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Theorem sitgval 24600
Description: Value of the simple function integral builder for a given space  W and measure  M. (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 )
Assertion
Ref Expression
sitgval  |-  ( 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 ) ) ) ) )
Distinct variable groups:    B, f    f, g, x    f, H   
f, M, g, x    S, f, g    f, W, g, x    .0. , f,
g, x    .x. , f
Allowed substitution hints:    ph( x, f, g)    B( x, g)    S( x)    .x. ( x, g)    H( x, g)    J( x, f, g)    V( x, f, g)

Proof of Theorem sitgval
Dummy variables  m  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
2 elex 2924 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
31, 2syl 16 . 2  |-  ( ph  ->  W  e.  _V )
4 sitgval.2 . 2  |-  ( ph  ->  M  e.  U. ran measures )
5 fveq2 5687 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
65fveq2d 5691 . . . . . . 7  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  (sigaGen `  ( TopOpen
`  W ) ) )
7 sitgval.s . . . . . . . 8  |-  S  =  (sigaGen `  J )
8 sitgval.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
98fveq2i 5690 . . . . . . . 8  |-  (sigaGen `  J
)  =  (sigaGen `  ( TopOpen
`  W ) )
107, 9eqtri 2424 . . . . . . 7  |-  S  =  (sigaGen `  ( TopOpen `  W
) )
116, 10syl6eqr 2454 . . . . . 6  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  S )
1211oveq2d 6056 . . . . 5  |-  ( w  =  W  ->  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  =  ( dom  mMblFnM S
) )
13 fveq2 5687 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
14 sitgval.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
1513, 14syl6eqr 2454 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1615sneqd 3787 . . . . . . . 8  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1716difeq2d 3425 . . . . . . 7  |-  ( w  =  W  ->  ( ran  g  \  { ( 0g `  w ) } )  =  ( ran  g  \  {  .0.  } ) )
1817raleqdv 2870 . . . . . 6  |-  ( w  =  W  ->  ( A. x  e.  ( ran  g  \  { ( 0g `  w ) } ) ( m `
 ( `' g
" { x }
) )  e.  ( 0 [,)  +oo )  <->  A. x  e.  ( ran  g  \  {  .0.  } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) )
1918anbi2d 685 . . . . 5  |-  ( w  =  W  ->  (
( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  { ( 0g `  w ) } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) )  <->  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  }
) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) ) )
2012, 19rabeqbidv 2911 . . . 4  |-  ( w  =  W  ->  { g  e.  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  |  ( ran  g  e. 
Fin  /\  A. x  e.  ( ran  g  \  { ( 0g `  w ) } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) }  =  { g  e.  ( dom  mMblFnM S
)  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) } )
21 id 20 . . . . 5  |-  ( w  =  W  ->  w  =  W )
2216difeq2d 3425 . . . . . 6  |-  ( w  =  W  ->  ( ran  f  \  { ( 0g `  w ) } )  =  ( ran  f  \  {  .0.  } ) )
23 fveq2 5687 . . . . . . . 8  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
24 sitgval.x . . . . . . . 8  |-  .x.  =  ( .s `  W )
2523, 24syl6eqr 2454 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
26 fveq2 5687 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
2726fveq2d 5691 . . . . . . . . 9  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  (RRHom `  (Scalar `  W ) ) )
28 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
2927, 28syl6eqr 2454 . . . . . . . 8  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  H )
3029fveq1d 5689 . . . . . . 7  |-  ( w  =  W  ->  (
(RRHom `  (Scalar `  w
) ) `  (
m `  ( `' f " { x }
) ) )  =  ( H `  (
m `  ( `' f " { x }
) ) ) )
31 eqidd 2405 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
3225, 30, 31oveq123d 6061 . . . . . 6  |-  ( w  =  W  ->  (
( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x )  =  ( ( H `
 ( m `  ( `' f " {
x } ) ) )  .x.  x ) )
3322, 32mpteq12dv 4247 . . . . 5  |-  ( w  =  W  ->  (
x  e.  ( ran  f  \  { ( 0g `  w ) } )  |->  ( ( (RRHom `  (Scalar `  w
) ) `  (
m `  ( `' f " { x }
) ) ) ( .s `  w ) x ) )  =  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `  ( m `
 ( `' f
" { x }
) ) )  .x.  x ) ) )
3421, 33oveq12d 6058 . . . 4  |-  ( w  =  W  ->  (
w  gsumg  ( x  e.  ( ran  f  \  {
( 0g `  w
) } )  |->  ( ( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x ) ) )  =  ( W  gsumg  ( x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `  ( m `
 ( `' f
" { x }
) ) )  .x.  x ) ) ) )
3520, 34mpteq12dv 4247 . . 3  |-  ( w  =  W  ->  (
f  e.  { g  e.  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  |  ( ran  g  e. 
