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Theorem sibf0 28540
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 28409 . . . 4  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
31, 2syl 16 . . 3  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
4 sitgval.s . . . 4  |-  S  =  (sigaGen `  J )
5 sitgval.j . . . . . . 7  |-  J  =  ( TopOpen `  W )
6 fvex 5858 . . . . . . 7  |-  ( TopOpen `  W )  e.  _V
75, 6eqeltri 2538 . . . . . 6  |-  J  e. 
_V
87a1i 11 . . . . 5  |-  ( ph  ->  J  e.  _V )
98sgsiga 28372 . . . 4  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
104, 9syl5eqel 2546 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
11 fconstmpt 5032 . . . 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 16137 . . . . 5  |-  ( W  e.  Mnd  ->  .0.  e.  B )
1713, 16syl 16 . . . 4  |-  ( ph  ->  .0.  e.  B )
18 sibf0.1 . . . . . 6  |-  ( ph  ->  W  e.  TopSp )
1914, 5tpsuni 19606 . . . . . 6  |-  ( W  e.  TopSp  ->  B  =  U. J )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  B  =  U. J
)
214unieqi 4244 . . . . . 6  |-  U. S  =  U. (sigaGen `  J
)
22 unisg 28373 . . . . . . 7  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
237, 22mp1i 12 . . . . . 6  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
2421, 23syl5eq 2507 . . . . 5  |-  ( ph  ->  U. S  =  U. J )
2520, 24eqtr4d 2498 . . . 4  |-  ( ph  ->  B  =  U. S
)
2617, 25eleqtrd 2544 . . 3  |-  ( ph  ->  .0.  e.  U. S
)
273, 10, 12, 26mbfmcst 28467 . 2  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S ) )
28 xpeq1 5002 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (
(/)  X.  {  .0.  } ) )
29 0xp 5069 . . . . . . . 8  |-  ( (/)  X. 
{  .0.  } )  =  (/)
3028, 29syl6eq 2511 . . . . . . 7  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (/) )
3130rneqd 5219 . . . . . 6  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  ran  (/) )
32 rn0 5243 . . . . . 6  |-  ran  (/)  =  (/)
3331, 32syl6eq 2511 . . . . 5  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  (/) )
34 0fin 7740 . . . . 5  |-  (/)  e.  Fin
3533, 34syl6eqel 2550 . . . 4  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
36 rnxp 5422 . . . . 5  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  {  .0.  } )
37 snfi 7589 . . . . 5  |-  {  .0.  }  e.  Fin
3836, 37syl6eqel 2550 . . . 4  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
3935, 38pm2.61ine 2767 . . 3  |-  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin
4039a1i 11 . 2  |-  ( ph  ->  ran  ( U. dom  M  X.  {  .0.  }
)  e.  Fin )
41 noel 3787 . . . . . 6  |-  -.  x  e.  (/)
4233difeq1d 3607 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( (/)  \  {  .0.  } ) )
43 0dif 3887 . . . . . . . . 9  |-  ( (/)  \  {  .0.  } )  =  (/)
4442, 43syl6eq 2511 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4536difeq1d 3607 . . . . . . . . 9  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( {  .0.  }  \  {  .0.  } ) )
46 difid 3884 . . . . . . . . 9  |-  ( {  .0.  }  \  {  .0.  } )  =  (/)
4745, 46syl6eq 2511 . . . . . . . 8  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4844, 47pm2.61ine 2767 . . . . . . 7  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
4948eleq2i 2532 . . . . . 6  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  <->  x  e.  (/) )
5041, 49mtbir 297 . . . . 5  |-  -.  x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )
5150pm2.21i 131 . . . 4  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5251adantl 464 . . 3  |-  ( (
ph  /\  x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5352ralrimiva 2868 . 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 28539 . 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 1177 1  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    \ cdif 3458   (/)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   Fincfn 7509   0cc0 9481   +oocpnf 9614   [,)cico 11534   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   TopOpenctopn 14911   0gc0g 14929   Mndcmnd 16118   TopSpctps 19564  RRHomcrrh 28208  sigAlgebracsiga 28337  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-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-1o 7122  df-map 7414  df-en 7510  df-fin 7513  df-0g 14931  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:  sitg0  28552
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