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Theorem sitg0 28552
Description: The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-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 )
sitg0.1  |-  ( ph  ->  W  e.  TopSp )
sitg0.2  |-  ( ph  ->  W  e.  Mnd )
Assertion
Ref Expression
sitg0  |-  ( ph  ->  ( ( Wsitg M
) `  ( U. dom  M  X.  {  .0.  } ) )  =  .0.  )

Proof of Theorem sitg0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . 3  |-  B  =  ( Base `  W
)
2 sitgval.j . . 3  |-  J  =  ( TopOpen `  W )
3 sitgval.s . . 3  |-  S  =  (sigaGen `  J )
4 sitgval.0 . . 3  |-  .0.  =  ( 0g `  W )
5 sitgval.x . . 3  |-  .x.  =  ( .s `  W )
6 sitgval.h . . 3  |-  H  =  (RRHom `  (Scalar `  W
) )
7 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
8 sitgval.2 . . 3  |-  ( ph  ->  M  e.  U. ran measures )
9 sitg0.1 . . . 4  |-  ( ph  ->  W  e.  TopSp )
10 sitg0.2 . . . 4  |-  ( ph  ->  W  e.  Mnd )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10sibf0 28540 . . 3  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
121, 2, 3, 4, 5, 6, 7, 8, 11sitgfval 28547 . 2  |-  ( ph  ->  ( ( Wsitg M
) `  ( U. dom  M  X.  {  .0.  } ) )  =  ( W  gsumg  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) ) ) )
13 rnxpss 5424 . . . . . . 7  |-  ran  ( U. dom  M  X.  {  .0.  } )  C_  {  .0.  }
14 ssdif0 3873 . . . . . . 7  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
C_  {  .0.  }  <->  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  =  (/) )
1513, 14mpbi 208 . . . . . 6  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
16 mpteq1 4519 . . . . . 6  |-  ( ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } )  =  (/)  ->  (
x  e.  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) )  =  ( x  e.  (/)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) ) )
1715, 16ax-mp 5 . . . . 5  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) ) )  .x.  x
) )  =  ( x  e.  (/)  |->  ( ( H `  ( M `
 ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) ) )  .x.  x
) )
18 mpt0 5690 . . . . 5  |-  ( x  e.  (/)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) )  =  (/)
1917, 18eqtri 2483 . . . 4  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) ) )  .x.  x
) )  =  (/)
2019oveq2i 6281 . . 3  |-  ( W 
gsumg  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) ) )  =  ( W 
gsumg  (/) )
214gsum0 16104 . . 3  |-  ( W 
gsumg  (/) )  =  .0.
2220, 21eqtri 2483 . 2  |-  ( W 
gsumg  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) ) )  =  .0.
2312, 22syl6eq 2511 1  |-  ( ph  ->  ( ( Wsitg M
) `  ( U. dom  M  X.  {  .0.  } ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   TopOpenctopn 14911   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118   TopSpctps 19564  RRHomcrrh 28208  sigaGencsigagen 28368  measurescmeas 28403  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-8 1825  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-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  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-rmo 2812  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-map 7414  df-en 7510  df-fin 7513  df-seq 12090  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-top 19566  df-topon 19569  df-topsp 19570  df-esum 28257  df-siga 28338  df-sigagen 28369  df-meas 28404  df-mbfm 28459  df-sitg 28536
This theorem is referenced by: (None)
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