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

Theorem brae 26669
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 461 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  m  =  M )
21dmeqd 5054 . . . . . . 7  |-  ( ( a  =  A  /\  m  =  M )  ->  dom  m  =  dom  M )
32unieqd 4113 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  U. dom  m  = 
U. dom  M )
4 simpl 457 . . . . . 6  |-  ( ( a  =  A  /\  m  =  M )  ->  a  =  A )
53, 4difeq12d 3487 . . . . 5  |-  ( ( a  =  A  /\  m  =  M )  ->  ( U. dom  m  \  a )  =  ( U. dom  M  \  A ) )
61, 5fveq12d 5709 . . . 4  |-  ( ( a  =  A  /\  m  =  M )  ->  ( m `  ( U. dom  m  \  a
) )  =  ( M `  ( U. dom  M  \  A ) ) )
76eqeq1d 2451 . . 3  |-  ( ( a  =  A  /\  m  =  M )  ->  ( ( m `  ( U. dom  m  \ 
a ) )  =  0  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
8 df-ae 26667 . . 3  |- a.e.  =  { <. a ,  m >.  |  ( m `  ( U. dom  m  \  a
) )  =  0 }
97, 8brabga 4615 . 2  |-  ( ( A  e.  dom  M  /\  M  e.  U. ran measures )  ->  ( Aa.e. M  <->  ( M `  ( U. dom  M  \  A ) )  =  0 ) )
109ancoms 453 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 369    = wceq 1369    e. wcel 1756    \ cdif 3337   U.cuni 4103   class class class wbr 4304   dom cdm 4852   ran crn 4853   ` cfv 5430   0cc0 9294  measurescmeas 26621  a.e.cae 26665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-dm 4862  df-iota 5393  df-fv 5438  df-ae 26667
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator