Theorem measinb 26587

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 26563	. . . . . 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 26530	. . . . 5 |- ( ( S e. U. ran sigAlgebra /\ x e. S /\ A e. S ) -> ( x i^i A ) e. S )
7	3, 4, 5, 6	syl3anc 1218	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( x i^i A ) e. S )
8		measvxrge0 26571	. . . 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 2438	. . 3 |- ( x e. S |-> ( M ` ( x i^i A ) ) ) = ( x e. S |-> ( M ` ( x i^i A ) ) )
11	9, 10	fmptd 5862	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( x e. S |-> ( M ` ( x i^i A ) ) ) : S --> ( 0 [,] +oo ) )
12		eqidd 2439	. . 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 3540	. . . . . . 7 |- ( x = (/) -> ( x i^i A ) = ( (/) i^i A ) )
14		incom 3538	. . . . . . . 8 |- ( A i^i (/) ) = ( (/) i^i A )
15		in0 3658	. . . . . . . 8 |- ( A i^i (/) ) = (/)
16	14, 15	eqtr3i 2460	. . . . . . 7 |- ( (/) i^i A ) = (/)
17	13, 16	syl6eq 2486	. . . . . 6 |- ( x = (/) -> ( x i^i A ) = (/) )
18	17	fveq2d 5690	. . . . 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 26572	. . . . 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 2470	. . 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 26509	. . . 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 9378	. . . 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 5774	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 )
29		measinblem 26586	. . . . 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 2439	. . . . . 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 3540	. . . . . . . 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 5690	. . . . . 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 26512	. . . . . . 7 |- ( ( S e. U. ran sigAlgebra /\ z e. ~P S /\ z ~<_ om ) -> U. z e. S )
39	35, 36, 37, 38	syl3anc 1218	. . . . . 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 26530	. . . . . . . 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 1218	. . . . . . 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 26571	. . . . . . 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 5774	. . . . 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 2439	. . . . . . . 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 3540	. . . . . . . . . 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 5690	. . . . . . . 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 3864	. . . . . . . . . 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 3352	. . . . . . . 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 26530	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. S /\ A e. S ) -> ( y i^i A ) e. S )
58	55, 53, 56, 57	syl3anc 1218	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( y i^i A ) e. S )
59		measvxrge0 26571	. . . . . . . . 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 5774	. . . . . . 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 26446	. . . . . 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 2480	. . . 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 2794	. 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 26565	. . 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 1369   e. wcel 1756   A.wral 2710   i^i cin 3322   C_ wss 3323   (/)c0 3632   ~Pcpw 3855   U.cuni 4086  Disj wdisj 4257   class class class wbr 4287   e. cmpt 4345   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   omcom 6471   ~<_ cdom 7300   RRcr 9273   0cc0 9274   +oocpnf 9407   [,]cicc 11295  Σ^*cesum 26435  sigAlgebracsiga 26502  measurescmeas 26561

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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-ac2 8624  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-ac 8278  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-ordt 14431  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-ps 15362  df-tsr 15363  df-mnd 15407  df-plusf 15408  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-cntz 15826  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-subrg 16841  df-abv 16880  df-lmod 16928  df-scaf 16929  df-sra 17230  df-rgmod 17231  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-tmd 19618  df-tgp 19619  df-tsms 19672  df-trg 19709  df-xms 19870  df-ms 19871  df-tms 19872  df-nm 20150  df-ngp 20151  df-nrg 20153  df-nlm 20154  df-ii 20428  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-esum 26436  df-siga 26503  df-meas 26562

This theorem is referenced by:  measinb2  26589  totprobd  26761  probmeasb  26765