Theorem measinb 26781

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 753	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> M e. (measures ` S ) )
2		measbase 26757	. . . . . 6 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
3	2	ad2antrr 725	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> S e. U. ran sigAlgebra )
4		simpr 461	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> x e. S )
5		simplr 754	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> A e. S )
6		inelsiga 26724	. . . . 5 |- ( ( S e. U. ran sigAlgebra /\ x e. S /\ A e. S ) -> ( x i^i A ) e. S )
7	3, 4, 5, 6	syl3anc 1219	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( x i^i A ) e. S )
8		measvxrge0 26765	. . . 4 |- ( ( M e. (measures ` S ) /\ ( x i^i A ) e. S ) -> ( M ` ( x i^i A ) ) e. ( 0 [,] +oo ) )
9	1, 7, 8	syl2anc 661	. . 3 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( M ` ( x i^i A ) ) e. ( 0 [,] +oo ) )
10		eqid 2454	. . 3 |- ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) )
11	9, 10	fmptd 5977	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) : S --> ( 0 [,] +oo ) )
12		eqidd 2455	. . 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 3654	. . . . . . 7 |- ( x = (/) -> ( x i^i A ) = ( (/) i^i A ) )
14		incom 3652	. . . . . . . 8 |- ( A i^i (/) ) = ( (/) i^i A )
15		in0 3772	. . . . . . . 8 |- ( A i^i (/) ) = (/)
16	14, 15	eqtr3i 2485	. . . . . . 7 |- ( (/) i^i A ) = (/)
17	13, 16	syl6eq 2511	. . . . . 6 |- ( x = (/) -> ( x i^i A ) = (/) )
18	17	fveq2d 5804	. . . . 5 |- ( x = (/) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
19	18	adantl 466	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
20		measvnul 26766	. . . . 5 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
21	20	ad2antrr 725	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` (/) ) = 0 )
22	19, 21	eqtrd 2495	. . 3 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` ( x i^i A ) ) = 0 )
23	2	adantr 465	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. S ) -> S e. U. ran sigAlgebra )
24		0elsiga 26703	. . . 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 9498	. . . 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 5889	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 )
29		measinblem 26780	. . . . 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 2455	. . . . . 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 3654	. . . . . . . 8 |- ( x = U. z -> ( x i^i A ) = ( U. z i^i A ) )
32	31	adantl 466	. . . . . . 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 5804	. . . . . 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 757	. . . . . . . 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 754	. . . . . . 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 755	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> z ~<_ om )
38		sigaclcu 26706	. . . . . . 7 |- ( ( S e. U. ran sigAlgebra /\ z e. ~P S /\ z ~<_ om ) -> U. z e. S )
39	35, 36, 37, 38	syl3anc 1219	. . . . . 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 758	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> A e. S )
41		inelsiga 26724	. . . . . . . 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 1219	. . . . . . 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 26765	. . . . . . 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 661	. . . . . 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 5889	. . . . 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 2455	. . . . . . . 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 3654	. . . . . . . . . 10 |- ( x = y -> ( x i^i A ) = ( y i^i A ) )
48	47	adantl 466	. . . . . . . . 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 5804	. . . . . . . 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 3978	. . . . . . . . . 10 |- ( z e. ~P S -> z C_ S )
51	50	ad2antlr 726	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> z C_ S )
52		simpr 461	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. z )
53	51, 52	sseldd 3466	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. S )
54		simplll 757	. . . . . . . . 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 758	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> A e. S )
57		inelsiga 26724	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. S /\ A e. S ) -> ( y i^i A ) e. S )
58	55, 53, 56, 57	syl3anc 1219	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( y i^i A ) e. S )
59		measvxrge0 26765	. . . . . . . . 9 |- ( ( M e. (measures ` S ) /\ ( y i^i A ) e. S ) -> ( M ` ( y i^i A ) ) e. ( 0 [,] +oo ) )
60	54, 58, 59	syl2anc 661	. . . . . . . 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 5889	. . . . . . 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 26640	. . . . . 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 465	. . . . 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 2505	. . . 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 434	. . 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 2830	. 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 26759	. . 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 1171	1 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) e. (measures ` S ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2799   i^i cin 3436   C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200  Disj wdisj 4371   class class class wbr 4401   |-> cmpt 4459   ran crn 4950   -->wf 5523   ` cfv 5527  (class class class)co 6201   omcom 6587   ~<_ cdom 7419   RRcr 9393   0cc0 9394   +oocpnf 9527   [,]cicc 11415  Σ^*cesum 26629  sigAlgebracsiga 26696  measurescmeas 26755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-ac2 8744  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473  ax-mulf 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-fi 7773  df-sup 7803  df-oi 7836  df-card 8221  df-acn 8224  df-ac 8398  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ioo 11416  df-ioc 11417  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-fac 12170  df-bc 12197  df-hash 12222  df-shft 12675  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-limsup 13068  df-clim 13085  df-rlim 13086  df-sum 13283  df-ef 13472  df-sin 13474  df-cos 13475  df-pi 13477  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-hom 14382  df-cco 14383  df-rest 14481  df-topn 14482  df-0g 14500  df-gsum 14501  df-topgen 14502  df-pt 14503  df-prds 14506  df-ordt 14559  df-xrs 14560  df-qtop 14565  df-imas 14566  df-xps 14568  df-mre 14644  df-mrc 14645  df-acs 14647  df-ps 15490  df-tsr 15491  df-mnd 15535  df-plusf 15536  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-subrg 16987  df-abv 17026  df-lmod 17074  df-scaf 17075  df-sra 17377  df-rgmod 17378  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-fg 17941  df-cnfld 17945  df-top 18636  df-bases 18638  df-topon 18639  df-topsp 18640  df-cld 18756  df-ntr 18757  df-cls 18758  df-nei 18835  df-lp 18873  df-perf 18874  df-cn 18964  df-cnp 18965  df-haus 19052  df-tx 19268  df-hmeo 19461  df-fil 19552  df-fm 19644  df-flim 19645  df-flf 19646  df-tmd 19776  df-tgp 19777  df-tsms 19830  df-trg 19867  df-xms 20028  df-ms 20029  df-tms 20030  df-nm 20308  df-ngp 20309  df-nrg 20311  df-nlm 20312  df-ii 20586  df-cncf 20587  df-limc 21475  df-dv 21476  df-log 22142  df-esum 26630  df-siga 26697  df-meas 26756

This theorem is referenced by:  measinb2  26783  totprobd  26954  probmeasb  26958