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Theorem truae 27966
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 2879 . . . . . 6  |-  ( ph  ->  A. x  e.  O  ( -.  ps  ->  x  e.  (/) ) )
4 rabss 3577 . . . . . 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 3816 . . . . 5  |-  ( { x  e.  O  |  -.  ps }  C_  (/)  ->  { x  e.  O  |  -.  ps }  =  (/) )
75, 6syl 16 . . . 4  |-  ( ph  ->  { x  e.  O  |  -.  ps }  =  (/) )
87fveq2d 5870 . . 3  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  ( M `  (/) ) )
9 truae.2 . . . 4  |-  ( ph  ->  M  e.  U. ran measures )
10 measbasedom 27924 . . . . 5  |-  ( M  e.  U. ran measures  <->  M  e.  (measures `  dom  M ) )
11 measvnul 27928 . . . . 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 2508 . 2  |-  ( ph  ->  ( M `  {
x  e.  O  |  -.  ps } )  =  0 )
15 truae.1 . . . 4  |-  U. dom  M  =  O
1615braew 27965 . . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818    C_ wss 3476   (/)c0 3785   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000   ` cfv 5588   0cc0 9493  measurescmeas 27917  a.e.cae 27960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-esum 27792  df-meas 27918  df-ae 27962
This theorem is referenced by: (None)
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