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Theorem measval 29094
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 1030 . . . 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 3487 . . 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 6336 . . . 4  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7496 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : S --> ( 0 [,] +oo ) }  e.  _V )
53, 4mpan2 685 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  { m  |  m : S --> ( 0 [,] +oo ) }  e.  _V )
6 ssexg 4542 . . 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 676 . 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 5721 . . . . 5  |-  ( s  =  S  ->  (
m : s --> ( 0 [,] +oo )  <->  m : S --> ( 0 [,] +oo ) ) )
9 pweq 3945 . . . . . 6  |-  ( s  =  S  ->  ~P s  =  ~P S
)
109raleqdv 2979 . . . . 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 1367 . . . 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 2589 . . 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 29092 . . 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 5961 . 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 681 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190  Disj wdisj 4366   class class class wbr 4395   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   omcom 6711    ~<_ cdom 7585   0cc0 9557   +oocpnf 9690   [,]cicc 11663  Σ*cesum 28922  sigAlgebracsiga 29003  measurescmeas 29091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-meas 29092
This theorem is referenced by: (None)
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