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Theorem sitgval 27914
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 3122 . . 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 5864 . . . . . . . 8  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
65fveq2d 5868 . . . . . . 7  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  (sigaGen `  ( TopOpen
`  W ) ) )
7 sitgval.s . . . . . . . 8  |-  S  =  (sigaGen `  J )
8 sitgval.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
98fveq2i 5867 . . . . . . . 8  |-  (sigaGen `  J
)  =  (sigaGen `  ( TopOpen
`  W ) )
107, 9eqtri 2496 . . . . . . 7  |-  S  =  (sigaGen `  ( TopOpen `  W
) )
116, 10syl6eqr 2526 . . . . . 6  |-  ( w  =  W  ->  (sigaGen `  ( TopOpen `  w )
)  =  S )
1211oveq2d 6298 . . . . 5  |-  ( w  =  W  ->  ( dom  mMblFnM (sigaGen `  ( TopOpen `  w
) ) )  =  ( dom  mMblFnM S
) )
13 fveq2 5864 . . . . . . . . . 10  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
14 sitgval.0 . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
1513, 14syl6eqr 2526 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1615sneqd 4039 . . . . . . . 8  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1716difeq2d 3622 . . . . . . 7  |-  ( w  =  W  ->  ( ran  g  \  { ( 0g `  w ) } )  =  ( ran  g  \  {  .0.  } ) )
1817raleqdv 3064 . . . . . 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 703 . . . . 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 3108 . . . 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 3622 . . . . . 6  |-  ( w  =  W  ->  ( ran  f  \  { ( 0g `  w ) } )  =  ( ran  f  \  {  .0.  } ) )
23 fveq2 5864 . . . . . . . 8  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
24 sitgval.x . . . . . . . 8  |-  .x.  =  ( .s `  W )
2523, 24syl6eqr 2526 . . . . . . 7  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
26 fveq2 5864 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
2726fveq2d 5868 . . . . . . . . 9  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  (RRHom `  (Scalar `  W ) ) )
28 sitgval.h . . . . . . . . 9  |-  H  =  (RRHom `  (Scalar `  W
) )
2927, 28syl6eqr 2526 . . . . . . . 8  |-  ( w  =  W  ->  (RRHom `  (Scalar `  w )
)  =  H )
3029fveq1d 5866 . . . . . . 7  |-  ( w  =  W  ->  (
(RRHom `  (Scalar `  w
) ) `  (
m `  ( `' f " { x }
) ) )  =  ( H `  (
m `  ( `' f " { x }
) ) ) )
31 eqidd 2468 . . . . . . 7  |-  ( w  =  W  ->  x  =  x )
3225, 30, 31oveq123d 6303 . . . . . 6  |-  ( w  =  W  ->  (
( (RRHom `  (Scalar `  w ) ) `  ( m `  ( `' f " {
x } ) ) ) ( .s `  w ) x )  =  ( ( H `
 ( m `  ( `' f " {
x } ) ) )  .x.  x ) )
3322, 32mpteq12dv 4525 . . . . 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 6300 . . . 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 4525 . . 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 5201 . . . . . 6  |-  ( m  =  M  ->  dom  m  =  dom  M )
3736oveq1d 6297 . . . . 5  |-  ( m  =  M  ->  ( dom  mMblFnM S )  =  ( dom  MMblFnM S
) )
38 fveq1 5863 . . . . . . . 8  |-  ( m  =  M  ->  (
m `  ( `' g " { x }
) )  =  ( M `  ( `' g " { x } ) ) )
3938eleq1d 2536 . . . . . . 7  |-  ( m  =  M  ->  (
( m `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo )  <->  ( M `  ( `' g " {
x } ) )  e.  ( 0 [,) +oo ) ) )
4039ralbidv 2903 . . . . . 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 703 . . . . 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 3108 . . . 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 457 . . . . . . . . 9  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  m  =  M )
4443fveq1d 5866 . . . . . . . 8  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
m `  ( `' f " { x }
) )  =  ( M `  ( `' f " { x } ) ) )
4544fveq2d 5868 . . . . . . 7  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  ( H `  ( m `  ( `' f " { x } ) ) )  =  ( H `  ( M `
 ( `' f
" { x }
) ) ) )
4645oveq1d 6297 . . . . . 6  |-  ( ( m  =  M  /\  x  e.  ( ran  f  \  {  .0.  }
) )  ->  (
( H `  (
m `  ( `' f " { x }
) ) )  .x.  x )  =  ( ( H `  ( M `  ( `' f " { x }
) ) )  .x.  x ) )
4746mpteq2dva 4533 . . . . 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 6298 . . . 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 4525 . . 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 27912 . . 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 6307 . . . . 5  |-  ( dom 
MMblFnM S )  e.  _V
5251rabex 4598 . . . 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 6129 . . 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 6420 . 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 661 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 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473   {csn 4027   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   ` cfv 5586  (class class class)co 6282   Fincfn 7513   0cc0 9488   +oocpnf 9621   [,)cico 11527   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   TopOpenctopn 14673   0gc0g 14691    gsumg cgsu 14692  RRHomcrrh 27610  sigaGencsigagen 27778  measurescmeas 27806  MblFnMcmbfm 27861  sitgcsitg 27911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-sitg 27912
This theorem is referenced by:  issibf  27915  sitgfval  27923  sitgf  27929
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