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Theorem measval 28342
Description: The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
Assertion
Ref Expression
measval  |-  ( S  e.  U. ran sigAlgebra  ->  (measures `  S )  =  {
m  |  ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
Distinct variable groups:    x, m, y    S, m, x
Allowed substitution hint:    S( y)

Proof of Theorem measval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . 4  |-  ( ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) )  ->  m : S --> ( 0 [,] +oo ) )
21ss2abi 3568 . . 3  |-  { m  |  ( m : S --> ( 0 [,] +oo )  /\  (
m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  C_  { m  |  m : S --> ( 0 [,] +oo ) }
3 ovex 6324 . . . 4  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7444 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : S --> ( 0 [,] +oo ) }  e.  _V )
53, 4mpan2 671 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  { m  |  m : S --> ( 0 [,] +oo ) }  e.  _V )
6 ssexg 4602 . . 3  |-  ( ( { m  |  ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  C_  { m  |  m : S --> ( 0 [,] +oo ) }  /\  { m  |  m : S --> ( 0 [,] +oo ) }  e.  _V )  ->  { m  |  (
m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  e.  _V )
72, 5, 6sylancr 663 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  { m  |  ( m : S --> ( 0 [,] +oo )  /\  (
m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  e.  _V )
8 feq2 5720 . . . . 5  |-  ( s  =  S  ->  (
m : s --> ( 0 [,] +oo )  <->  m : S --> ( 0 [,] +oo ) ) )
9 pweq 4018 . . . . . 6  |-  ( s  =  S  ->  ~P s  =  ~P S
)
109raleqdv 3060 . . . . 5  |-  ( s  =  S  ->  ( A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  (
m `  U. x )  = Σ* y  e.  x ( m `  y ) )  <->  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  (
m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) )
118, 103anbi13d 1301 . . . 4  |-  ( s  =  S  ->  (
( m : s --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\ 
A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  ->  (
m `  U. x )  = Σ* y  e.  x ( m `  y ) ) )  <->  ( m : S --> ( 0 [,] +oo )  /\  (
m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) ) )
1211abbidv 2593 . . 3  |-  ( s  =  S  ->  { m  |  ( m : s --> ( 0 [,] +oo )  /\  (
m `  (/) )  =  0  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  =  { m  |  ( m : S --> ( 0 [,] +oo )  /\  (
m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
13 df-meas 28340 . . 3  |- measures  =  ( s  e.  U. ran sigAlgebra  |->  { m  |  ( m : s --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  s ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
1412, 13fvmptg 5954 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  { m  |  ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) }  e.  _V )  ->  (measures `  S )  =  { m  |  ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
157, 14mpdan 668 1  |-  ( S  e.  U. ran sigAlgebra  ->  (measures `  S )  =  {
m  |  ( m : S --> ( 0 [,] +oo )  /\  ( m `  (/) )  =  0  /\  A. x  e.  ~P  S ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( m `  U. x )  = Σ* y  e.  x ( m `  y ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   _Vcvv 3109    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251  Disj wdisj 4427   class class class wbr 4456   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   omcom 6699    ~<_ cdom 7533   0cc0 9509   +oocpnf 9642   [,]cicc 11557  Σ*cesum 28201  sigAlgebracsiga 28280  measurescmeas 28339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-meas 28340
This theorem is referenced by: (None)
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