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Theorem sibf0 27766
Description: The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-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 )
sibf0.1  |-  ( ph  ->  W  e.  TopSp )
sibf0.2  |-  ( ph  ->  W  e.  Mnd )
Assertion
Ref Expression
sibf0  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )

Proof of Theorem sibf0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sitgval.2 . . . . 5  |-  ( ph  ->  M  e.  U. ran measures )
2 dmmeas 27662 . . . . 5  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
31, 2syl 16 . . . 4  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
4 sitgval.s . . . . 5  |-  S  =  (sigaGen `  J )
5 sitgval.j . . . . . . . 8  |-  J  =  ( TopOpen `  W )
6 fvex 5867 . . . . . . . 8  |-  ( TopOpen `  W )  e.  _V
75, 6eqeltri 2544 . . . . . . 7  |-  J  e. 
_V
87a1i 11 . . . . . 6  |-  ( ph  ->  J  e.  _V )
98sgsiga 27632 . . . . 5  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
104, 9syl5eqel 2552 . . . 4  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
11 fconstmpt 5035 . . . . 5  |-  ( U. dom  M  X.  {  .0.  } )  =  ( x  e.  U. dom  M  |->  .0.  )
1211a1i 11 . . . 4  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  =  ( x  e. 
U. dom  M  |->  .0.  ) )
13 sibf0.2 . . . . . 6  |-  ( ph  ->  W  e.  Mnd )
14 sitgval.b . . . . . . 7  |-  B  =  ( Base `  W
)
15 sitgval.0 . . . . . . 7  |-  .0.  =  ( 0g `  W )
1614, 15mndidcl 15745 . . . . . 6  |-  ( W  e.  Mnd  ->  .0.  e.  B )
1713, 16syl 16 . . . . 5  |-  ( ph  ->  .0.  e.  B )
18 sibf0.1 . . . . . . 7  |-  ( ph  ->  W  e.  TopSp )
1914, 5tpsuni 19199 . . . . . . 7  |-  ( W  e.  TopSp  ->  B  =  U. J )
2018, 19syl 16 . . . . . 6  |-  ( ph  ->  B  =  U. J
)
214unieqi 4247 . . . . . . 7  |-  U. S  =  U. (sigaGen `  J
)
22 unisg 27633 . . . . . . . 8  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
237, 22mp1i 12 . . . . . . 7  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
2421, 23syl5eq 2513 . . . . . 6  |-  ( ph  ->  U. S  =  U. J )
2520, 24eqtr4d 2504 . . . . 5  |-  ( ph  ->  B  =  U. S
)
2617, 25eleqtrd 2550 . . . 4  |-  ( ph  ->  .0.  e.  U. S
)
273, 10, 12, 26mbfmcst 27720 . . 3  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S ) )
28 xpeq1 5006 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (
(/)  X.  {  .0.  } ) )
29 0xp 5071 . . . . . . . . 9  |-  ( (/)  X. 
{  .0.  } )  =  (/)
3028, 29syl6eq 2517 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (/) )
3130rneqd 5221 . . . . . . 7  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  ran  (/) )
32 rn0 5245 . . . . . . 7  |-  ran  (/)  =  (/)
3331, 32syl6eq 2517 . . . . . 6  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  (/) )
34 0fin 7737 . . . . . 6  |-  (/)  e.  Fin
3533, 34syl6eqel 2556 . . . . 5  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
36 rnxp 5428 . . . . . 6  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  {  .0.  } )
37 snfi 7586 . . . . . 6  |-  {  .0.  }  e.  Fin
3836, 37syl6eqel 2556 . . . . 5  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
3935, 38pm2.61ine 2773 . . . 4  |-  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin
4039a1i 11 . . 3  |-  ( ph  ->  ran  ( U. dom  M  X.  {  .0.  }
)  e.  Fin )
41 noel 3782 . . . . . . 7  |-  -.  x  e.  (/)
4233difeq1d 3614 . . . . . . . . . 10  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( (/)  \  {  .0.  } ) )
43 0dif 3891 . . . . . . . . . 10  |-  ( (/)  \  {  .0.  } )  =  (/)
4442, 43syl6eq 2517 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4536difeq1d 3614 . . . . . . . . . 10  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( {  .0.  }  \  {  .0.  } ) )
46 difid 3888 . . . . . . . . . 10  |-  ( {  .0.  }  \  {  .0.  } )  =  (/)
4745, 46syl6eq 2517 . . . . . . . . 9  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4844, 47pm2.61ine 2773 . . . . . . . 8  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
4948eleq2i 2538 . . . . . . 7  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  <->  x  e.  (/) )
5041, 49mtbir 299 . . . . . 6  |-  -.  x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )
5150pm2.21i 131 . . . . 5  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5251adantl 466 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5352ralrimiva 2871 . . 3  |-  ( ph  ->  A. x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) )  e.  ( 0 [,) +oo ) )
5427, 40, 533jca 1171 . 2  |-  ( ph  ->  ( ( U. dom  M  X.  {  .0.  }
)  e.  ( dom 
MMblFnM S )  /\  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin  /\  A. x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } ) ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) )  e.  ( 0 [,) +oo ) ) )
55 sitgval.x . . 3  |-  .x.  =  ( .s `  W )
56 sitgval.h . . 3  |-  H  =  (RRHom `  (Scalar `  W
) )
57 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
5814, 5, 4, 15, 55, 56, 57, 1issibf 27765 . 2  |-  ( ph  ->  ( ( U. dom  M  X.  {  .0.  }
)  e.  dom  ( Wsitg M )  <->  ( ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S )  /\  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin  /\  A. x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } ) ( M `  ( `' ( U. dom  M  X.  {  .0.  } )
" { x }
) )  e.  ( 0 [,) +oo )
) ) )
5954, 58mpbird 232 1  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   _Vcvv 3106    \ cdif 3466   (/)c0 3778   {csn 4020   U.cuni 4238    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   dom cdm 4992   ran crn 4993   "cima 4995   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481   +oocpnf 9614   [,)cico 11520   Basecbs 14479  Scalarcsca 14547   .scvsca 14548   TopOpenctopn 14666   0gc0g 14684   Mndcmnd 15715   TopSpctps 19157  RRHomcrrh 27460  sigAlgebracsiga 27597  sigaGencsigagen 27628  measurescmeas 27656  MblFnMcmbfm 27711  sitgcsitg 27761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1o 7120  df-map 7412  df-en 7507  df-fin 7510  df-0g 14686  df-mnd 15721  df-top 19159  df-topon 19162  df-topsp 19163  df-esum 27531  df-siga 27598  df-sigagen 27629  df-meas 27657  df-mbfm 27712  df-sitg 27762
This theorem is referenced by:  sitg0  27778
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