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Theorem measres 28366
Description: Building a measure restricted to a smaller sigma algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measres  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M  |`  T )  e.  (measures `  T ) )

Proof of Theorem measres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 997 . 2  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  T  e.  U.
ran sigAlgebra )
2 measfrge0 28347 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
323ad2ant1 1017 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  M : S
--> ( 0 [,] +oo ) )
4 simp3 998 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  T  C_  S
)
53, 4fssresd 5758 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M  |`  T ) : T --> ( 0 [,] +oo ) )
6 0elsiga 28287 . . . . 5  |-  ( T  e.  U. ran sigAlgebra  ->  (/)  e.  T
)
7 fvres 5886 . . . . 5  |-  ( (/)  e.  T  ->  ( ( M  |`  T ) `  (/) )  =  ( M `  (/) ) )
81, 6, 73syl 20 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) `  (/) )  =  ( M `
 (/) ) )
9 measvnul 28350 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
1093ad2ant1 1017 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M `  (/) )  =  0 )
118, 10eqtrd 2498 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) `  (/) )  =  0 )
12 simp11 1026 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  M  e.  (measures `  S ) )
13 simp13 1028 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  T  C_  S
)
14 simp2 997 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  x  e.  ~P T )
15 sspwb 4705 . . . . . . . . 9  |-  ( T 
C_  S  <->  ~P T  C_ 
~P S )
16 ssel2 3494 . . . . . . . . 9  |-  ( ( ~P T  C_  ~P S  /\  x  e.  ~P T )  ->  x  e.  ~P S )
1715, 16sylanb 472 . . . . . . . 8  |-  ( ( T  C_  S  /\  x  e.  ~P T
)  ->  x  e.  ~P S )
1813, 14, 17syl2anc 661 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  x  e.  ~P S )
19 simp3 998 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  ( x  ~<_  om 
/\ Disj  y  e.  x  y ) )
20 measvun 28353 . . . . . . 7  |-  ( ( M  e.  (measures `  S
)  /\  x  e.  ~P S  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) )
2112, 18, 19, 20syl3anc 1228 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  ( M `  U. x )  = Σ* y  e.  x ( M `  y ) )
2213ad2ant1 1017 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  T  e.  U. ran sigAlgebra )
23 simp3l 1024 . . . . . . . 8  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  x  ~<_  om )
24 sigaclcu 28290 . . . . . . . 8  |-  ( ( T  e.  U. ran sigAlgebra  /\  x  e.  ~P T  /\  x  ~<_  om )  ->  U. x  e.  T
)
2522, 14, 23, 24syl3anc 1228 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  U. x  e.  T
)
26 fvres 5886 . . . . . . 7  |-  ( U. x  e.  T  ->  ( ( M  |`  T ) `
 U. x )  =  ( M `  U. x ) )
2725, 26syl 16 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  ( ( M  |`  T ) `  U. x )  =  ( M `  U. x
) )
28 elpwi 4024 . . . . . . . . . . 11  |-  ( x  e.  ~P T  ->  x  C_  T )
2928sselda 3499 . . . . . . . . . 10  |-  ( ( x  e.  ~P T  /\  y  e.  x
)  ->  y  e.  T )
3029adantll 713 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T
)  /\  y  e.  x )  ->  y  e.  T )
31 fvres 5886 . . . . . . . . 9  |-  ( y  e.  T  ->  (
( M  |`  T ) `
 y )  =  ( M `  y
) )
3230, 31syl 16 . . . . . . . 8  |-  ( ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T
)  /\  y  e.  x )  ->  (
( M  |`  T ) `
 y )  =  ( M `  y
) )
3332esumeq2dv 28212 . . . . . . 7  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T )  -> Σ* y  e.  x ( ( M  |`  T ) `  y
)  = Σ* y  e.  x
( M `  y
) )
34333adant3 1016 . . . . . 6  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  -> Σ* y  e.  x ( ( M  |`  T ) `
 y )  = Σ* y  e.  x ( M `
 y ) )
3521, 27, 343eqtr4d 2508 . . . . 5  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T  /\  (
x  ~<_  om  /\ Disj  y  e.  x  y ) )  ->  ( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) )
36353expia 1198 . . . 4  |-  ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T )  -> 
( ( x  ~<_  om 
/\ Disj  y  e.  x  y )  ->  ( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) ) )
3736ralrimiva 2871 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  A. x  e.  ~P  T ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) ) )
385, 11, 373jca 1176 . 2  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) : T --> ( 0 [,] +oo )  /\  (
( M  |`  T ) `
 (/) )  =  0  /\  A. x  e. 
~P  T ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) ) ) )
39 ismeas 28343 . . 3  |-  ( T  e.  U. ran sigAlgebra  ->  (
( M  |`  T )  e.  (measures `  T
)  <->  ( ( M  |`  T ) : T --> ( 0 [,] +oo )  /\  ( ( M  |`  T ) `  (/) )  =  0  /\  A. x  e.  ~P  T ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) ) ) ) )
4039biimprd 223 . 2  |-  ( T  e.  U. ran sigAlgebra  ->  (
( ( M  |`  T ) : T --> ( 0 [,] +oo )  /\  ( ( M  |`  T ) `  (/) )  =  0  /\  A. x  e.  ~P  T ( ( x  ~<_  om  /\ Disj  y  e.  x  y )  -> 
( ( M  |`  T ) `  U. x )  = Σ* y  e.  x ( ( M  |`  T ) `  y
) ) )  -> 
( M  |`  T )  e.  (measures `  T
) ) )
411, 38, 40sylc 60 1  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M  |`  T )  e.  (measures `  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251  Disj wdisj 4427   class class class wbr 4456   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   omcom 6699    ~<_ cdom 7533   0cc0 9509   +oocpnf 9642   [,]cicc 11557  Σ*cesum 28201  sigAlgebracsiga 28280  measurescmeas 28339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-esum 28202  df-siga 28281  df-meas 28340
This theorem is referenced by:  measinb2  28367
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