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Theorem ismeas 24545
Description: The property of being a measure (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
Assertion
Ref Expression
ismeas  |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  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 ) ) ) ) )
Distinct variable groups:    x, y, M    x, S
Allowed substitution hint:    S( y)

Proof of Theorem ismeas
Dummy variables  m  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2956 . . 3  |-  ( M  e.  (measures `  S
)  ->  M  e.  _V )
21a1i 11 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  S
)  ->  M  e.  _V ) )
3 simp1 957 . . 3  |-  ( ( 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 ) )
4 ovex 6098 . . . 4  |-  ( 0 [,]  +oo )  e.  _V
5 fex2 5595 . . . . . 6  |-  ( ( M : S --> ( 0 [,]  +oo )  /\  S  e.  U. ran sigAlgebra  /\  ( 0 [,]  +oo )  e.  _V )  ->  M  e.  _V )
653expb 1154 . . . . 5  |-  ( ( M : S --> ( 0 [,]  +oo )  /\  ( S  e.  U. ran sigAlgebra  /\  (
0 [,]  +oo )  e. 
_V ) )  ->  M  e.  _V )
76expcom 425 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  ( 0 [,]  +oo )  e.  _V )  ->  ( M : S --> ( 0 [,]  +oo )  ->  M  e.  _V ) )
84, 7mpan2 653 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  ( M : S --> ( 0 [,]  +oo )  ->  M  e.  _V ) )
93, 8syl5 30 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  (
( 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  e.  _V ) )
10 df-meas 24542 . . . 4  |- 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 ) ) ) } )
11 vex 2951 . . . . . 6  |-  s  e. 
_V
12 mapex 7016 . . . . . 6  |-  ( ( s  e.  _V  /\  ( 0 [,]  +oo )  e.  _V )  ->  { m  |  m : s --> ( 0 [,]  +oo ) }  e.  _V )
1311, 4, 12mp2an 654 . . . . 5  |-  { m  |  m : s --> ( 0 [,]  +oo ) }  e.  _V
14 simp1 957 . . . . . 6  |-  ( ( 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 ) )
1514ss2abi 3407 . . . . 5  |-  { 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 ) }
1613, 15ssexi 4340 . . . 4  |-  { 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
17 simpr 448 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  m  =  M )
18 simpl 444 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  s  =  S )
1917, 18feq12d 5574 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( m : s --> ( 0 [,]  +oo ) 
<->  M : S --> ( 0 [,]  +oo ) ) )
20 fveq1 5719 . . . . . . 7  |-  ( m  =  M  ->  (
m `  (/) )  =  ( M `  (/) ) )
2120eqeq1d 2443 . . . . . 6  |-  ( m  =  M  ->  (
( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2221adantl 453 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2318pweqd 3796 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ~P s  =  ~P S )
24 fveq1 5719 . . . . . . . . 9  |-  ( m  =  M  ->  (
m `  U. x )  =  ( M `  U. x ) )
25 fveq1 5719 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m `  y )  =  ( M `  y ) )
2625esumeq2sdv 24428 . . . . . . . . 9  |-  ( m  =  M  -> Σ* y  e.  x
( m `  y
)  = Σ* y  e.  x
( M `  y
) )
2724, 26eqeq12d 2449 . . . . . . . 8  |-  ( m  =  M  ->  (
( m `  U. x )  = Σ* y  e.  x ( m `  y )  <->  ( M `  U. x )  = Σ* y  e.  x ( M `
 y ) ) )
2827imbi2d 308 . . . . . . 7  |-  ( m  =  M  ->  (
( ( x  ~<_  om 
/\ Disj  y  e.  x y )  ->  ( m `  U. x )  = Σ* y  e.  x ( m `
 y ) )  <-> 
( ( x  ~<_  om 
/\ Disj  y  e.  x y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `
 y ) ) ) )
2928adantl 453 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( ( x  ~<_  om  /\ Disj  y  e.  x
y )  ->  (
m `  U. x )  = Σ* y  e.  x ( m `  y ) )  <->  ( ( x  ~<_  om  /\ Disj  y  e.  x
y )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) ) ) )
3023, 29raleqbidv 2908 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( 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 ) ) ) )
3119, 22, 303anbi123d 1254 . . . 4  |-  ( ( s  =  S  /\  m  =  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 : 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 ) ) ) ) )
3210, 16, 31abfmpel 24059 . . 3  |-  ( ( S  e.  U. ran sigAlgebra  /\  M  e.  _V )  ->  ( M  e.  (measures `  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 ) ) ) ) )
3332ex 424 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  _V  ->  ( M  e.  (measures `  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 ) ) ) ) ) )
342, 9, 33pm5.21ndd 344 1  |-  ( S  e.  U. ran sigAlgebra  ->  ( M  e.  (measures `  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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   _Vcvv 2948   (/)c0 3620   ~Pcpw 3791   U.cuni 4007  Disj wdisj 4174   class class class wbr 4204   omcom 4837   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099   0cc0 8982    +oocpnf 9109   [,]cicc 10911  Σ*cesum 24416  sigAlgebracsiga 24482  measurescmeas 24541
This theorem is referenced by:  measbasedom  24548  measfrge0  24549  measvnul  24552  measvun  24555  measinb  24567  measres  24568  measdivcstOLD  24570  measdivcst  24571  cntmeas  24572  volmeas  24579  dstrvprob  24721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-esum 24417  df-meas 24542
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