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Theorem sibf0 29119
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 . . . 4  |-  ( ph  ->  M  e.  U. ran measures )
2 dmmeas 28975 . . . 4  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
31, 2syl 17 . . 3  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
4 sitgval.s . . . 4  |-  S  =  (sigaGen `  J )
5 sitgval.j . . . . . . 7  |-  J  =  ( TopOpen `  W )
6 fvex 5835 . . . . . . 7  |-  ( TopOpen `  W )  e.  _V
75, 6eqeltri 2502 . . . . . 6  |-  J  e. 
_V
87a1i 11 . . . . 5  |-  ( ph  ->  J  e.  _V )
98sgsiga 28916 . . . 4  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
104, 9syl5eqel 2510 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
11 fconstmpt 4840 . . . 4  |-  ( U. dom  M  X.  {  .0.  } )  =  ( x  e.  U. dom  M  |->  .0.  )
1211a1i 11 . . 3  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  =  ( x  e. 
U. dom  M  |->  .0.  ) )
13 sibf0.2 . . . . 5  |-  ( ph  ->  W  e.  Mnd )
14 sitgval.b . . . . . 6  |-  B  =  ( Base `  W
)
15 sitgval.0 . . . . . 6  |-  .0.  =  ( 0g `  W )
1614, 15mndidcl 16497 . . . . 5  |-  ( W  e.  Mnd  ->  .0.  e.  B )
1713, 16syl 17 . . . 4  |-  ( ph  ->  .0.  e.  B )
18 sibf0.1 . . . . . 6  |-  ( ph  ->  W  e.  TopSp )
1914, 5tpsuni 19895 . . . . . 6  |-  ( W  e.  TopSp  ->  B  =  U. J )
2018, 19syl 17 . . . . 5  |-  ( ph  ->  B  =  U. J
)
214unieqi 4171 . . . . . 6  |-  U. S  =  U. (sigaGen `  J
)
22 unisg 28917 . . . . . . 7  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
237, 22mp1i 13 . . . . . 6  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
2421, 23syl5eq 2474 . . . . 5  |-  ( ph  ->  U. S  =  U. J )
2520, 24eqtr4d 2465 . . . 4  |-  ( ph  ->  B  =  U. S
)
2617, 25eleqtrd 2508 . . 3  |-  ( ph  ->  .0.  e.  U. S
)
273, 10, 12, 26mbfmcst 29033 . 2  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S ) )
28 xpeq1 4810 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (
(/)  X.  {  .0.  } ) )
29 0xp 4877 . . . . . . . 8  |-  ( (/)  X. 
{  .0.  } )  =  (/)
3028, 29syl6eq 2478 . . . . . . 7  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (/) )
3130rneqd 5024 . . . . . 6  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  ran  (/) )
32 rn0 5048 . . . . . 6  |-  ran  (/)  =  (/)
3331, 32syl6eq 2478 . . . . 5  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  (/) )
34 0fin 7752 . . . . 5  |-  (/)  e.  Fin
3533, 34syl6eqel 2514 . . . 4  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
36 rnxp 5229 . . . . 5  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  {  .0.  } )
37 snfi 7604 . . . . 5  |-  {  .0.  }  e.  Fin
3836, 37syl6eqel 2514 . . . 4  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
3935, 38pm2.61ine 2684 . . 3  |-  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin
4039a1i 11 . 2  |-  ( ph  ->  ran  ( U. dom  M  X.  {  .0.  }
)  e.  Fin )
41 noel 3708 . . . . . 6  |-  -.  x  e.  (/)
4233difeq1d 3525 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( (/)  \  {  .0.  } ) )
43 0dif 3811 . . . . . . . . 9  |-  ( (/)  \  {  .0.  } )  =  (/)
4442, 43syl6eq 2478 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4536difeq1d 3525 . . . . . . . . 9  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( {  .0.  }  \  {  .0.  } ) )
46 difid 3808 . . . . . . . . 9  |-  ( {  .0.  }  \  {  .0.  } )  =  (/)
4745, 46syl6eq 2478 . . . . . . . 8  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4844, 47pm2.61ine 2684 . . . . . . 7  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
4948eleq2i 2498 . . . . . 6  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  <->  x  e.  (/) )
5041, 49mtbir 300 . . . . 5  |-  -.  x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )
5150pm2.21i 134 . . . 4  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5251adantl 467 . . 3  |-  ( (
ph  /\  x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5352ralrimiva 2779 . 2  |-  ( ph  ->  A. x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) ( M `  ( `' ( U. dom  M  X.  {  .0.  }
) " { x } ) )  e.  ( 0 [,) +oo ) )
54 sitgval.x . . 3  |-  .x.  =  ( .s `  W )
55 sitgval.h . . 3  |-  H  =  (RRHom `  (Scalar `  W
) )
56 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
5714, 5, 4, 15, 54, 55, 56, 1issibf 29118 . 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 )
) ) )
5827, 40, 53, 57mpbir3and 1188 1  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   _Vcvv 3022    \ cdif 3376   (/)c0 3704   {csn 3941   U.cuni 4162    |-> cmpt 4425    X. cxp 4794   `'ccnv 4795   dom cdm 4796   ran crn 4797   "cima 4799   ` cfv 5544  (class class class)co 6249   Fincfn 7524   0cc0 9490   +oocpnf 9623   [,)cico 11588   Basecbs 15064  Scalarcsca 15136   .scvsca 15137   TopOpenctopn 15263   0gc0g 15281   Mndcmnd 16478   TopSpctps 19861  RRHomcrrh 28749  sigAlgebracsiga 28881  sigaGencsigagen 28912  measurescmeas 28969  MblFnMcmbfm 29024  sitgcsitg 29114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1o 7137  df-map 7429  df-en 7525  df-fin 7528  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-top 19863  df-topon 19865  df-topsp 19866  df-esum 28801  df-siga 28882  df-sigagen 28913  df-meas 28970  df-mbfm 29025  df-sitg 29115
This theorem is referenced by:  sitg0  29131
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