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Theorem sibf0 28149
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 28045 . . . 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 5866 . . . . . . 7  |-  ( TopOpen `  W )  e.  _V
75, 6eqeltri 2527 . . . . . 6  |-  J  e. 
_V
87a1i 11 . . . . 5  |-  ( ph  ->  J  e.  _V )
98sgsiga 28015 . . . 4  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
104, 9syl5eqel 2535 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
11 fconstmpt 5033 . . . 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 15812 . . . . 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 19312 . . . . . 6  |-  ( W  e.  TopSp  ->  B  =  U. J )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  B  =  U. J
)
214unieqi 4243 . . . . . 6  |-  U. S  =  U. (sigaGen `  J
)
22 unisg 28016 . . . . . . 7  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
237, 22mp1i 12 . . . . . 6  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
2421, 23syl5eq 2496 . . . . 5  |-  ( ph  ->  U. S  =  U. J )
2520, 24eqtr4d 2487 . . . 4  |-  ( ph  ->  B  =  U. S
)
2617, 25eleqtrd 2533 . . 3  |-  ( ph  ->  .0.  e.  U. S
)
273, 10, 12, 26mbfmcst 28103 . 2  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S ) )
28 xpeq1 5003 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (
(/)  X.  {  .0.  } ) )
29 0xp 5070 . . . . . . . 8  |-  ( (/)  X. 
{  .0.  } )  =  (/)
3028, 29syl6eq 2500 . . . . . . 7  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (/) )
3130rneqd 5220 . . . . . 6  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  ran  (/) )
32 rn0 5244 . . . . . 6  |-  ran  (/)  =  (/)
3331, 32syl6eq 2500 . . . . 5  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  (/) )
34 0fin 7749 . . . . 5  |-  (/)  e.  Fin
3533, 34syl6eqel 2539 . . . 4  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
36 rnxp 5427 . . . . 5  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  {  .0.  } )
37 snfi 7598 . . . . 5  |-  {  .0.  }  e.  Fin
3836, 37syl6eqel 2539 . . . 4  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
3935, 38pm2.61ine 2756 . . 3  |-  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin
4039a1i 11 . 2  |-  ( ph  ->  ran  ( U. dom  M  X.  {  .0.  }
)  e.  Fin )
41 noel 3774 . . . . . 6  |-  -.  x  e.  (/)
4233difeq1d 3606 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( (/)  \  {  .0.  } ) )
43 0dif 3885 . . . . . . . . 9  |-  ( (/)  \  {  .0.  } )  =  (/)
4442, 43syl6eq 2500 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4536difeq1d 3606 . . . . . . . . 9  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( {  .0.  }  \  {  .0.  } ) )
46 difid 3882 . . . . . . . . 9  |-  ( {  .0.  }  \  {  .0.  } )  =  (/)
4745, 46syl6eq 2500 . . . . . . . 8  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4844, 47pm2.61ine 2756 . . . . . . 7  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
4948eleq2i 2521 . . . . . 6  |-  ( x  e.  ( ran  ( U. dom  M  X.  {  .0.  } )  \  {  .0.  } )  <->  x  e.  (/) )
5041, 49mtbir 299 . . . . 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 466 . . 3  |-  ( (
ph  /\  x  e.  ( ran  ( U. dom  M  X.  {  .0.  }
)  \  {  .0.  } ) )  ->  ( M `  ( `' ( U. dom  M  X.  {  .0.  } ) " { x } ) )  e.  ( 0 [,) +oo ) )
5352ralrimiva 2857 . 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 28148 . 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 1180 1  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  dom  ( Wsitg M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458   (/)c0 3770   {csn 4014   U.cuni 4234    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992   ` cfv 5578  (class class class)co 6281   Fincfn 7518   0cc0 9495   +oocpnf 9628   [,)cico 11540   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   TopOpenctopn 14696   0gc0g 14714   Mndcmnd 15793   TopSpctps 19270  RRHomcrrh 27847  sigAlgebracsiga 27980  sigaGencsigagen 28011  measurescmeas 28039  MblFnMcmbfm 28094  sitgcsitg 28144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1o 7132  df-map 7424  df-en 7519  df-fin 7522  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-top 19272  df-topon 19275  df-topsp 19276  df-esum 27914  df-siga 27981  df-sigagen 28012  df-meas 28040  df-mbfm 28095  df-sitg 28145
This theorem is referenced by:  sitg0  28161
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