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Theorem sitgval 26648
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 2979 . . 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 5688 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
65fveq2d 5692 . . . . . . 7  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  (sigaGen `  ( TopOpen
`  W ) ) )
7 sitgval.s . . . . . . . 8  |-  S  =  (sigaGen `  J )
8 sitgval.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
98fveq2i 5691 . . . . . . . 8  |-  (sigaGen `  J
)  =  (sigaGen `  ( TopOpen
`  W ) )
107, 9eqtri 2461 . . . . . . 7  |-  S  =  (sigaGen `  ( TopOpen `  W
) )
116, 10syl6eqr 2491 . . . . . 6  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  S )
1211oveq2d 6106 . . . . 5  |-  ( w  =  W  ->  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  =  ( dom  mMblFnM S
) )
13 fveq2 5688 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
14 sitgval.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
1513, 14syl6eqr 2491 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1615sneqd 3886 . . . . . . . 8  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1716difeq2d 3471 . . . . . . 7  |-  ( w  =  W  ->  ( ran  g  \  { ( 0g `  w ) } )  =  ( ran  g  \  {  .0.  } ) )
1817raleqdv 2921 . . . . . 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 698 . . . . 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 2965 . . . 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 22 . . . . 5  |-  ( w  =  W  ->  w  =  W )
2216difeq2d 3471 . . . . . 6  |-  ( w  =  W  ->  ( ran  f  \  { ( 0g `  w ) } )  =  ( ran  f  \  {  .0.  } ) )
23 fveq2 5688 . . . . . . . 8  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
24 sitgval.x . . . . . . . 8  |-  .x.  =  ( .s `  W )
2523, 24syl6eqr 2491 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
26 fveq2 5688 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
2726fveq2d 5692 . . . . . . . . 9  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  (RRHom `  (Scalar `  W ) ) )
28 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
2927, 28syl6eqr 2491 . . . . . . . 8  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  H )
3029fveq1d 5690 . . . . . . 7  |-  ( w  =  W  ->  (
(RRHom `  (Scalar `  w
) ) `  (
m `  ( `' f " { x }
) ) )  =  ( H `  (
m `  ( `' f " { x }
) ) ) )
31 eqidd 2442 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
3225, 30, 31oveq123d 6111 . . . . . 6  |-  ( w  =  W  ->  (
( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x )  =  ( ( H `
 ( m `  ( `' f " {
x } ) ) )  .x.  x ) )
3322, 32mpteq12dv 4367 . . . . 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 6108 . . . 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 4367 . . 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 5036 . . . . . 6  |-  ( m  =  M  ->  dom  m  =  dom  M )
3736oveq1d 6105 . . . . 5  |-  ( m  =  M  ->  ( dom  mMblFnM S )  =  ( dom  MMblFnM S
) )
38 fveq1 5687 . . . . . . . 8  |-  ( m  =  M  ->  (
m `  ( `' g " { x }
) )  =  ( M `  ( `' g " { x } ) ) )
3938eleq1d 2507 . . . . . . 7  |-  ( m  =  M  ->  (
( m `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo ) ) )
4039ralbidv 2733 . . . . . 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 698 . . . . 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 2965 . . . 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 454 . . . . . . . . 9  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  m  =  M )
4443fveq1d 5690 . . . . . . . 8  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
m `  ( `' f " { x }
) )  =  ( M `  ( `' f " { x } ) ) )
4544fveq2d 5692 . . . . . . 7  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  ( H `  ( m `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' f
" { x }
) ) ) )
4645oveq1d 6105 . . . . . 6  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
( H `  (
m `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' f " { x }
) ) )  .x.  x ) )
4746mpteq2dva 4375 . . . . 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 6106 . . . 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 4367 . . 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 26646 . . 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 6115 . . . . 5  |-  ( dom 
MMblFnM S )  e.  _V
5251rabex 4440 . . . 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 5945 . . 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 6225 . 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 656 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717   _Vcvv 2970    \ cdif 3322   {csn 3874   U.cuni 4088    e. cmpt 4347   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839   ` cfv 5415  (class class class)co 6090   Fincfn 7306   0cc0 9278   +oocpnf 9411   [,)cico 11298   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   TopOpenctopn 14356   0gc0g 14374    gsumg cgsu 14375  RRHomcrrh 26358  sigaGencsigagen 26517  measurescmeas 26545  MblFnMcmbfm 26601  sitgcsitg 26645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-sitg 26646
This theorem is referenced by:  issibf  26649  sitgfval  26657  sitgf  26663
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