Theorem measdivcst 30288

Description: Division of a measure by a positive constant is a measure. (Contributed by Thierry Arnoux, 25-Dec-2016.) (Revised by Thierry Arnoux, 30-Jan-2017.)

Assertion

Ref	Expression
measdivcst	|- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) e. (measures ` S ) )

Proof of Theorem measdivcst

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

Step	Hyp	Ref	Expression
1		ofcfval3 30164	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. dom M |-> ( ( M ` x ) /𝑒 A ) ) )
2		measfrge0 30266	. . . . . 6 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )
3		fdm 6051	. . . . . 6 |- ( M : S --> ( 0 [,] +oo ) -> dom M = S )
4	2, 3	syl 17	. . . . 5 |- ( M e. (measures ` S ) -> dom M = S )
5	4	adantr 481	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> dom M = S )
6	5	mpteq1d 4738	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. dom M |-> ( ( M ` x ) /𝑒 A ) ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) )
7	1, 6	eqtrd 2656	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) )
8		measvxrge0 30268	. . . . . 6 |- ( ( M e. (measures ` S ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
9	8	adantlr 751	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
10		simplr 792	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> A e. RR+ )
11	9, 10	xrpxdivcld 29643	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( ( M ` x ) /𝑒 A ) e. ( 0 [,] +oo ) )
12		eqid 2622	. . . 4 |- ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) )
13	11, 12	fmptd 6385	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) )
14		measbase 30260	. . . . . . 7 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
15		0elsiga 30177	. . . . . . 7 |- ( S e. U. ran sigAlgebra -> (/) e. S )
16	14, 15	syl 17	. . . . . 6 |- ( M e. (measures ` S ) -> (/) e. S )
17	16	adantr 481	. . . . 5 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> (/) e. S )
18		ovex 6678	. . . . 5 |- ( ( M ` (/) ) /𝑒 A ) e. _V
19		fveq2 6191	. . . . . . 7 |- ( x = (/) -> ( M ` x ) = ( M ` (/) ) )
20	19	oveq1d 6665	. . . . . 6 |- ( x = (/) -> ( ( M ` x ) /𝑒 A ) = ( ( M ` (/) ) /𝑒 A ) )
21	20, 12	fvmptg 6280	. . . . 5 |- ( ( (/) e. S /\ ( ( M ` (/) ) /𝑒 A ) e. _V ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
22	17, 18, 21	sylancl 694	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
23		measvnul 30269	. . . . . 6 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
24	23	oveq1d 6665	. . . . 5 |- ( M e. (measures ` S ) -> ( ( M ` (/) ) /𝑒 A ) = ( 0 /𝑒 A ) )
25		xdiv0rp 29638	. . . . 5 |- ( A e. RR+ -> ( 0 /𝑒 A ) = 0 )
26	24, 25	sylan9eq 2676	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( M ` (/) ) /𝑒 A ) = 0 )
27	22, 26	eqtrd 2656	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 )
28		simpll 790	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> ( M e. (measures ` S ) /\ A e. RR+ ) )
29		simplr 792	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> y e. ~P S )
30		simprl 794	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
31		simprr 796	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> Disj z e. y z )
32		vex 3203	. . . . . . . . . 10 |- y e. _V
33	32	a1i 11	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> y e. _V )
34		simplll 798	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> M e. (measures ` S ) )
35		selpw 4165	. . . . . . . . . . . 12 |- ( y e. ~P S <-> y C_ S )
36		ssel2 3598	. . . . . . . . . . . 12 |- ( ( y C_ S /\ z e. y ) -> z e. S )
37	35, 36	sylanb 489	. . . . . . . . . . 11 |- ( ( y e. ~P S /\ z e. y ) -> z e. S )
38	37	adantll 750	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> z e. S )
39		measvxrge0 30268	. . . . . . . . . 10 |- ( ( M e. (measures ` S ) /\ z e. S ) -> ( M ` z ) e. ( 0 [,] +oo ) )
40	34, 38, 39	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> ( M ` z ) e. ( 0 [,] +oo ) )
41		simplr 792	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> A e. RR+ )
42	33, 40, 41	esumdivc 30145	. . . . . . . 8 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> (Σ* z e. y ( M ` z ) /𝑒 A ) = Σ* z e. y ( ( M ` z ) /𝑒 A ) )
43	42	3ad2antr1 1226	. . . . . . 7 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> (Σ* z e. y ( M ` z ) /𝑒 A ) = Σ* z e. y ( ( M ` z ) /𝑒 A ) )
