Theorem measres 30285

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.

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> T e. U. ran sigAlgebra )
2		measfrge0 30266	. . . . 5 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )
3	2	3ad2ant1 1082	. . . 4 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> M : S --> ( 0 [,] +oo ) )
4		simp3 1063	. . . 4 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> T C_ S )
5	3, 4	fssresd 6071	. . 3 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> ( M |` T ) : T --> ( 0 [,] +oo ) )
6		0elsiga 30177	. . . . 5 |- ( T e. U. ran sigAlgebra -> (/) e. T )
7		fvres 6207	. . . . 5 |- ( (/) e. T -> ( ( M |` T ) ` (/) ) = ( M ` (/) ) )
8	1, 6, 7	3syl 18	. . . 4 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> ( ( M |` T ) ` (/) ) = ( M ` (/) ) )
9		measvnul 30269	. . . . 5 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
10	9	3ad2ant1 1082	. . . 4 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> ( M ` (/) ) = 0 )
11	8, 10	eqtrd 2656	. . 3 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> ( ( M |` T ) ` (/) ) = 0 )
12		simp11 1091	. . . . . . 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 1093	. . . . . . . 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 1062	. . . . . . . 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 4917	. . . . . . . . 9 |- ( T C_ S <-> ~P T C_ ~P S )
16		ssel2 3598	. . . . . . . . 9 |- ( ( ~P T C_ ~P S /\ x e. ~P T ) -> x e. ~P S )
17	15, 16	sylanb 489	. . . . . . . 8 |- ( ( T C_ S /\ x e. ~P T ) -> x e. ~P S )
18	13, 14, 17	syl2anc 693	. . . . . . 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 1063	. . . . . . 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 30272	. . . . . . 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 ) )
21	12, 18, 19, 20	syl3anc 1326	. . . . . 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 ) )
22	1	3ad2ant1 1082	. . . . . . . 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 1089	. . . . . . . 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 30180	. . . . . . . 8 |- ( ( T e. U. ran sigAlgebra /\ x e. ~P T /\ x ~<_ om ) -> U. x e. T )
25	22, 14, 23, 24	syl3anc 1326	. . . . . . 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 6207	. . . . . . 7 |- ( U. x e. T -> ( ( M |` T ) ` U. x ) = ( M ` U. x ) )
27	25, 26	syl 17	. . . . . 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 4168	. . . . . . . . . . 11 |- ( x e. ~P T -> x C_ T )
29	28	sselda 3603	. . . . . . . . . 10 |- ( ( x e. ~P T /\ y e. x ) -> y e. T )
30	29	adantll 750	. . . . . . . . 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 6207	. . . . . . . . 9 |- ( y e. T -> ( ( M |` T ) ` y ) = ( M ` y ) )
32	30, 31	syl 17	. . . . . . . 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 ) )
33	32	esumeq2dv 30100	. . . . . . 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 ) )
34	33	3adant3 1081	. . . . . 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 ) )
35	21, 27, 34	3eqtr4d 2666	. . . . 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 ) )
36	35	3expia 1267	. . . 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 ) ) )
37	36	ralrimiva 2966	. . 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 ) ) )
38	5, 11, 37	3jca 1242	. 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 30262	. . 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 ) ) ) ) )
40	39	biimprd 238	. 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 ) ) )
41	1, 38, 40	sylc 65	1 |- ( ( M e. (measures ` S ) /\ T e. U. ran sigAlgebra /\ T C_ S ) -> ( M |` T ) e. (measures ` T ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  Σ^*cesum 30089  sigAlgebracsiga 30170  measurescmeas 30258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-esum 30090  df-siga 30171  df-meas 30259

This theorem is referenced by:  measinb2  30286