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Theorem measvun 26622
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 989 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  A  e.  ~P S )
2 measbase 26610 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
3 ismeas 26612 . . . . . 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 1002 . . 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 1009 . 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 990 . 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 4294 . . . . 5  |-  ( y  =  A  ->  (
y  ~<_  om  <->  A  ~<_  om )
)
10 disjeq1 4268 . . . . 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 4098 . . . . . 6  |-  ( y  =  A  ->  U. y  =  U. A )
1312fveq2d 5694 . . . . 5  |-  ( y  =  A  ->  ( M `  U. y )  =  ( M `  U. A ) )
14 esumeq1 26489 . . . . 5  |-  ( y  =  A  -> Σ* x  e.  y
( M `  x
)  = Σ* x  e.  A
( M `  x
) )
1513, 14eqeq12d 2456 . . . 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 3068 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   (/)c0 3636   ~Pcpw 3859   U.cuni 4090  Disj wdisj 4261   class class class wbr 4291   ran crn 4840   -->wf 5413   ` cfv 5417  (class class class)co 6090   omcom 6475    ~<_ cdom 7307   0cc0 9281   +oocpnf 9414   [,]cicc 11302  Σ*cesum 26482  sigAlgebracsiga 26549  measurescmeas 26608
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-esum 26483  df-meas 26609
This theorem is referenced by:  measxun2  26623  measvunilem  26625  measssd  26628  measres  26635  measdivcstOLD  26637  measdivcst  26638  probcun  26800  totprobd  26808
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