Theorem measdivcst 26493

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 26398	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. dom M |-> ( ( M ` x ) /𝑒 A ) ) )
2		measfrge0 26471	. . . . . 6 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )
3		fdm 5551	. . . . . 6 |- ( M : S --> ( 0 [,] +oo ) -> dom M = S )
4	2, 3	syl 16	. . . . 5 |- ( M e. (measures ` S ) -> dom M = S )
5	4	adantr 462	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> dom M = S )
6	5	mpteq1d 4361	. . 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 2465	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) )
8		measvxrge0 26473	. . . . . 6 |- ( ( M e. (measures ` S ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
9	8	adantlr 707	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( M ` x ) e. ( 0 [,] +oo ) )
10		simplr 747	. . . . 5 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> A e. RR+ )
11	9, 10	xrpxdivcld 25933	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( ( M ` x ) /𝑒 A ) e. ( 0 [,] +oo ) )
12		eqid 2433	. . . 4 |- ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) )
13	11, 12	fmptd 5855	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) )
14		measbase 26465	. . . . . . 7 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
15		0elsiga 26411	. . . . . . 7 |- ( S e. U. ran sigAlgebra -> (/) e. S )
16	14, 15	syl 16	. . . . . 6 |- ( M e. (measures ` S ) -> (/) e. S )
17	16	adantr 462	. . . . 5 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> (/) e. S )
18		ovex 6105	. . . . 5 |- ( ( M ` (/) ) /𝑒 A ) e. _V
19		fveq2 5679	. . . . . . 7 |- ( x = (/) -> ( M ` x ) = ( M ` (/) ) )
20	19	oveq1d 6095	. . . . . 6 |- ( x = (/) -> ( ( M ` x ) /𝑒 A ) = ( ( M ` (/) ) /𝑒 A ) )
21	20, 12	fvmptg 5760	. . . . 5 |- ( ( (/) e. S /\ ( ( M ` (/) ) /𝑒 A ) e. _V ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
22	17, 18, 21	sylancl 655	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = ( ( M ` (/) ) /𝑒 A ) )
23		measvnul 26474	. . . . . 6 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
24	23	oveq1d 6095	. . . . 5 |- ( M e. (measures ` S ) -> ( ( M ` (/) ) /𝑒 A ) = ( 0 /𝑒 A ) )
25		xdiv0rp 25928	. . . . 5 |- ( A e. RR+ -> ( 0 /𝑒 A ) = 0 )
26	24, 25	sylan9eq 2485	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( M ` (/) ) /𝑒 A ) = 0 )
27	22, 26	eqtrd 2465	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) ` (/) ) = 0 )
28		simpll 746	. . . . . 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 747	. . . . . . 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 748	. . . . . . 7 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ ( y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
31		simprr 749	. . . . . . 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 1161	. . . . . 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 2965	. . . . . . . . . 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 750	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> M e. (measures ` S ) )
36		selpw 3855	. . . . . . . . . . . 12 |- ( y e. ~P S <-> y C_ S )
37		ssel2 3339	. . . . . . . . . . . 12 |- ( ( y C_ S /\ z e. y ) -> z e. S )
38	36, 37	sylanb 469	. . . . . . . . . . 11 |- ( ( y e. ~P S /\ z e. y ) -> z e. S )
39	38	adantll 706	. . . . . . . . . 10 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> z e. S )
40		measvxrge0 26473	. . . . . . . . . 10 |- ( ( M e. (measures ` S ) /\ z e. S ) -> ( M ` z ) e. ( 0 [,] +oo ) )
41	35, 39, 40	syl2anc 654	. . . . . . . . 9 |- ( ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) /\ z e. y ) -> ( M ` z ) e. ( 0 [,] +oo ) )
42		simplr 747	. . . . . . . . 9 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ y e. ~P S ) -> A e. RR+ )
43	34, 41, 42	esumdivc 26386	. . . . . . . 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 1146	. . . . . . 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 718	. . . . . . . . . 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 987	. . . . . . . . . 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 988	. . . . . . . . . 10 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ ( y e. ~P S /\ y ~<_ om /\ Disj z e. y z ) ) -> y ~<_ om )
48		sigaclcu 26414	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. ~P S /\ y ~<_ om ) -> U. y e. S )
49	45, 46, 47, 48	syl3anc 1211	. . . . . . . . 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 5679	. . . . . . . . . . 11 |- ( x = U. y -> ( M ` x ) = ( M ` U. y ) )
51	50	oveq1d 6095	. . . . . . . . . 10 |- ( x = U. y -> ( ( M ` x ) /𝑒 A ) = ( ( M ` U. y ) /𝑒 A ) )
52		ovex 6105	. . . . . . . . . 10 |- ( ( M ` x ) /𝑒 A ) e. _V
53	51, 12, 52	fvmpt3i 5766	. . . . . . . . 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 746	. . . . . . . . . 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 989	. . . . . . . . . 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 26477	. . . . . . . . . 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 1215	. . . . . . . . 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 6095	. . . . . . . 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 2465	. . . . . . 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 5679	. . . . . . . . . . . 12 |- ( x = z -> ( M ` x ) = ( M ` z ) )
62	61	oveq1d 6095	. . . . . . . . . . 11 |- ( x = z -> ( ( M ` x ) /𝑒 A ) = ( ( M ` z ) /𝑒 A ) )
63	62, 12, 52	fvmpt3i 5766	. . . . . . . . . 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 26348	. . . . . . . 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 2475	. . . . . 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 654	. . . . 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 2789	. . 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 26467	. . . . . 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 462	. . 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 1310	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) )
76	7, 75	eqeltrd 2507	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 958   = wceq 1362   e. wcel 1755   A.wral 2705   _Vcvv 2962   C_ wss 3316   (/)c0 3625   ~Pcpw 3848   U.cuni 4079  Disj wdisj 4250   class class class wbr 4280   e. cmpt 4338   dom cdm 4827   ran crn 4828   -->wf 5402   ` cfv 5406  (class class class)co 6080   omcom 6465   ~<_ cdom 7296   0cc0 9270   +oocpnf 9403   RR+crp 10979   [,]cicc 11291   /_𝑒 cxdiv 25915  Σ^*cesum 26337  ∘_𝑓/𝑐cofc 26391  sigAlgebracsiga 26404  measurescmeas 26463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-tset 14240  df-ple 14241  df-ds 14243  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-ordt 14422  df-xrs 14423  df-mre 14507  df-mrc 14508  df-acs 14510  df-ps 15353  df-tsr 15354  df-mnd 15398  df-mhm 15447  df-submnd 15448  df-cntz 15815  df-cmn 16259  df-fbas 17658  df-fg 17659  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-ntr 18466  df-nei 18544  df-cn 18673  df-cnp 18674  df-haus 18761  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-tsms 19539  df-xdiv 25916  df-esum 26338  df-ofc 26392  df-siga 26405  df-meas 26464

This theorem is referenced by:  probfinmeasb  26660