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Theorem sitgf 24613
Description: The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-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 )
sitgf.1  |-  ( (
ph  /\  f  e.  dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f
)  e.  B )
Assertion
Ref Expression
sitgf  |-  ( ph  ->  ( Wsitg M ) : dom  ( Wsitg M ) --> B )
Distinct variable groups:    B, f    f, H    f, M    S, f    f, W    .0. , f    .x. , f    ph, f
Allowed substitution hints:    J( f)    V( f)

Proof of Theorem sitgf
Dummy variables  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 5448 . . . . 5  |-  Fun  (
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 ) ) ) )
2 sitgval.b . . . . . . 7  |-  B  =  ( Base `  W
)
3 sitgval.j . . . . . . 7  |-  J  =  ( TopOpen `  W )
4 sitgval.s . . . . . . 7  |-  S  =  (sigaGen `  J )
5 sitgval.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
6 sitgval.x . . . . . . 7  |-  .x.  =  ( .s `  W )
7 sitgval.h . . . . . . 7  |-  H  =  (RRHom `  (Scalar `  W
) )
8 sitgval.1 . . . . . . 7  |-  ( ph  ->  W  e.  V )
9 sitgval.2 . . . . . . 7  |-  ( ph  ->  M  e.  U. ran measures )
102, 3, 4, 5, 6, 7, 8, 9sitgval 24600 . . . . . 6  |-  ( 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 ) ) ) ) )
1110funeqd 5434 . . . . 5  |-  ( ph  ->  ( Fun  ( Wsitg M )  <->  Fun  ( 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 ) ) ) ) ) )
121, 11mpbiri 225 . . . 4  |-  ( ph  ->  Fun  ( Wsitg M
) )
13 funfn 5441 . . . 4  |-  ( Fun  ( Wsitg M )  <-> 
( Wsitg M )  Fn  dom  ( Wsitg M ) )
1412, 13sylib 189 . . 3  |-  ( ph  ->  ( Wsitg M )  Fn  dom  ( Wsitg M ) )
15 sitgf.1 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f
)  e.  B )
1615ralrimiva 2749 . . . . . 6  |-  ( ph  ->  A. f  e.  dom  ( Wsitg M ) ( ( Wsitg M ) `
 f )  e.  B )
1716r19.21bi 2764 . . . . 5  |-  ( (
ph  /\  f  e.  dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f
)  e.  B )
1817ralrimiva 2749 . . . 4  |-  ( ph  ->  A. f  e.  dom  ( Wsitg M ) ( ( Wsitg M ) `
 f )  e.  B )
19 fnfvrnss 5855 . . . 4  |-  ( ( ( Wsitg M )  Fn  dom  ( Wsitg M )  /\  A. f  e.  dom  ( Wsitg M ) ( ( Wsitg M ) `  f )  e.  B
)  ->  ran  ( Wsitg M )  C_  B
)
2014, 18, 19syl2anc 643 . . 3  |-  ( ph  ->  ran  ( Wsitg M
)  C_  B )
2114, 20jca 519 . 2  |-  ( ph  ->  ( ( Wsitg M
)  Fn  dom  ( Wsitg M )  /\  ran  ( Wsitg M )  C_  B ) )
22 df-f 5417 . 2  |-  ( ( Wsitg M ) : dom  ( Wsitg M
) --> B  <->  ( ( Wsitg M )  Fn  dom  ( Wsitg M )  /\  ran  ( Wsitg M ) 
C_  B ) )
2321, 22sylibr 204 1  |-  ( ph  ->  ( Wsitg M ) : dom  ( Wsitg M ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    \ cdif 3277    C_ wss 3280   {csn 3774   U.cuni 3975    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` 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 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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