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Theorem sitg0 28039
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 28027 . . 3  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
121, 2, 3, 4, 5, 6, 7, 8, 11sitgfval 28034 . 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 5439 . . . . . . 7  |-  ran  ( U. dom  M  X.  {  .0.  } )  C_  {  .0.  }
14 ssdif0 3885 . . . . . . 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 4527 . . . . . 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 5708 . . . . 5  |-  ( x  e.  (/)  |->  ( ( H `
 ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) ) ) 
.x.  x ) )  =  (/)
1917, 18eqtri 2496 . . . 4  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) ) )  .x.  x
) )  =  (/)
2019oveq2i 6296 . . 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 15835 . . 3  |-  ( W 
gsumg  (/) )  =  .0.
2220, 21eqtri 2496 . 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 2524 1  |-  ( ph  ->  ( ( Wsitg M
) `  ( U. dom  M  X.  {  .0.  } ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6285   Basecbs 14493  Scalarcsca 14561   .scvsca 14562   TopOpenctopn 14680   0gc0g 14698    gsumg cgsu 14699   Mndcmnd 15729   TopSpctps 19204  RRHomcrrh 27725  sigaGencsigagen 27889  measurescmeas 27917  sitgcsitg 28022
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-rmo 2822  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-1o 7131  df-map 7423  df-en 7518  df-fin 7521  df-seq 12077  df-0g 14700  df-gsum 14701  df-mnd 15735  df-top 19206  df-topon 19209  df-topsp 19210  df-esum 27792  df-siga 27859  df-sigagen 27890  df-meas 27918  df-mbfm 27973  df-sitg 28023
This theorem is referenced by: (None)
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