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Theorem sitgf 28553
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 5606 . . . 4  |-  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 . . . . . 6  |-  B  =  ( Base `  W
)
3 sitgval.j . . . . . 6  |-  J  =  ( TopOpen `  W )
4 sitgval.s . . . . . 6  |-  S  =  (sigaGen `  J )
5 sitgval.0 . . . . . 6  |-  .0.  =  ( 0g `  W )
6 sitgval.x . . . . . 6  |-  .x.  =  ( .s `  W )
7 sitgval.h . . . . . 6  |-  H  =  (RRHom `  (Scalar `  W
) )
8 sitgval.1 . . . . . 6  |-  ( ph  ->  W  e.  V )
9 sitgval.2 . . . . . 6  |-  ( ph  ->  M  e.  U. ran measures )
102, 3, 4, 5, 6, 7, 8, 9sitgval 28538 . . . . 5  |-  ( 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 5591 . . . 4  |-  ( 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 233 . . 3  |-  ( ph  ->  Fun  ( Wsitg M
) )
13 funfn 5599 . . 3  |-  ( Fun  ( Wsitg M )  <-> 
( Wsitg M )  Fn  dom  ( Wsitg M ) )
1412, 13sylib 196 . 2  |-  ( ph  ->  ( Wsitg M )  Fn  dom  ( Wsitg M ) )
15 sitgf.1 . . . 4  |-  ( (
ph  /\  f  e.  dom  ( Wsitg M ) )  ->  ( ( Wsitg M ) `  f
)  e.  B )
1615ralrimiva 2868 . . 3  |-  ( ph  ->  A. f  e.  dom  ( Wsitg M ) ( ( Wsitg M ) `
 f )  e.  B )
17 fnfvrnss 6035 . . 3  |-  ( ( ( Wsitg M )  Fn  dom  ( Wsitg M )  /\  A. f  e.  dom  ( Wsitg M ) ( ( Wsitg M ) `  f )  e.  B
)  ->  ran  ( Wsitg M )  C_  B
)
1814, 16, 17syl2anc 659 . 2  |-  ( ph  ->  ran  ( Wsitg M
)  C_  B )
19 df-f 5574 . 2  |-  ( ( Wsitg M ) : dom  ( Wsitg M
) --> B  <->  ( ( Wsitg M )  Fn  dom  ( Wsitg M )  /\  ran  ( Wsitg M ) 
C_  B ) )
2014, 18, 19sylanbrc 662 1  |-  ( ph  ->  ( Wsitg M ) : dom  ( Wsitg M ) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    \ cdif 3458    C_ wss 3461   {csn 4016   U.cuni 4235    |-> cmpt 4497   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Fincfn 7509   0cc0 9481   +oocpnf 9614   [,)cico 11534   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   TopOpenctopn 14911   0gc0g 14929    gsumg cgsu 14930  RRHomcrrh 28208  sigaGencsigagen 28368  measurescmeas 28403  MblFnMcmbfm 28458  sitgcsitg 28535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-sitg 28536
This theorem is referenced by: (None)
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