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

Theorem brae 28450
Description: 'almost everywhere' relation for a measure and a measurable set  A. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Assertion
Ref Expression
brae  |-  ( ( M  e.  U. ran measures  /\  A  e.  dom  M )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )

Proof of Theorem brae
Dummy variables  m  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  m  =  M )
21dmeqd 5194 . . . . . . 7  |-  ( ( a  =  A  /\  m  =  M )  ->  dom  m  =  dom  M )
32unieqd 4245 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  U. dom  m  = 
U. dom  M )
4 simpl 455 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  a  =  A )
53, 4difeq12d 3609 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  ( U. dom  m  \  a )  =  ( U. dom  M  \  A ) )
61, 5fveq12d 5854 . . . 4  |-  ( ( a  =  A  /\  m  =  M )  ->  ( m `  ( U. dom  m  \  a
) )  =  ( M `  ( U. dom  M  \  A ) ) )
76eqeq1d 2456 . . 3  |-  ( ( a  =  A  /\  m  =  M )  ->  ( ( m `  ( U. dom  m  \ 
a ) )  =  0  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
8 df-ae 28448 . . 3  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
97, 8brabga 4750 . 2  |-  ( ( A  e.  dom  M  /\  M  e.  U. ran measures )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
109ancoms 451 1  |-  ( ( M  e.  U. ran measures  /\  A  e.  dom  M )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    \ cdif 3458   U.cuni 4235   class class class wbr 4439   dom cdm 4988   ran crn 4989   ` cfv 5570   0cc0 9481  measurescmeas 28403  a.e.cae 28446
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  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-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-dm 4998  df-iota 5534  df-fv 5578  df-ae 28448
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator