Theorem vitalilem5 23381

Description: Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
vitali.1	|- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
vitali.2	|- S = ( ( 0 [,] 1 ) /. .~ )
vitali.3	|- ( ph -> F Fn S )
vitali.4	|- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
vitali.5	|- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
vitali.6	|- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
vitali.7	|- ( ph -> -. ran F e. ( ~P RR \ dom vol ) )

Assertion

Ref	Expression
vitalilem5	|- -. ph

Distinct variable groups:   n, s, x, y, z, G   ph, n, x, z   z, S   x, T   n, F, s, x, y, z   .~ , n, s, x, y, z

Allowed substitution hints:   ph( y, s)   S( x, y, n, s)   T( y, z, n, s)

Proof of Theorem vitalilem5

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1 10550	. . . 4 |- 0 < 1
2		0re 10040	. . . . . 6 |- 0 e. RR
3		1re 10039	. . . . . 6 |- 1 e. RR
4		0le1 10551	. . . . . 6 |- 0 <_ 1
5		ovolicc 23291	. . . . . 6 |- ( ( 0 e. RR /\ 1 e. RR /\ 0 <_ 1 ) -> ( vol* ` ( 0 [,] 1 ) ) = ( 1 - 0 ) )
6	2, 3, 4, 5	mp3an 1424	. . . . 5 |- ( vol* ` ( 0 [,] 1 ) ) = ( 1 - 0 )
7		1m0e1 11131	. . . . 5 |- ( 1 - 0 ) = 1
8	6, 7	eqtri 2644	. . . 4 |- ( vol* ` ( 0 [,] 1 ) ) = 1
9	1, 8	breqtrri 4680	. . 3 |- 0 < ( vol* ` ( 0 [,] 1 ) )
10	8, 3	eqeltri 2697	. . . 4 |- ( vol* ` ( 0 [,] 1 ) ) e. RR
11	2, 10	ltnlei 10158	. . 3 |- ( 0 < ( vol* ` ( 0 [,] 1 ) ) <-> -. ( vol* ` ( 0 [,] 1 ) ) <_ 0 )
12	9, 11	mpbi 220	. 2 |- -. ( vol* ` ( 0 [,] 1 ) ) <_ 0
13		vitali.1	. . . . . 6 |- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
14		vitali.2	. . . . . 6 |- S = ( ( 0 [,] 1 ) /. .~ )
15		vitali.3	. . . . . 6 |- ( ph -> F Fn S )
16		vitali.4	. . . . . 6 |- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
17		vitali.5	. . . . . 6 |- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
18		vitali.6	. . . . . 6 |- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
19		vitali.7	. . . . . 6 |- ( ph -> -. ran F e. ( ~P RR \ dom vol ) )
20	13, 14, 15, 16, 17, 18, 19	vitalilem2 23378	. . . . 5 |- ( ph -> ( ran F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) /\ U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) ) )
21	20	simp2d 1074	. . . 4 |- ( ph -> ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) )
22	13	vitalilem1 23376	. . . . . . . . . . . . . . . . . . . . . . . 24 |- .~ Er ( 0 [,] 1 )
23		erdm 7752	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( .~ Er ( 0 [,] 1 ) -> dom .~ = ( 0 [,] 1 ) )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 |- dom .~ = ( 0 [,] 1 )
25		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ph /\ z e. S ) -> z e. S )
26	25, 14	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ z e. S ) -> z e. ( ( 0 [,] 1 ) /. .~ ) )
27		elqsn0 7816	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( dom .~ = ( 0 [,] 1 ) /\ z e. ( ( 0 [,] 1 ) /. .~ ) ) -> z =/= (/) )
28	24, 26, 27	sylancr 695	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ z e. S ) -> z =/= (/) )
29	22	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ph -> .~ Er ( 0 [,] 1 ) )
30	29	qsss 7808	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ph -> ( ( 0 [,] 1 ) /. .~ ) C_ ~P ( 0 [,] 1 ) )
31	14, 30	syl5eqss 3649	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ph -> S C_ ~P ( 0 [,] 1 ) )
32	31	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ph /\ z e. S ) -> z e. ~P ( 0 [,] 1 ) )
33	32	elpwid 4170	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ph /\ z e. S ) -> z C_ ( 0 [,] 1 ) )
34	33	sseld 3602	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ z e. S ) -> ( ( F ` z ) e. z -> ( F ` z ) e. ( 0 [,] 1 ) ) )
35	28, 34	embantd 59	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ z e. S ) -> ( ( z =/= (/) -> ( F ` z ) e. z ) -> ( F ` z ) e. ( 0 [,] 1 ) ) )
36	35	ralimdva 2962	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) -> A. z e. S ( F ` z ) e. ( 0 [,] 1 ) ) )
37	16, 36	mpd 15	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A. z e. S ( F ` z ) e. ( 0 [,] 1 ) )
38		ffnfv 6388	. . . . . . . . . . . . . . . . . . 19 |- ( F : S --> ( 0 [,] 1 ) <-> ( F Fn S /\ A. z e. S ( F ` z ) e. ( 0 [,] 1 ) ) )
39	15, 37, 38	sylanbrc 698	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F : S --> ( 0 [,] 1 ) )
