Theorem measdivcst 26783

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 26688	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. dom M |-> ( ( M ` x ) /𝑒 A ) ) )
2		measfrge0 26761	. . . . . 6 |- ( M e. (measures ` S ) -> M : S --> ( 0 [,] +oo ) )
3		fdm 5670	. . . . . 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 4480	. . 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 2495	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( M∘𝑓/𝑐 /𝑒 A ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) )
8		measvxrge0 26763	. . . . . 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 26254	. . . 4 |- ( ( ( M e. (measures ` S ) /\ A e. RR+ ) /\ x e. S ) -> ( ( M ` x ) /𝑒 A ) e. ( 0 [,] +oo ) )
12		eqid 2454	. . . 4 |- ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) = ( x e. S |-> ( ( M ` x ) /𝑒 A ) )
13	11, 12	fmptd 5975	. . 3 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) : S --> ( 0 [,] +oo ) )
14		measbase 26755	. . . . . . 7 |- ( M e. (measures ` S ) -> S e. U. ran sigAlgebra )
15		0elsiga 26701	. . . . . . 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 6224	. . . . 5 |- ( ( M ` (/) ) /𝑒 A ) e. _V
19		fveq2 5798	. . . . . . 7 |- ( x = (/) -> ( M ` x ) = ( M ` (/) ) )
20	19	oveq1d 6214	. . . . . 6 |- ( x = (/) -> ( ( M ` x ) /𝑒 A ) = ( ( M ` (/) ) /𝑒 A ) )
21	20, 12	fvmptg 5880	. . . . 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 26764	. . . . . 6 |- ( M e. (measures ` S ) -> ( M ` (/) ) = 0 )
24	23	oveq1d 6214	. . . . 5 |- ( M e. (measures ` S ) -> ( ( M ` (/) ) /𝑒 A ) = ( 0 /𝑒 A ) )
25		xdiv0rp 26249	. . . . 5 |- ( A e. RR+ -> ( 0 /𝑒 A ) = 0 )
26	24, 25	sylan9eq 2515	. . . 4 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( ( M ` (/) ) /𝑒 A ) = 0 )
27	22, 26	eqtrd 2495	. . 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 3079	. . . . . . . . . 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 3974	. . . . . . . . . . . 12 |- ( y e. ~P S <-> y C_ S )
37		ssel2 3458	. . . . . . . . . . . 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 26763	. . . . . . . . . 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 26676	. . . . . . . 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 26704	. . . . . . . . . 10 |- ( ( S e. U. ran sigAlgebra /\ y e. ~P S /\ y ~<_ om ) -> U. y e. S )
49	45, 46, 47, 48	syl3anc 1219	. . . . . . . . 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 5798	. . . . . . . . . . 11 |- ( x = U. y -> ( M ` x ) = ( M ` U. y ) )
51	50	oveq1d 6214	. . . . . . . . . 10 |- ( x = U. y -> ( ( M ` x ) /𝑒 A ) = ( ( M ` U. y ) /𝑒 A ) )
52		ovex 6224	. . . . . . . . . 10 |- ( ( M ` x ) /𝑒 A ) e. _V
53	51, 12, 52	fvmpt3i 5886	. . . . . . . . 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 26767	. . . . . . . . . 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 1223	. . . . . . . . 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 6214	. . . . . . . 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 2495	. . . . . . 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 5798	. . . . . . . . . . . 12 |- ( x = z -> ( M ` x ) = ( M ` z ) )
62	61	oveq1d 6214	. . . . . . . . . . 11 |- ( x = z -> ( ( M ` x ) /𝑒 A ) = ( ( M ` z ) /𝑒 A ) )
63	62, 12, 52	fvmpt3i 5886	. . . . . . . . . 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 26638	. . . . . . . 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 2505	. . . . . 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 2829	. . 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 26757	. . . . . 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 1318	. 2 |- ( ( M e. (measures ` S ) /\ A e. RR+ ) -> ( x e. S |-> ( ( M ` x ) /𝑒 A ) ) e. (measures ` S ) )
76	7, 75	eqeltrd 2542	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 1370   e. wcel 1758   A.wral 2798   _Vcvv 3076   C_ wss 3435   (/)c0 3744   ~Pcpw 3967   U.cuni 4198  Disj wdisj 4369   class class class wbr 4399   |-> cmpt 4457   dom cdm 4947   ran crn 4948   -->wf 5521   ` cfv 5525  (class class class)co 6199   omcom 6585   ~<_ cdom 7417   0cc0 9392   +oocpnf 9525   RR+crp 11101   [,]cicc 11413   /_𝑒 cxdiv 26236  Σ^*cesum 26627  ∘_𝑓/𝑐cofc 26681  sigAlgebracsiga 26694  measurescmeas 26753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-disj 4370  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-seq 11923  df-hash 12220  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-tset 14375  df-ple 14376  df-ds 14378  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-ordt 14557  df-xrs 14558  df-mre 14642  df-mrc 14643  df-acs 14645  df-ps 15488  df-tsr 15489  df-mnd 15533  df-mhm 15582  df-submnd 15583  df-cntz 15953  df-cmn 16399  df-fbas 17938  df-fg 17939  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-ntr 18755  df-nei 18833  df-cn 18962  df-cnp 18963  df-haus 19050  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-tsms 19828  df-xdiv 26237  df-esum 26628  df-ofc 26682  df-siga 26695  df-meas 26754

This theorem is referenced by:  probfinmeasb  26955