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Theorem measvnul 26556
Description: The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvnul  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )

Proof of Theorem measvnul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 26547 . . . 4  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
2 ismeas 26549 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  S
)  <->  ( M : S
--> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\ 
A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  ->  ( M `  U. y )  = Σ* x  e.  y ( M `  x ) ) ) ) )
31, 2syl 16 . . 3  |-  ( M  e.  (measures `  S
)  ->  ( M  e.  (measures `  S )  <->  ( M : S --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  -> 
( M `  U. y )  = Σ* x  e.  y ( M `  x ) ) ) ) )
43ibi 241 . 2  |-  ( M  e.  (measures `  S
)  ->  ( M : S --> ( 0 [,] +oo )  /\  ( M `  (/) )  =  0  /\  A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  -> 
( M `  U. y )  = Σ* x  e.  y ( M `  x ) ) ) )
54simp2d 996 1  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   (/)c0 3634   ~Pcpw 3857   U.cuni 4088  Disj wdisj 4259   class class class wbr 4289   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090   omcom 6475    ~<_ cdom 7304   0cc0 9278   +oocpnf 9411   [,]cicc 11299  Σ*cesum 26419  sigAlgebracsiga 26486  measurescmeas 26545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-esum 26420  df-meas 26546
This theorem is referenced by:  measxun2  26560  measvunilem0  26563  measssd  26565  measinb  26571  measres  26572  measdivcstOLD  26574  measdivcst  26575  truae  26595  probnul  26727
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