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Theorem sitgval 28661
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 3065 . . 3  |-  ( W  e.  V  ->  W  e.  _V )
31, 2syl 17 . 2  |-  ( ph  ->  W  e.  _V )
4 sitgval.2 . 2  |-  ( ph  ->  M  e.  U. ran measures )
5 fveq2 5803 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
65fveq2d 5807 . . . . . . 7  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  (sigaGen `  ( TopOpen
`  W ) ) )
7 sitgval.s . . . . . . . 8  |-  S  =  (sigaGen `  J )
8 sitgval.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
98fveq2i 5806 . . . . . . . 8  |-  (sigaGen `  J
)  =  (sigaGen `  ( TopOpen
`  W ) )
107, 9eqtri 2429 . . . . . . 7  |-  S  =  (sigaGen `  ( TopOpen `  W
) )
116, 10syl6eqr 2459 . . . . . 6  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  S )
1211oveq2d 6248 . . . . 5  |-  ( w  =  W  ->  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  =  ( dom  mMblFnM S
) )
13 fveq2 5803 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
14 sitgval.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
1513, 14syl6eqr 2459 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1615sneqd 3981 . . . . . . . 8  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1716difeq2d 3558 . . . . . . 7  |-  ( w  =  W  ->  ( ran  g  \  { ( 0g `  w ) } )  =  ( ran  g  \  {  .0.  } ) )
1817raleqdv 3007 . . . . . 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 702 . . . . 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 3051 . . . 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 3558 . . . . . 6  |-  ( w  =  W  ->  ( ran  f  \  { ( 0g `  w ) } )  =  ( ran  f  \  {  .0.  } ) )
23 fveq2 5803 . . . . . . . 8  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
24 sitgval.x . . . . . . . 8  |-  .x.  =  ( .s `  W )
2523, 24syl6eqr 2459 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
26 fveq2 5803 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
2726fveq2d 5807 . . . . . . . . 9  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  (RRHom `  (Scalar `  W ) ) )
28 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
2927, 28syl6eqr 2459 . . . . . . . 8  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  H )
3029fveq1d 5805 . . . . . . 7  |-  ( w  =  W  ->  (
(RRHom `  (Scalar `  w
) ) `  (
m `  ( `' f " { x }
) ) )  =  ( H `  (
m `  ( `' f " { x }
) ) ) )
31 eqidd 2401 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
3225, 30, 31oveq123d 6253 . . . . . 6  |-  ( w  =  W  ->  (
( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x )  =  ( ( H `
 ( m `  ( `' f " {
x } ) ) )  .x.  x ) )
3322, 32mpteq12dv 4470 . . . . 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 6250 . . . 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 4470 . . 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 5143 . . . . . 6  |-  ( m  =  M  ->  dom  m  =  dom  M )
3736oveq1d 6247 . . . . 5  |-  ( m  =  M  ->  ( dom  mMblFnM S )  =  ( dom  MMblFnM S
) )
38 fveq1 5802 . . . . . . . 8  |-  ( m  =  M  ->  (
m `  ( `' g " { x }
) )  =  ( M `  ( `' g " { x } ) ) )
3938eleq1d 2469 . . . . . . 7  |-  ( m  =  M  ->  (
( m `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo ) ) )
4039ralbidv 2840 . . . . . 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 702 . . . . 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 3051 . . . 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 455 . . . . . . . . 9  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  m  =  M )
4443fveq1d 5805 . . . . . . . 8  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
m `  ( `' f " { x }
) )  =  ( M `  ( `' f " { x } ) ) )
4544fveq2d 5807 . . . . . . 7  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  ( H `  ( m `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' f
" { x }
) ) ) )
4645oveq1d 6247 . . . . . 6  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
( H `  (
m `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' f " { x }
) ) )  .x.  x ) )
4746mpteq2dva 4478 . . . . 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 6248 . . . 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 4470 . . 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 28659 . . 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 6260 . . . 4  |-  ( dom 
MMblFnM S )  e.  _V
5251mptrabex 6079 . . 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
5335, 49, 50, 52ovmpt2 6373 . 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 ) ) ) ) )
543, 4, 53syl2anc 659 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 367    = wceq 1403    e. wcel 1840   A.wral 2751   {crab 2755   _Vcvv 3056    \ cdif 3408   {csn 3969   U.cuni 4188    |-> cmpt 4450   `'ccnv 4939   dom cdm 4940   ran crn 4941   "cima 4943   ` cfv 5523  (class class class)co 6232   Fincfn 7472   0cc0 9440   +oocpnf 9573   [,)cico 11500   Basecbs 14731  Scalarcsca 14802   .scvsca 14803   TopOpenctopn 14926   0gc0g 14944    gsumg cgsu 14945  RRHomcrrh 28307  sigaGencsigagen 28467  measurescmeas 28524  MblFnMcmbfm 28579  sitgcsitg 28658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-sitg 28659
This theorem is referenced by:  issibf  28662  sitgfval  28670  sitgf  28676
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