 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 962 . 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   {crab 2709   _Vcvv 2962    \ cdif 3313   {csn 3865   U.cuni 4079    e. cmpt 4338   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830   ` cfv 5406  (class class class)co 6080   Fincfn 7298   0cc0 9270   +oocpnf 9403   [,)cico 11290   Basecbs 14157  Scalarcsca 14224   .scvsca 14225   TopOpenctopn 14343   0gc0g 14361    gsumg cgsu 14362  RRHomcrrh 26276  sigaGencsigagen 26435  measurescmeas 26463  MblFnMcmbfm 26519  sitgcsitg 26563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-sitg 26564
This theorem is referenced by:  sibf0  26568  sibfmbl  26569  sibfrn  26571  sibfima  26572  sibfof  26574
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