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Theorem ismeas 28036
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 3102 . . 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 995 . . 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 6305 . . . 4  |-  ( 0 [,] +oo )  e. 
_V
5 fex2 6736 . . . . . 6  |-  ( ( M : S --> ( 0 [,] +oo )  /\  S  e.  U. ran sigAlgebra  /\  (
0 [,] +oo )  e.  _V )  ->  M  e.  _V )
653expb 1196 . . . . 5  |-  ( ( M : S --> ( 0 [,] +oo )  /\  ( S  e.  U. ran sigAlgebra  /\  ( 0 [,] +oo )  e.  _V )
)  ->  M  e.  _V )
76expcom 435 . . . 4  |-  ( ( S  e.  U. ran sigAlgebra  /\  ( 0 [,] +oo )  e.  _V )  ->  ( M : S --> ( 0 [,] +oo )  ->  M  e.  _V ) )
84, 7mpan2 671 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  ( M : S --> ( 0 [,] +oo )  ->  M  e.  _V )
)
93, 8syl5 32 . 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 28033 . . . 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 3096 . . . . . 6  |-  s  e. 
_V
12 mapex 7424 . . . . . 6  |-  ( ( s  e.  _V  /\  ( 0 [,] +oo )  e.  _V )  ->  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V )
1311, 4, 12mp2an 672 . . . . 5  |-  { m  |  m : s --> ( 0 [,] +oo ) }  e.  _V
14 simp1 995 . . . . . 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 3554 . . . . 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 4578 . . . 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 461 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  m  =  M )
18 simpl 457 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  s  =  S )
1917, 18feq12d 5706 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( m : s --> ( 0 [,] +oo ) 
<->  M : S --> ( 0 [,] +oo ) ) )
20 fveq1 5851 . . . . . . 7  |-  ( m  =  M  ->  (
m `  (/) )  =  ( M `  (/) ) )
2120eqeq1d 2443 . . . . . 6  |-  ( m  =  M  ->  (
( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2221adantl 466 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2318pweqd 3998 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ~P s  =  ~P S )
24 fveq1 5851 . . . . . . . . 9  |-  ( m  =  M  ->  (
m `  U. x )  =  ( M `  U. x ) )
25 fveq1 5851 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m `  y )  =  ( M `  y ) )
2625esumeq2sdv 27918 . . . . . . . . 9  |-  ( m  =  M  -> Σ* y  e.  x
( m `  y
)  = Σ* y  e.  x
( M `  y
) )
2724, 26eqeq12d 2463 . . . . . . . 8  |-  ( m  =  M  ->  (
( m `  U. x )  = Σ* y  e.  x ( m `  y )  <->  ( M `  U. x )  = Σ* y  e.  x ( M `
 y ) ) )
2827imbi2d 316 . . . . . . 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 466 . . . . . 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 3052 . . . . 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 1298 . . . 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 27358 . . 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 434 . 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 354 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   {cab 2426   A.wral 2791   _Vcvv 3093   (/)c0 3767   ~Pcpw 3993   U.cuni 4230  Disj wdisj 4403   class class class wbr 4433   ran crn 4986   -->wf 5570   ` cfv 5574  (class class class)co 6277   omcom 6681    ~<_ cdom 7512   0cc0 9490   +oocpnf 9623   [,]cicc 11536  Σ*cesum 27906  sigAlgebracsiga 27973  measurescmeas 28032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-esum 27907  df-meas 28033
This theorem is referenced by:  measbasedom  28039  measfrge0  28040  measvnul  28043  measvun  28046  measinb  28058  measres  28059  measdivcstOLD  28061  measdivcst  28062  cntmeas  28063  volmeas  28069  ddemeas  28074  dstrvprob  28276
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