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Theorem measval 27837
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 3572 . . 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 6309 . . . 4  |-  ( 0 [,] +oo )  e. 
_V
4 mapex 7426 . . . 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 4593 . . 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 5714 . . . . 5  |-  ( s  =  S  ->  (
m : s --> ( 0 [,] +oo )  <->  m : S --> ( 0 [,] +oo ) ) )
9 pweq 4013 . . . . . 6  |-  ( s  =  S  ->  ~P s  =  ~P S
)
109raleqdv 3064 . . . . 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 2603 . . 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 27835 . . 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 5948 . 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245  Disj wdisj 4417   class class class wbr 4447   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   omcom 6684    ~<_ cdom 7514   0cc0 9492   +oocpnf 9625   [,]cicc 11532  Σ*cesum 27708  sigAlgebracsiga 27775  measurescmeas 27834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-meas 27835
This theorem is referenced by: (None)
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