Theorem measinb 24528

Description: Building a measure restricted to the intersection with a given set. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
measinb	|- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) e. (measures ` S ) )

Distinct variable groups:   x, A   x, S   x, M

Proof of Theorem measinb

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 731	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> M e. (measures ` S ) )
2		measbase 24504	. . . . . 6 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
3	2	ad2antrr 707	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> S e. U. ran sigAlgebra )
4		simpr 448	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> x e. S )
5		simplr 732	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> A e. S )
6		inelsiga 24471	. . . . 5 |- ( ( S e. U. ran sigAlgebra /\ x e. S /\ A e. S ) -> ( x i^i A ) e. S )
7	3, 4, 5, 6	syl3anc 1184	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( x i^i A ) e. S )
8		measvxrge0 24512	. . . 4 |- ( ( M e. (measures ` S ) /\ ( x i^i A ) e. S ) -> ( M ` ( x i^i A ) ) e. ( 0 [,] +oo ) )
9	1, 7, 8	syl2anc 643	. . 3 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( M ` ( x i^i A ) ) e. ( 0 [,] +oo ) )
10		eqid 2404	. . 3 |- ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) )
11	9, 10	fmptd 5852	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) : S --> ( 0 [,] +oo ) )
12		eqidd 2405	. . 3 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) ) )
13		ineq1 3495	. . . . . . 7 |- ( x = (/) -> ( x i^i A ) = ( (/) i^i A ) )
14		incom 3493	. . . . . . . 8 |- ( A i^i (/) ) = ( (/) i^i A )
15		in0 3613	. . . . . . . 8 |- ( A i^i (/) ) = (/)
16	14, 15	eqtr3i 2426	. . . . . . 7 |- ( (/) i^i A ) = (/)
17	13, 16	syl6eq 2452	. . . . . 6 |- ( x = (/) -> ( x i^i A ) = (/) )
18	17	fveq2d 5691	. . . . 5 |- ( x = (/) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
19	18	adantl 453	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
20		measvnul 24513	. . . . 5 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
21	20	ad2antrr 707	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` (/) ) = 0 )
22	19, 21	eqtrd 2436	. . 3 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` ( x i^i A ) ) = 0 )
23	2	adantr 452	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. S ) -> S e. U. ran sigAlgebra )
24		0elsiga 24450	. . . 4 |- ( S e. U. ran sigAlgebra -> (/) e. S )
25	23, 24	syl 16	. . 3 |- ( ( M e. (measures ` S ) /\ A e. S ) -> (/) e. S )
26		0re 9047	. . . 4 |- 0 e. RR
27	26	a1i 11	. . 3 |- ( ( M e. (measures ` S ) /\ A e. S ) -> 0 e. RR )
28	12, 22, 25, 27	fvmptd 5769	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 )
29		measinblem 24527	. . . . 5 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( M ` ( U. z i^i A ) ) = Σ* y e. z ( M ` ( y i^i A ) ) )
30		eqidd 2405	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) ) )
31		ineq1 3495	. . . . . . . 8 |- ( x = U. z -> ( x i^i A ) = ( U. z i^i A ) )
32	31	adantl 453	. . . . . . 7 |- ( ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) /\ x = U. z ) -> ( x i^i A ) = ( U. z i^i A ) )
33	32	fveq2d 5691	. . . . . 6 |- ( ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) /\ x = U. z ) -> ( M ` ( x i^i A ) ) = ( M ` ( U. z i^i A ) ) )
34		simplll 735	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> M e. (measures ` S ) )
35	34, 2	syl 16	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> S e. U. ran sigAlgebra )
36		simplr 732	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> z e. ~P S )
37		simprl 733	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> z ~<_ om )
38		sigaclcu 24453	. . . . . . 7 |- ( ( S e. U. ran sigAlgebra /\ z e. ~P S /\ z ~<_ om ) -> U. z e. S )
39	35, 36, 37, 38	syl3anc 1184	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> U. z e. S )
40		simpllr 736	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> A e. S )
41		inelsiga 24471	. . . . . . . 8 |- ( ( S e. U. ran sigAlgebra /\ U. z e. S /\ A e. S ) -> ( U. z i^i A ) e. S )
42	35, 39, 40, 41	syl3anc 1184	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( U. z i^i A ) e. S )
