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Theorem measvun 28417
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 995 . 2  |-  ( ( M  e.  (measures `  S
)  /\  A  e.  ~P S  /\  ( A  ~<_  om  /\ Disj  x  e.  A  x ) )  ->  A  e.  ~P S )
2 measbase 28405 . . . . . 6  |-  ( M  e.  (measures `  S
)  ->  S  e.  U.
ran sigAlgebra )
3 ismeas 28407 . . . . . 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 1008 . . 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 1015 . 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 996 . 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 4442 . . . . 5  |-  ( y  =  A  ->  (
y  ~<_  om  <->  A  ~<_  om )
)
10 disjeq1 4417 . . . . 5  |-  ( y  =  A  ->  (Disj  x  e.  y  x  <-> Disj  x  e.  A  x ) )
119, 10anbi12d 708 . . . 4  |-  ( y  =  A  ->  (
( y  ~<_  om  /\ Disj  x  e.  y  x )  <-> 
( A  ~<_  om  /\ Disj  x  e.  A  x ) ) )
12 unieq 4243 . . . . . 6  |-  ( y  =  A  ->  U. y  =  U. A )
1312fveq2d 5852 . . . . 5  |-  ( y  =  A  ->  ( M `  U. y )  =  ( M `  U. A ) )
14 esumeq1 28263 . . . . 5  |-  ( y  =  A  -> Σ* x  e.  y
( M `  x
)  = Σ* x  e.  A
( M `  x
) )
1513, 14eqeq12d 2476 . . . 4  |-  ( y  =  A  ->  (
( M `  U. y )  = Σ* x  e.  y ( M `  x )  <->  ( M `  U. A )  = Σ* x  e.  A ( M `
 x ) ) )
1611, 15imbi12d 318 . . 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 3203 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   (/)c0 3783   ~Pcpw 3999   U.cuni 4235  Disj wdisj 4410   class class class wbr 4439   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   omcom 6673    ~<_ cdom 7507   0cc0 9481   +oocpnf 9614   [,]cicc 11535  Σ*cesum 28256  sigAlgebracsiga 28337  measurescmeas 28403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-esum 28257  df-meas 28404
This theorem is referenced by:  measxun2  28418  measvunilem  28420  measssd  28423  measres  28430  measdivcstOLD  28432  measdivcst  28433  probcun  28621  totprobd  28629
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