Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sibf0 Structured version   Unicode version

Theorem sibf0 26840
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 26735 . . . . 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 5785 . . . . . . . 8  |-  ( TopOpen `  W )  e.  _V
75, 6eqeltri 2532 . . . . . . 7  |-  J  e. 
_V
87a1i 11 . . . . . 6  |-  ( ph  ->  J  e.  _V )
98sgsiga 26705 . . . . 5  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
104, 9syl5eqel 2540 . . . 4  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
11 fconstmpt 4966 . . . . 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 15527 . . . . . 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 18645 . . . . . . 7  |-  ( W  e.  TopSp  ->  B  =  U. J )
2018, 19syl 16 . . . . . 6  |-  ( ph  ->  B  =  U. J
)
214unieqi 4184 . . . . . . 7  |-  U. S  =  U. (sigaGen `  J
)
22 unisg 26706 . . . . . . . 8  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
237, 22mp1i 12 . . . . . . 7  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
2421, 23syl5eq 2502 . . . . . 6  |-  ( ph  ->  U. S  =  U. J )
2520, 24eqtr4d 2493 . . . . 5  |-  ( ph  ->  B  =  U. S
)
2617, 25eleqtrd 2538 . . . 4  |-  ( ph  ->  .0.  e.  U. S
)
273, 10, 12, 26mbfmcst 26794 . . 3  |-  ( ph  ->  ( U. dom  M  X.  {  .0.  } )  e.  ( dom  MMblFnM S ) )
28 xpeq1 4938 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (
(/)  X.  {  .0.  } ) )
29 0xp 5001 . . . . . . . . 9  |-  ( (/)  X. 
{  .0.  } )  =  (/)
3028, 29syl6eq 2506 . . . . . . . 8  |-  ( U. dom  M  =  (/)  ->  ( U. dom  M  X.  {  .0.  } )  =  (/) )
3130rneqd 5151 . . . . . . 7  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  ran  (/) )
32 rn0 5175 . . . . . . 7  |-  ran  (/)  =  (/)
3331, 32syl6eq 2506 . . . . . 6  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  (/) )
34 0fin 7627 . . . . . 6  |-  (/)  e.  Fin
3533, 34syl6eqel 2544 . . . . 5  |-  ( U. dom  M  =  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
36 rnxp 5352 . . . . . 6  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  =  {  .0.  } )
37 snfi 7476 . . . . . 6  |-  {  .0.  }  e.  Fin
3836, 37syl6eqel 2544 . . . . 5  |-  ( U. dom  M  =/=  (/)  ->  ran  ( U. dom  M  X.  {  .0.  } )  e. 
Fin )
3935, 38pm2.61ine 2758 . . . 4  |-  ran  ( U. dom  M  X.  {  .0.  } )  e.  Fin
4039a1i 11 . . 3  |-  ( ph  ->  ran  ( U. dom  M  X.  {  .0.  }
)  e.  Fin )
41 noel 3725 . . . . . . 7  |-  -.  x  e.  (/)
4233difeq1d 3557 . . . . . . . . . 10  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( (/)  \  {  .0.  } ) )
43 0dif 3834 . . . . . . . . . 10  |-  ( (/)  \  {  .0.  } )  =  (/)
4442, 43syl6eq 2506 . . . . . . . . 9  |-  ( U. dom  M  =  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4536difeq1d 3557 . . . . . . . . . 10  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  ( {  .0.  }  \  {  .0.  } ) )
46 difid 3831 . . . . . . . . . 10  |-  ( {  .0.  }  \  {  .0.  } )  =  (/)
4745, 46syl6eq 2506 . . . . . . . . 9  |-  ( U. dom  M  =/=  (/)  ->  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/) )
4844, 47pm2.61ine 2758 . . . . . . . 8  |-  ( ran  ( U. dom  M  X.  {  .0.  } ) 
\  {  .0.  }
)  =  (/)
4948eleq2i 2526 . . . . . . 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 2881 . . 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 1168 . 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 26839 . 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 965    = wceq 1370    e. wcel 1757    =/= wne 2641   A.wral 2792   _Vcvv 3054    \ cdif 3409   (/)c0 3721   {csn 3961   U.cuni 4175    |-> cmpt 4434    X. cxp 4922   `'ccnv 4923   dom cdm 4924   ran crn 4925   "cima 4927   ` cfv 5502  (class class class)co 6176   Fincfn 7396   0cc0 9369   +oocpnf 9502   [,)cico 11389   Basecbs 14262  Scalarcsca 14329   .scvsca 14330   TopOpenctopn 14448   0gc0g 14466   Mndcmnd 15497   TopSpctps 18603  RRHomcrrh 26542  sigAlgebracsiga 26670  sigaGencsigagen 26701  measurescmeas 26729  MblFnMcmbfm 26785  sitgcsitg 26835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1o 7006  df-map 7302  df-en 7397  df-fin 7400  df-0g 14468  df-mnd 15503  df-top 18605  df-topon 18608  df-topsp 18609  df-esum 26604  df-siga 26671  df-sigagen 26702  df-meas 26730  df-mbfm 26786  df-sitg 26836
This theorem is referenced by:  sitg0  26852
  Copyright terms: Public domain W3C validator