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Theorem ismeas 26618
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 2986 . . 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 988 . . 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 6121 . . . 4  |-  ( 0 [,] +oo )  e. 
_V
5 fex2 6537 . . . . . 6  |-  ( ( M : S --> ( 0 [,] +oo )  /\  S  e.  U. ran sigAlgebra  /\  (
0 [,] +oo )  e.  _V )  ->  M  e.  _V )
653expb 1188 . . . . 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 26615 . . . 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 2980 . . . . . 6  |-  s  e. 
_V
12 mapex 7225 . . . . . 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 988 . . . . . 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 3429 . . . . 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 4442 . . . 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 5553 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( m : s --> ( 0 [,] +oo ) 
<->  M : S --> ( 0 [,] +oo ) ) )
20 fveq1 5695 . . . . . . 7  |-  ( m  =  M  ->  (
m `  (/) )  =  ( M `  (/) ) )
2120eqeq1d 2451 . . . . . 6  |-  ( m  =  M  ->  (
( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2221adantl 466 . . . . 5  |-  ( ( s  =  S  /\  m  =  M )  ->  ( ( m `  (/) )  =  0  <->  ( M `  (/) )  =  0 ) )
2318pweqd 3870 . . . . . 6  |-  ( ( s  =  S  /\  m  =  M )  ->  ~P s  =  ~P S )
24 fveq1 5695 . . . . . . . . 9  |-  ( m  =  M  ->  (
m `  U. x )  =  ( M `  U. x ) )
25 fveq1 5695 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m `  y )  =  ( M `  y ) )
2625esumeq2sdv 26500 . . . . . . . . 9  |-  ( m  =  M  -> Σ* y  e.  x
( m `  y
)  = Σ* y  e.  x
( M `  y
) )
2724, 26eqeq12d 2457 . . . . . . . 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 2936 . . . . 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 1289 . . . 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 25975 . . 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 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   _Vcvv 2977   (/)c0 3642   ~Pcpw 3865   U.cuni 4096  Disj wdisj 4267   class class class wbr 4297   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096   omcom 6481    ~<_ cdom 7313   0cc0 9287   +oocpnf 9420   [,]cicc 11308  Σ*cesum 26488  sigAlgebracsiga 26555  measurescmeas 26614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-esum 26489  df-meas 26615
This theorem is referenced by:  measbasedom  26621  measfrge0  26622  measvnul  26625  measvun  26628  measinb  26640  measres  26641  measdivcstOLD  26643  measdivcst  26644  cntmeas  26645  volmeas  26652  ddemeas  26657  dstrvprob  26859
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