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Theorem measvun 28053
Description: The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvun  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  ( M `  U. A )  = Σ* x  e.  A ( M `  x ) )
Distinct variable groups:    x, A    x, M
Allowed substitution hint:    S( x)

Proof of Theorem measvun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp2 998 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  A  e.  ~P S )
2 measbase 28041 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
3 ismeas 28043 . . . . . 6  |-  ( 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 ) ) ) ) )
42, 3syl 16 . . . . 5  |-  ( 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 ) ) ) ) )
54ibi 241 . . . 4  |-  ( 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 ) ) ) )
65simp3d 1011 . . 3  |-  ( M  e.  (measures `  S
)  ->  A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  -> 
( M `  U. y )  = Σ* x  e.  y ( M `  x ) ) )
763ad2ant1 1018 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  A. y  e.  ~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  ->  ( M `  U. y )  = Σ* x  e.  y ( M `  x ) ) )
8 simp3 999 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  ( A  ~<_  om 
/\ Disj  x  e.  A  x ) )
9 breq1 4440 . . . . 5  |-  ( y  =  A  ->  (
y  ~<_  om  <->  A  ~<_  om )
)
10 disjeq1 4414 . . . . 5  |-  ( y  =  A  ->  (Disj  x  e.  y  x  <-> Disj  x  e.  A  x ) )
119, 10anbi12d 710 . . . 4  |-  ( y  =  A  ->  (
( y  ~<_  om  /\ Disj  x  e.  y  x )  <-> 
( A  ~<_  om  /\ Disj  x  e.  A  x ) ) )
12 unieq 4242 . . . . . 6  |-  ( y  =  A  ->  U. y  =  U. A )
1312fveq2d 5860 . . . . 5  |-  ( y  =  A  ->  ( M `  U. y )  =  ( M `  U. A ) )
14 esumeq1 27920 . . . . 5  |-  ( y  =  A  -> Σ* x  e.  y
( M `  x
)  = Σ* x  e.  A
( M `  x
) )
1513, 14eqeq12d 2465 . . . 4  |-  ( y  =  A  ->  (
( M `  U. y )  = Σ* x  e.  y ( M `  x )  <->  ( M `  U. A )  = Σ* x  e.  A ( M `
 x ) ) )
1611, 15imbi12d 320 . . 3  |-  ( y  =  A  ->  (
( ( y  ~<_  om 
/\ Disj  x  e.  y  x )  ->  ( M `  U. y )  = Σ* x  e.  y ( M `
 x ) )  <-> 
( ( A  ~<_  om 
/\ Disj  x  e.  A  x )  ->  ( M `  U. A )  = Σ* x  e.  A ( M `
 x ) ) ) )
1716rspcv 3192 . 2  |-  ( A  e.  ~P S  -> 
( A. y  e. 
~P  S ( ( y  ~<_  om  /\ Disj  x  e.  y  x )  -> 
( M `  U. y )  = Σ* x  e.  y ( M `  x ) )  -> 
( ( A  ~<_  om 
/\ Disj  x  e.  A  x )  ->  ( M `  U. A )  = Σ* x  e.  A ( M `
 x ) ) ) )
181, 7, 8, 17syl3c 61 1  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  ( M `  U. A )  = Σ* x  e.  A ( M `  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   (/)c0 3770   ~Pcpw 3997   U.cuni 4234  Disj wdisj 4407   class class class wbr 4437   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   omcom 6685    ~<_ cdom 7516   0cc0 9495   +oocpnf 9628   [,]cicc 11541  Σ*cesum 27913  sigAlgebracsiga 27980  measurescmeas 28039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-esum 27914  df-meas 28040
This theorem is referenced by:  measxun2  28054  measvunilem  28056  measssd  28059  measres  28066  measdivcstOLD  28068  measdivcst  28069  probcun  28230  totprobd  28238
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