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Theorem measres 26655
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 989 . 2  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  T  e.  U.
ran sigAlgebra )
2 measfrge0 26636 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  M : S
--> ( 0 [,] +oo ) )
323ad2ant1 1009 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  M : S
--> ( 0 [,] +oo ) )
4 simp3 990 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  T  C_  S
)
5 fssres 5597 . . . 4  |-  ( ( M : S --> ( 0 [,] +oo )  /\  T  C_  S )  -> 
( M  |`  T ) : T --> ( 0 [,] +oo ) )
63, 4, 5syl2anc 661 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M  |`  T ) : T --> ( 0 [,] +oo ) )
7 0elsiga 26576 . . . . 5  |-  ( T  e.  U. ran sigAlgebra  ->  (/)  e.  T
)
8 fvres 5723 . . . . 5  |-  ( (/)  e.  T  ->  ( ( M  |`  T ) `  (/) )  =  ( M `  (/) ) )
91, 7, 83syl 20 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) `  (/) )  =  ( M `
 (/) ) )
10 measvnul 26639 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
11103ad2ant1 1009 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M `  (/) )  =  0 )
129, 11eqtrd 2475 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) `  (/) )  =  0 )
13 simp11 1018 . . . . . . 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 ) )
14 simp13 1020 . . . . . . . 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
)
15 simp2 989 . . . . . . . 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 )
16 sspwb 4560 . . . . . . . . 9  |-  ( T 
C_  S  <->  ~P T  C_ 
~P S )
17 ssel2 3370 . . . . . . . . 9  |-  ( ( ~P T  C_  ~P S  /\  x  e.  ~P T )  ->  x  e.  ~P S )
1816, 17sylanb 472 . . . . . . . 8  |-  ( ( T  C_  S  /\  x  e.  ~P T
)  ->  x  e.  ~P S )
1914, 15, 18syl2anc 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 )
20 simp3 990 . . . . . . 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 ) )
21 measvun 26642 . . . . . . 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 ) )
2213, 19, 20, 21syl3anc 1218 . . . . . 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 ) )
2313ad2ant1 1009 . . . . . . . 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 )
24 simp3l 1016 . . . . . . . 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 )
25 sigaclcu 26579 . . . . . . . 8  |-  ( ( T  e.  U. ran sigAlgebra  /\  x  e.  ~P T  /\  x  ~<_  om )  ->  U. x  e.  T
)
2623, 15, 24, 25syl3anc 1218 . . . . . . 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
)
27 fvres 5723 . . . . . . 7  |-  ( U. x  e.  T  ->  ( ( M  |`  T ) `
 U. x )  =  ( M `  U. x ) )
2826, 27syl 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
) )
29 elpwi 3888 . . . . . . . . . . 11  |-  ( x  e.  ~P T  ->  x  C_  T )
3029sselda 3375 . . . . . . . . . 10  |-  ( ( x  e.  ~P T  /\  y  e.  x
)  ->  y  e.  T )
3130adantll 713 . . . . . . . . 9  |-  ( ( ( ( M  e.  (measures `  S )  /\  T  e.  U. ran sigAlgebra  /\  T  C_  S )  /\  x  e.  ~P T
)  /\  y  e.  x )  ->  y  e.  T )
32 fvres 5723 . . . . . . . . 9  |-  ( y  e.  T  ->  (
( M  |`  T ) `
 y )  =  ( M `  y
) )
3331, 32syl 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
) )
3433esumeq2dv 26513 . . . . . . 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
) )
35343adant3 1008 . . . . . 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 ) )
3622, 28, 353eqtr4d 2485 . . . . 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
) )
37363expia 1189 . . . 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
) ) )
3837ralrimiva 2818 . . 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
) ) )
396, 12, 383jca 1168 . 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
) ) ) )
40 ismeas 26632 . . 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
) ) ) ) )
4140biimprd 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
) ) )
421, 39, 41sylc 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 965    = wceq 1369    e. wcel 1756   A.wral 2734    C_ wss 3347   (/)c0 3656   ~Pcpw 3879   U.cuni 4110  Disj wdisj 4281   class class class wbr 4311   ran crn 4860    |` cres 4861   -->wf 5433   ` cfv 5437  (class class class)co 6110   omcom 6495    ~<_ cdom 7327   0cc0 9301   +oocpnf 9434   [,]cicc 11322  Σ*cesum 26502  sigAlgebracsiga 26569  measurescmeas 26628
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
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 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-disj 4282  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fv 5445  df-ov 6113  df-esum 26503  df-siga 26570  df-meas 26629
This theorem is referenced by:  measinb2  26656
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