Theorem measdivcst 26608

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 26513	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. dom M |-> ( ( M ` x ) /𝑒 A ) ) )
2		measfrge0 26586	. . . . . 6 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )
3		fdm 5558	. . . . . 6 |- ( M : S --> ( 0 [,] +oo ) -> dom M = S )
4	2, 3	syl 16	. . . . 5 |- ( M e. (measures ` S ) -> dom M = S )
5	4	adantr 465	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> dom M = S )
6	5	mpteq1d 4368	. . 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 2470	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) )
8		measvxrge0 26588	. . . . . 6 |- ( ( M e. (measures ` S ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
9	8	adantlr 714	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
10		simplr 754	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> A e. RR+ )
11	9, 10	xrpxdivcld 26078	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( ( M ` x ) /𝑒 A ) e. ( 0 [,] +oo ) )
12		eqid 2438	. . . 4 |- ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) )
13	11, 12	fmptd 5862	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) )
14		measbase 26580	. . . . . . 7 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
15		0elsiga 26526	. . . . . . 7 |- ( S e. U. ran sigAlgebra -> (/) e. S )
16	14, 15	syl 16	. . . . . 6 |- ( M e. (measures ` S ) -> (/) e. S )
17	16	adantr 465	. . . . 5 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> (/) e. S )
18		ovex 6111	. . . . 5 |- ( ( M ` (/) ) /𝑒 A ) e. _V
19		fveq2 5686	. . . . . . 7 |- ( x = (/) -> ( M ` x ) = ( M ` (/) ) )
20	19	oveq1d 6101	. . . . . 6 |- ( x = (/) -> ( ( M ` x ) /𝑒 A ) = ( ( M ` (/) ) /𝑒 A ) )
21	20, 12	fvmptg 5767	. . . . 5 |- ( ( (/) e. S /\ ( ( M ` (/) ) /𝑒 A ) e. _V ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
22	17, 18, 21	sylancl 662	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
23		measvnul 26589	. . . . . 6 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
24	23	oveq1d 6101	. . . . 5 |- ( M e. (measures ` S ) -> ( ( M ` (/) ) /𝑒 A ) = ( 0 /𝑒 A ) )
25		xdiv0rp 26073	. . . . 5 |- ( A e. RR+ -> ( 0 /𝑒 A ) = 0 )
26	24, 25	sylan9eq 2490	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( M ` (/) ) /𝑒 A ) = 0 )
27	22, 26	eqtrd 2470	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 )
28		simpll 753	. . . . . 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 754	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> y e. ~P S )
30		simprl 755	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
31		simprr 756	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> Disj z e. y z )
32	29, 30, 31	3jca 1168	. . . . . 6 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) )
33		vex 2970	. . . . . . . . . 10 |- y e. _V
34	33	a1i 11	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> y e. _V )
35		simplll 757	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> M e. (measures ` S ) )
36		selpw 3862	. . . . . . . . . . . 12 |- ( y e. ~P S <-> y C_ S )
37		ssel2 3346	. . . . . . . . . . . 12 |- ( ( y C_ S /\ z e. y ) -> z e. S )
38	36, 37	sylanb 472	. . . . . . . . . . 11 |- ( ( y e. ~P S /\ z e. y ) -> z e. S )
39	38	adantll 713	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> z e. S )
40		measvxrge0 26588	. . . . . . . . . 10 |- ( ( M e. (measures ` S ) /\ z e. S ) -> ( M ` z ) e. ( 0 [,] +oo ) )
41	35, 39, 40	syl2anc 661	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> ( M ` z ) e. ( 0 [,] +oo ) )
42		simplr 754	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> A e. RR+ )
43	34, 41, 42	esumdivc 26501	. . . . . . . 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 ) )
44	43	3ad2antr1 1153	. . . . . . 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 ) )
45	14	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> S e. U. ran sigAlgebra )
46		simpr1 994	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> y e. ~P S )
47		simpr2 995	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
48		sigaclcu 26529	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. ~P S /\ y ~<_ om ) -> U. y e. S )
49	45, 46, 47, 48	syl3anc 1218	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> U. y e. S )
50		fveq2 5686	. . . . . . . . . . 11 |- ( x = U. y -> ( M ` x ) = ( M ` U. y ) )
51	50	oveq1d 6101	. . . . . . . . . 10 |- ( x = U. y -> ( ( M ` x ) /𝑒 A ) = ( ( M ` U. y ) /𝑒 A ) )
52		ovex 6111	. . . . . . . . . 10 |- ( ( M ` x ) /𝑒 A ) e. _V
53	51, 12, 52	fvmpt3i 5773	. . . . . . . . 9 |- ( U. y e. S -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` U. y ) = ( ( M ` U. y ) /𝑒 A ) )
54	49, 53	syl 16	. . . . . . . 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 ) )
55		simpll 753	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> M e. (measures ` S ) )
56		simpr3 996	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> Disj z e. y z )
57		measvun 26592	. . . . . . . . . 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 ) )
58	55, 46, 47, 56, 57	syl112anc 1222	. . . . . . . . 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 ) )
59	58	oveq1d 6101	. . . . . . . 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 ) )
60	54, 59	eqtrd 2470	. . . . . . 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 ) )
61		fveq2 5686	. . . . . . . . . . . 12 |- ( x = z -> ( M ` x ) = ( M ` z ) )
62	61	oveq1d 6101	. . . . . . . . . . 11 |- ( x = z -> ( ( M ` x ) /𝑒 A ) = ( ( M ` z ) /𝑒 A ) )
63	62, 12, 52	fvmpt3i 5773	. . . . . . . . . 10 |- ( z e. S -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = ( ( M ` z ) /𝑒 A ) )
64	38, 63	syl 16	. . . . . . . . 9 |- ( ( y e. ~P S /\ z e. y ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` z ) = ( ( M ` z ) /𝑒 A ) )
65	64	esumeq2dv 26463	. . . . . . . 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 ) )
66	46, 65	syl 16	. . . . . . 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 ) )
67	44, 60, 66	3eqtr4d 2480	. . . . . 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 ) )
68	28, 32, 67	syl2anc 661	. . . . 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 ) )
69	68	ex 434	. . . 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 ) ) )
70	69	ralrimiva 2794	. . 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 ) ) )
71		ismeas 26582	. . . . . 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 ) ) ) ) )
72	14, 71	syl 16	. . . . 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 ) ) ) ) )
73	72	biimprd 223	. . . 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 ) ) )
74	73	adantr 465	. . 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 ) ) )
75	13, 27, 70, 74	mp3and 1317	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) )
76	7, 75	eqeltrd 2512	1 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) e. (measures ` S ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2710   _Vcvv 2967   C_ wss 3323   (/)c0 3632   ~Pcpw 3855   U.cuni 4086  Disj wdisj 4257   class class class wbr 4287   e. cmpt 4345   dom cdm 4835   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   omcom 6471   ~<_ cdom 7300   0cc0 9274   +oocpnf 9407   RR+crp 10983   [,]cicc 11295   /_𝑒 cxdiv 26060  Σ^*cesum 26452  ∘_𝑓/𝑐cofc 26506  sigAlgebracsiga 26519  measurescmeas 26578

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-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

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-oadd 6916  df-er 7093  df-map 7208  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-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-seq 11799  df-hash 12096  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-tset 14249  df-ple 14250  df-ds 14252  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-ordt 14431  df-xrs 14432  df-mre 14516  df-mrc 14517  df-acs 14519  df-ps 15362  df-tsr 15363  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-cntz 15826  df-cmn 16270  df-fbas 17794  df-fg 17795  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-ntr 18604  df-nei 18682  df-cn 18811  df-cnp 18812  df-haus 18899  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-tsms 19677  df-xdiv 26061  df-esum 26453  df-ofc 26507  df-siga 26520  df-meas 26579

This theorem is referenced by:  probfinmeasb  26781