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

Theorem truae 29139
Description: A truth holds almost everywhere. (Contributed by Thierry Arnoux, 20-Oct-2017.)
Hypotheses
Ref Expression
truae.1  |-  U. dom  M  =  O
truae.2  |-  ( ph  ->  M  e.  U. ran measures )
truae.3  |-  ( ph  ->  ps )
Assertion
Ref Expression
truae  |-  ( ph  ->  { x  e.  O  |  ps }a.e. M )
Distinct variable groups:    x, O    ph, x
Allowed substitution hints:    ps( x)    M( x)

Proof of Theorem truae
StepHypRef Expression
1 truae.3 . . . . . . . 8  |-  ( ph  ->  ps )
21pm2.24d 139 . . . . . . 7  |-  ( ph  ->  ( -.  ps  ->  x  e.  (/) ) )
32ralrimivw 2810 . . . . . 6  |-  ( ph  ->  A. x  e.  O  ( -.  ps  ->  x  e.  (/) ) )
4 rabss 3492 . . . . . 6  |-  ( { x  e.  O  |  -.  ps }  C_  (/)  <->  A. x  e.  O  ( -.  ps  ->  x  e.  (/) ) )
53, 4sylibr 217 . . . . 5  |-  ( ph  ->  { x  e.  O  |  -.  ps }  C_  (/) )
6 ss0 3768 . . . . 5  |-  ( { x  e.  O  |  -.  ps }  C_  (/)  ->  { x  e.  O  |  -.  ps }  =  (/) )
75, 6syl 17 . . . 4  |-  ( ph  ->  { x  e.  O  |  -.  ps }  =  (/) )
87fveq2d 5883 . . 3  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  ( M `  (/) ) )
9 truae.2 . . . 4  |-  ( ph  ->  M  e.  U. ran measures )
10 measbasedom 29098 . . . . 5  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
11 measvnul 29102 . . . . 5  |-  ( M  e.  (measures `  dom  M )  ->  ( M `  (/) )  =  0 )
1210, 11sylbi 200 . . . 4  |-  ( M  e.  U. ran measures  ->  ( M `  (/) )  =  0 )
139, 12syl 17 . . 3  |-  ( ph  ->  ( M `  (/) )  =  0 )
148, 13eqtrd 2505 . 2  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  0 )
15 truae.1 . . . 4  |-  U. dom  M  =  O
1615braew 29138 . . 3  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ps }a.e. M  <->  ( M `  { x  e.  O  |  -.  ps } )  =  0 ) )
179, 16syl 17 . 2  |-  ( ph  ->  ( { x  e.  O  |  ps }a.e. M 
<->  ( M `  {
x  e.  O  |  -.  ps } )  =  0 ) )
1814, 17mpbird 240 1  |-  ( ph  ->  { x  e.  O  |  ps }a.e. M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    C_ wss 3390   (/)c0 3722   U.cuni 4190   class class class wbr 4395   dom cdm 4839   ran crn 4840   ` cfv 5589   0cc0 9557  measurescmeas 29091  a.e.cae 29133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-esum 28923  df-meas 29092  df-ae 29135
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator