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Theorem measres 28018
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 27999 . . . . 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
)
5 fssres 5757 . . . 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 27939 . . . . 5  |-  ( T  e.  U. ran sigAlgebra  ->  (/)  e.  T
)
8 fvres 5886 . . . . 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 28002 . . . . 5  |-  ( M  e.  (measures `  S
)  ->  ( M `  (/) )  =  0 )
11103ad2ant1 1017 . . . 4  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( M `  (/) )  =  0 )
129, 11eqtrd 2508 . . 3  |-  ( ( M  e.  (measures `  S
)  /\  T  e.  U.
ran sigAlgebra  /\  T  C_  S
)  ->  ( ( M  |`  T ) `  (/) )  =  0 )
13 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 ) )
14 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
)
15 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 )
16 sspwb 4702 . . . . . . . . 9  |-  ( T 
C_  S  <->  ~P T  C_ 
~P S )
17 ssel2 3504 . . . . . . . . 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 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 ) )
21 measvun 28005 . . . . . . 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 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 ) )
2313ad2ant1 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 )
24 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 )
25 sigaclcu 27942 . . . . . . . 8  |-  ( ( T  e.  U. ran sigAlgebra  /\  x  e.  ~P T  /\  x  ~<_  om )  ->  U. x  e.  T
)
2623, 15, 24, 25syl3anc 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
)
27 fvres 5886 . . . . . . 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 4025 . . . . . . . . . . 11  |-  ( x  e.  ~P T  ->  x  C_  T )
3029sselda 3509 . . . . . . . . . 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 5886 . . . . . . . . 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 27876 . . . . . . 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 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 ) )
3622, 28, 353eqtr4d 2518 . . . . 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 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
) ) )
3837ralrimiva 2881 . . 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 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
) ) ) )
40 ismeas 27995 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   U.cuni 4251  Disj wdisj 4423   class class class wbr 4453   ran crn 5006    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   omcom 6695    ~<_ cdom 7526   0cc0 9504   +oocpnf 9637   [,]cicc 11544  Σ*cesum 27865  sigAlgebracsiga 27932  measurescmeas 27991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-esum 27866  df-siga 27933  df-meas 27992
This theorem is referenced by:  measinb2  28019
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