Fin  /\  A. x  e.  ( ran  g  \  { ( 0g `  w ) } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) } 
|->  ( w  gsumg  ( x  e.  ( ran  f  \  {
( 0g `  w
) } )  |->  ( ( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) 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 ) ) ) ) )
36 dmeq 5029 . . . . . 6  |-  ( m  =  M  ->  dom  m  =  dom  M )
3736oveq1d 6055 . . . . 5  |-  ( m  =  M  ->  ( dom  mMblFnM S )  =  ( dom  MMblFnM S
) )
38 fveq1 5686 . . . . . . . 8  |-  ( m  =  M  ->  (
m `  ( `' g " { x }
) )  =  ( M `  ( `' g " { x } ) ) )
3938eleq1d 2470 . . . . . . 7  |-  ( m  =  M  ->  (
( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo )  <->  ( M `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) ) )
4039ralbidv 2686 . . . . . 6  |-  ( m  =  M  ->  ( A. x  e.  ( ran  g  \  {  .0.  } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo )  <->  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) ) )
4140anbi2d 685 . . . . 5  |-  ( m  =  M  ->  (
( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( m `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) )  <->  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  }
) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) ) )
4237, 41rabeqbidv 2911 . . . 4  |-  ( m  =  M  ->  { g  e.  ( dom  mMblFnM S )  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( m `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) }  =  { g  e.  ( dom  MMblFnM S
)  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) } )
43 simpl 444 . . . . . . . . 9  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  m  =  M )
4443fveq1d 5689 . . . . . . . 8  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
m `  ( `' f " { x }
) )  =  ( M `  ( `' f " { x } ) ) )
4544fveq2d 5691 . . . . . . 7  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  ( H `  ( m `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' f
" { x }
) ) ) )
4645oveq1d 6055 . . . . . 6  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
( H `  (
m `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' f " { x }
) ) )  .x.  x ) )
4746mpteq2dva 4255 . . . . 5  |-  ( m  =  M  ->  (
x  e.  ( ran  f  \  {  .0.  } )  |->  ( ( H `
 ( m `  ( `' f " {
x } ) ) )  .x.  x ) )  =  ( x  e.  ( ran  f  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' f " { x } ) ) ) 
.x.  x ) ) )
4847oveq2d 6056 . . . 4  |-  ( m  =  M  ->  ( 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 ) ) ) )
4942, 48mpteq12dv 4247 . . 3  |-  ( m  =  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 ) ) ) )  =  ( 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 ) ) ) ) )
50 df-sitg 24598 . . 3  |- sitg  =  ( w  e.  _V ,  m  e.  U. ran measures  |->  ( f  e.  { g  e.  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w )
) )  |  ( ran  g  e.  Fin  /\ 
A. x  e.  ( ran  g  \  {
( 0g `  w
) } ) ( m `  ( `' g " { x } ) )  e.  ( 0 [,)  +oo ) ) }  |->  ( w  gsumg  ( x  e.  ( ran  f  \  {
( 0g `  w
) } )  |->  ( ( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x ) ) ) ) )
51 ovex 6065 . . . . 5  |-  ( dom 
MMblFnM S )  e.  _V
5251rabex 4314 . . . 4  |-  { g  e.  ( dom  MMblFnM S )  |  ( ran  g  e.  Fin  /\  A. x  e.  ( ran  g  \  {  .0.  } ) ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) 
+oo ) ) }  e.  _V
5352mptex 5925 . . 3  |-  ( 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
5435, 49, 50, 53ovmpt2 6168 . 2  |-  ( ( W  e.  _V  /\  M  e.  U. ran measures )  -> 
( 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 ) ) ) ) )
553, 4, 54syl2anc 643 1  |-  ( 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = 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:  issibf  24601  sitgfval  24608  sitgf  24613
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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