Theorem measinb 28429

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 751	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> M e. (measures ` S ) )
2		measbase 28405	. . . . . 6 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
3	2	ad2antrr 723	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> S e. U. ran sigAlgebra )
4		simpr 459	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> x e. S )
5		simplr 753	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> A e. S )
6		inelsiga 28365	. . . . 5 |- ( ( S e. U. ran sigAlgebra /\ x e. S /\ A e. S ) -> ( x i^i A ) e. S )
7	3, 4, 5, 6	syl3anc 1226	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x e. S ) -> ( x i^i A ) e. S )
8		measvxrge0 28413	. . . 4 |- ( ( M e. (measures ` S ) /\ ( x i^i A ) e. S ) -> ( M ` ( x i^i A ) ) e. ( 0 [,] +oo ) )
9	1, 7, 8	syl2anc 659	. . 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 6031	. 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 3679	. . . . . . 7 |- ( x = (/) -> ( x i^i A ) = ( (/) i^i A ) )
14		incom 3677	. . . . . . . 8 |- ( A i^i (/) ) = ( (/) i^i A )
15		in0 3810	. . . . . . . 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 5852	. . . . 5 |- ( x = (/) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
19	18	adantl 464	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. S ) /\ x = (/) ) -> ( M ` ( x i^i A ) ) = ( M ` (/) ) )
20		measvnul 28414	. . . . 5 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
21	20	ad2antrr 723	. . . 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 463	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. S ) -> S e. U. ran sigAlgebra )
24		0elsiga 28344	. . . 4 |- ( S e. U. ran sigAlgebra -> (/) e. S )
25	23, 24	syl 16	. . 3 |- ( ( M e. (measures ` S ) /\ A e. S ) -> (/) e. S )
26		0red 9586	. . 3 |- ( ( M e. (measures ` S ) /\ A e. S ) -> 0 e. RR )
27	12, 22, 25, 26	fvmptd 5936	. 2 |- ( ( M e. (measures ` S ) /\ A e. S ) -> ( ( x e. S |-> ( M ` ( x i^i A ) ) ) ` (/) ) = 0 )
28		measinblem 28428	. . . . 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 ) ) )
29		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 ) ) ) )
30		ineq1 3679	. . . . . . . 8 |- ( x = U. z -> ( x i^i A ) = ( U. z i^i A ) )
31	30	adantl 464	. . . . . . 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 ) )
32	31	fveq2d 5852	. . . . . 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 ) ) )
33		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 ) )
34	33, 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 )
35		simplr 753	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> z e. ~P S )
36		simprl 754	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> z ~<_ om )
37		sigaclcu 28347	. . . . . . 7 |- ( ( S e. U. ran sigAlgebra /\ z e. ~P S /\ z ~<_ om ) -> U. z e. S )
38	34, 35, 36, 37	syl3anc 1226	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> U. z e. S )
39		simpllr 758	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ ( z ~<_ om /\ Disj y e. z y ) ) -> A e. S )
40		inelsiga 28365	. . . . . . . 8 |- ( ( S e. U. ran sigAlgebra /\ U. z e. S /\ A e. S ) -> ( U. z i^i A ) e. S )
41	34, 38, 39, 40	syl3anc 1226	. . . . . . 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 )
42		measvxrge0 28413	. . . . . . 7 |- ( ( M e. (measures ` S ) /\ ( U. z i^i A ) e. S ) -> ( M ` ( U. z i^i A ) ) e. ( 0 [,] +oo ) )
43	33, 41, 42	syl2anc 659	. . . . . 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 ) )
44	29, 32, 38, 43	fvmptd 5936	. . . . 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 ) ) )
45		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 ) ) ) )
46		ineq1 3679	. . . . . . . . . 10 |- ( x = y -> ( x i^i A ) = ( y i^i A ) )
47	46	adantl 464	. . . . . . . . 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 ) )
48	47	fveq2d 5852	. . . . . . . 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 ) ) )
49		elpwi 4008	. . . . . . . . . 10 |- ( z e. ~P S -> z C_ S )
50	49	ad2antlr 724	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> z C_ S )
51		simpr 459	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. z )
52	50, 51	sseldd 3490	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> y e. S )
53		simplll 757	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> M e. (measures ` S ) )
54	53, 2	syl 16	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> S e. U. ran sigAlgebra )
55		simpllr 758	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> A e. S )
56		inelsiga 28365	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. S /\ A e. S ) -> ( y i^i A ) e. S )
57	54, 52, 55, 56	syl3anc 1226	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( y i^i A ) e. S )
58		measvxrge0 28413	. . . . . . . . 9 |- ( ( M e. (measures ` S ) /\ ( y i^i A ) e. S ) -> ( M ` ( y i^i A ) ) e. ( 0 [,] +oo ) )
59	53, 57, 58	syl2anc 659	. . . . . . . 8 |- ( ( ( ( M e. (measures ` S ) /\ A e. S ) /\ z e. ~P S ) /\ y e. z ) -> ( M ` ( y i^i A ) ) e. ( 0 [,] +oo ) )
60	45, 48, 52, 59	fvmptd 5936	. . . . . . 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 ) ) )
61	60	esumeq2dv 28267	. . . . . 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 ) ) )
62	61	adantr 463	. . . . 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 ) ) )
63	28, 44, 62	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 ) )
64	63	ex 432	. . 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 ) ) )
65	64	ralrimiva 2868	. 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 ) ) )
66		ismeas 28407	. . 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 ) ) ) ) )
67	23, 66	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 ) ) ) ) )
68	11, 27, 65, 67	mpbir3and 1177	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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   i^i cin 3460   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235  Disj wdisj 4410   class class class wbr 4439   |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   omcom 6673   ~<_ cdom 7507   RRcr 9480   0cc0 9481   +oocpnf 9614   [,]cicc 11535  Σ^*cesum 28256  sigAlgebracsiga 28337  measurescmeas 28403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-ac2 8834  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-ac 8488  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-ordt 14990  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-ps 16029  df-tsr 16030  df-plusf 16070  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-subrg 17622  df-abv 17661  df-lmod 17709  df-scaf 17710  df-sra 18013  df-rgmod 18014  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-tmd 20737  df-tgp 20738  df-tsms 20791  df-trg 20828  df-xms 20989  df-ms 20990  df-tms 20991  df-nm 21269  df-ngp 21270  df-nrg 21272  df-nlm 21273  df-ii 21547  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-esum 28257  df-siga 28338  df-meas 28404

This theorem is referenced by:  measinb2  28431  totprobd  28629  probmeasb  28633