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Theorem measvnul 26635
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 26626 . . . 4  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
2 ismeas 26628 . . . 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 1001 1  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   (/)c0 3652   ~Pcpw 3875   U.cuni 4106  Disj wdisj 4277   class class class wbr 4307   ran crn 4856   -->wf 5429   ` cfv 5433  (class class class)co 6106   omcom 6491    ~<_ cdom 7323   0cc0 9297   +oocpnf 9430   [,]cicc 11318  Σ*cesum 26498  sigAlgebracsiga 26565  measurescmeas 26624
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  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 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-fv 5441  df-ov 6109  df-esum 26499  df-meas 26625
This theorem is referenced by:  measxun2  26639  measvunilem0  26642  measssd  26644  measinb  26650  measres  26651  measdivcstOLD  26653  measdivcst  26654  truae  26674  probnul  26812
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