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Theorem truae 26804
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 143 . . . . . . 7  |-  ( ph  ->  ( -.  ps  ->  x  e.  (/) ) )
32ralrimivw 2831 . . . . . 6  |-  ( ph  ->  A. x  e.  O  ( -.  ps  ->  x  e.  (/) ) )
4 rabss 3538 . . . . . 6  |-  ( { x  e.  O  |  -.  ps }  C_  (/)  <->  A. x  e.  O  ( -.  ps  ->  x  e.  (/) ) )
53, 4sylibr 212 . . . . 5  |-  ( ph  ->  { x  e.  O  |  -.  ps }  C_  (/) )
6 ss0 3777 . . . . 5  |-  ( { x  e.  O  |  -.  ps }  C_  (/)  ->  { x  e.  O  |  -.  ps }  =  (/) )
75, 6syl 16 . . . 4  |-  ( ph  ->  { x  e.  O  |  -.  ps }  =  (/) )
87fveq2d 5804 . . 3  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  ( M `  (/) ) )
9 truae.2 . . . 4  |-  ( ph  ->  M  e.  U. ran measures )
10 measbasedom 26762 . . . . 5  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
11 measvnul 26766 . . . . 5  |-  ( M  e.  (measures `  dom  M )  ->  ( M `  (/) )  =  0 )
1210, 11sylbi 195 . . . 4  |-  ( M  e.  U. ran measures  ->  ( M `  (/) )  =  0 )
139, 12syl 16 . . 3  |-  ( ph  ->  ( M `  (/) )  =  0 )
148, 13eqtrd 2495 . 2  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  0 )
15 truae.1 . . . 4  |-  U. dom  M  =  O
1615braew 26803 . . 3  |-  ( M  e.  U. ran measures  ->  ( { x  e.  O  |  ps }a.e. M  <->  ( M `  { x  e.  O  |  -.  ps } )  =  0 ) )
179, 16syl 16 . 2  |-  ( ph  ->  ( { x  e.  O  |  ps }a.e. M 
<->  ( M `  {
x  e.  O  |  -.  ps } )  =  0 ) )
1814, 17mpbird 232 1  |-  ( ph  ->  { x  e.  O  |  ps }a.e. M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803    C_ wss 3437   (/)c0 3746   U.cuni 4200   class class class wbr 4401   dom cdm 4949   ran crn 4950   ` cfv 5527   0cc0 9394  measurescmeas 26755  a.e.cae 26798
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-esum 26630  df-meas 26756  df-ae 26800
This theorem is referenced by: (None)
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