43		measvxrge0 24512	. . . . . . 7 |- ( ( M e. (measures ` S ) /\ ( U. z i^i A ) e. S ) -> ( M ` ( U. z i^i A ) ) e. ( 0 [,] +oo ) )
44	34, 42, 43	syl2anc 643	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( M ` ( U. z i^i A ) ) e. ( 0 [,] +oo ) )
45	30, 33, 39, 44	fvmptd 5769	. . . . 5 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = ( M ` ( U. z i^i A ) ) )
46		eqidd 2405	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) ) )
47		ineq1 3495	. . . . . . . . . 10 |- ( x = y -> ( x i^i A ) = ( y i^i A ) )
48	47	adantl 453	. . . . . . . . 9 |- ( ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) /\ x = y ) -> ( x i^i A ) = ( y i^i A ) )
49	48	fveq2d 5691	. . . . . . . 8 |- ( ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) /\ x = y ) -> ( M ` ( x i^i A ) ) = ( M ` ( y i^i A ) ) )
50		elpwi 3767	. . . . . . . . . 10 |- ( z e. ~P S -> z C_ S )
51	50	ad2antlr 708	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> z C_ S )
52		simpr 448	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. z )
53	51, 52	sseldd 3309	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. S )
54		simplll 735	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> M e. (measures ` S ) )
55	54, 2	syl 16	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> S e. U. ran sigAlgebra )
56		simpllr 736	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> A e. S )
57		inelsiga 24471	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. S /\ A e. S ) -> ( y i^i A ) e. S )
58	55, 53, 56, 57	syl3anc 1184	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( y i^i A ) e. S )
59		measvxrge0 24512	. . . . . . . . 9 |- ( ( M e. (measures ` S ) /\ ( y i^i A ) e. S ) -> ( M ` ( y i^i A ) ) e. ( 0 [,] +oo ) )
60	54, 58, 59	syl2anc 643	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( M ` ( y i^i A ) ) e. ( 0 [,] +oo ) )
61	46, 49, 53, 60	fvmptd 5769	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) = ( M ` ( y i^i A ) ) )
62	61	esumeq2dv 24388	. . . . . 6 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) -> Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) = Σ* y e. z ( M ` ( y i^i A ) ) )
63	62	adantr 452	. . . . 5 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) = Σ* y e. z ( M ` ( y i^i A ) ) )
64	29, 45, 63	3eqtr4d 2446	. . . 4 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) )
65	64	ex 424	. . 3 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) -> ( ( z ~<_ om /\ Disj y e. z y ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) ) )
66	65	ralrimiva 2749	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> A. z e. ~P S ( ( z ~<_ om /\ Disj y e. z y ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) ) )
67		ismeas 24506	. . 3 |- ( S e. U. ran sigAlgebra -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) e. (measures ` S ) <-> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 /\ A. z e. ~P S ( ( z ~<_ om /\ Disj y e. z y ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) ) ) ) )
68	23, 67	syl 16	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) e. (measures ` S ) <-> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 /\ A. z e. ~P S ( ( z ~<_ om /\ Disj y e. z y ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` U. z ) = Σ* y e. z ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` y ) ) ) ) )
69	11, 28, 66, 68	mpbir3and 1137	1 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) e. (measures ` S ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   i^i cin 3279   C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U.cuni 3975  Disj wdisj 4142   class class class wbr 4172   e. cmpt 4226   omcom 4804   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   ~<_ cdom 7066   RRcr 8945   0cc0 8946   +oocpnf 9073   [,]cicc 10875  Σ^*cesum 24377  sigAlgebracsiga 24443  measurescmeas 24502

This theorem is referenced by:  measinb2  24530  totprobd  24637  probmeasb  24641

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-tmd 18055  df-tgp 18056  df-tsms 18109  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-esum 24378  df-siga 24444  df-meas 24503