44	14	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> S e. U. ran sigAlgebra )
45		simpr1 1067	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> y e. ~P S )
46		simpr2 1068	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
47		sigaclcu 30180	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. ~P S /\ y ~<_ om ) -> U. y e. S )
48	44, 45, 46, 47	syl3anc 1326	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> U. y e. S )
49		fveq2 6191	. . . . . . . . . . 11 |- ( x = U. y -> ( M ` x ) = ( M ` U. y ) )
50	49	oveq1d 6665	. . . . . . . . . 10 |- ( x = U. y -> ( ( M ` x ) /𝑒 A ) = ( ( M ` U. y ) /𝑒 A ) )
51		ovex 6678	. . . . . . . . . 10 |- ( ( M ` x ) /𝑒 A ) e. _V
52	50, 12, 51	fvmpt3i 6287	. . . . . . . . 9 |- ( U. y e. S -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = ( ( M ` U. y ) /𝑒 A ) )
53	48, 52	syl 17	. . . . . . . 8 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = ( ( M ` U. y ) /𝑒 A ) )
54		simpll 790	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> M e. (measures ` S ) )
55		simpr3 1069	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> Disj z e. y z )
56		measvun 30272	. . . . . . . . . 10 |- ( ( M e. (measures ` S ) /\ y e. ~P S /\ ( y ~<_ om /\ Disj z e. y z ) ) -> ( M ` U. y ) = Σ* z e. y ( M ` z ) )
57	54, 45, 46, 55, 56	syl112anc 1330	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> ( M ` U. y ) = Σ* z e. y ( M ` z ) )
58	57	oveq1d 6665	. . . . . . . 8 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> ( ( M ` U. y ) /𝑒 A ) = (Σ* z e. y ( M ` z ) /𝑒 A ) )
59	53, 58	eqtrd 2656	. . . . . . 7 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = (Σ* z e. y ( M ` z ) /𝑒 A ) )
60		fveq2 6191	. . . . . . . . . . . 12 |- ( x = z -> ( M ` x ) = ( M ` z ) )
61	60	oveq1d 6665	. . . . . . . . . . 11 |- ( x = z -> ( ( M ` x ) /𝑒 A ) = ( ( M ` z ) /𝑒 A ) )
62	61, 12, 51	fvmpt3i 6287	. . . . . . . . . 10 |- ( z e. S -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = ( ( M ` z ) /𝑒 A ) )
63	37, 62	syl 17	. . . . . . . . 9 |- ( ( y e. ~P S /\ z e. y ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = ( ( M ` z ) /𝑒 A ) )
64	63	esumeq2dv 30100	. . . . . . . 8 |- ( y e. ~P S -> Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = Σ* z e. y ( ( M ` z ) /𝑒 A ) )
65	45, 64	syl 17	. . . . . . 7 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = Σ* z e. y ( ( M ` z ) /𝑒 A ) )
66	43, 59, 65	3eqtr4d 2666	. . . . . 6 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) )
67	28, 29, 30, 31, 66	syl13anc 1328	. . . . 5 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) )
68	67	ex 450	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) )
69	68	ralrimiva 2966	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> A. y e. ~P S ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) )
70		ismeas 30262	. . . . . 6 |- ( S e. U. ran sigAlgebra -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) <-> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) ) ) )
71	14, 70	syl 17	. . . . 5 |- ( M e. (measures ` S ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) <-> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) ) ) )
72	71	biimprd 238	. . . 4 |- ( M e. (measures ` S ) -> ( ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) ) )
73	72	adantr 481	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) /\ ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 /\ A. y e. ~P S ( ( y ~<_ om /\ Disj z e. y z ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = Σ* z e. y ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) ) ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) ) )
74	13, 27, 69, 73	mp3and 1427	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) )
75	7, 74	eqeltrd 2701	1 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) e. (measures ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   RR+crp 11832   [,]cicc 12178   /_𝑒 cxdiv 29625  Σ^*cesum 30089  ∘_𝑓/𝑐cofc 30157  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-xdiv 29626  df-esum 30090  df-ofc 30158  df-siga 30171  df-meas 30259

This theorem is referenced by:  probfinmeasb  30491