40		frn 6053	. . . . . . . . . . . . . . . . . 18 |- ( F : S --> ( 0 [,] 1 ) -> ran F C_ ( 0 [,] 1 ) )
41	39, 40	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F C_ ( 0 [,] 1 ) )
42		unitssre 12319	. . . . . . . . . . . . . . . . 17 |- ( 0 [,] 1 ) C_ RR
43	41, 42	syl6ss 3615	. . . . . . . . . . . . . . . 16 |- ( ph -> ran F C_ RR )
44		reex 10027	. . . . . . . . . . . . . . . . 17 |- RR e. _V
45	44	elpw2 4828	. . . . . . . . . . . . . . . 16 |- ( ran F e. ~P RR <-> ran F C_ RR )
46	43, 45	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ph -> ran F e. ~P RR )
47	46	anim1i 592	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. ran F e. dom vol ) -> ( ran F e. ~P RR /\ -. ran F e. dom vol ) )
48		eldif 3584	. . . . . . . . . . . . . 14 |- ( ran F e. ( ~P RR \ dom vol ) <-> ( ran F e. ~P RR /\ -. ran F e. dom vol ) )
49	47, 48	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ph /\ -. ran F e. dom vol ) -> ran F e. ( ~P RR \ dom vol ) )
50	49	ex 450	. . . . . . . . . . . 12 |- ( ph -> ( -. ran F e. dom vol -> ran F e. ( ~P RR \ dom vol ) ) )
51	19, 50	mt3d 140	. . . . . . . . . . 11 |- ( ph -> ran F e. dom vol )
52	51	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ran F e. dom vol )
53		f1of 6137	. . . . . . . . . . . . 13 |- ( G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
54	17, 53	syl 17	. . . . . . . . . . . 12 |- ( ph -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
55		inss1 3833	. . . . . . . . . . . . 13 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
56		qssre 11798	. . . . . . . . . . . . 13 |- QQ C_ RR
57	55, 56	sstri 3612	. . . . . . . . . . . 12 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ RR
58		fss 6056	. . . . . . . . . . . 12 |- ( ( G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) /\ ( QQ i^i ( -u 1 [,] 1 ) ) C_ RR ) -> G : NN --> RR )
59	54, 57, 58	sylancl 694	. . . . . . . . . . 11 |- ( ph -> G : NN --> RR )
60	59	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
61		shftmbl 23306	. . . . . . . . . 10 |- ( ( ran F e. dom vol /\ ( G ` n ) e. RR ) -> { s e. RR | ( s - ( G ` n ) ) e. ran F } e. dom vol )
62	52, 60, 61	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> { s e. RR | ( s - ( G ` n ) ) e. ran F } e. dom vol )
63	62, 18	fmptd 6385	. . . . . . . 8 |- ( ph -> T : NN --> dom vol )
64	63	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ m e. NN ) -> ( T ` m ) e. dom vol )
65	64	ralrimiva 2966	. . . . . 6 |- ( ph -> A. m e. NN ( T ` m ) e. dom vol )
66		iunmbl 23321	. . . . . 6 |- ( A. m e. NN ( T ` m ) e. dom vol -> U_ m e. NN ( T ` m ) e. dom vol )
67	65, 66	syl 17	. . . . 5 |- ( ph -> U_ m e. NN ( T ` m ) e. dom vol )
68		mblss 23299	. . . . 5 |- ( U_ m e. NN ( T ` m ) e. dom vol -> U_ m e. NN ( T ` m ) C_ RR )
69	67, 68	syl 17	. . . 4 |- ( ph -> U_ m e. NN ( T ` m ) C_ RR )
70		ovolss 23253	. . . 4 |- ( ( ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) /\ U_ m e. NN ( T ` m ) C_ RR ) -> ( vol* ` ( 0 [,] 1 ) ) <_ ( vol* ` U_ m e. NN ( T ` m ) ) )
71	21, 69, 70	syl2anc 693	. . 3 |- ( ph -> ( vol* ` ( 0 [,] 1 ) ) <_ ( vol* ` U_ m e. NN ( T ` m ) ) )
72		eqid 2622	. . . . . 6 |- seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) = seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) )
73		eqid 2622	. . . . . 6 |- ( m e. NN |-> ( vol* ` ( T ` m ) ) ) = ( m e. NN |-> ( vol* ` ( T ` m ) ) )
74		mblss 23299	. . . . . . 7 |- ( ( T ` m ) e. dom vol -> ( T ` m ) C_ RR )
75	64, 74	syl 17	. . . . . 6 |- ( ( ph /\ m e. NN ) -> ( T ` m ) C_ RR )
76	13, 14, 15, 16, 17, 18, 19	vitalilem4 23380	. . . . . . 7 |- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) = 0 )
77	76, 2	syl6eqel 2709	. . . . . 6 |- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) e. RR )
78	76	mpteq2dva 4744	. . . . . . . . . 10 |- ( ph -> ( m e. NN |-> ( vol* ` ( T ` m ) ) ) = ( m e. NN |-> 0 ) )
79		fconstmpt 5163	. . . . . . . . . . 11 |- ( NN X. { 0 } ) = ( m e. NN |-> 0 )
80		nnuz 11723	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
81	80	xpeq1i 5135	. . . . . . . . . . 11 |- ( NN X. { 0 } ) = ( ( ZZ>= ` 1 ) X. { 0 } )
82	79, 81	eqtr3i 2646	. . . . . . . . . 10 |- ( m e. NN |-> 0 ) = ( ( ZZ>= ` 1 ) X. { 0 } )
83	78, 82	syl6eq 2672	. . . . . . . . 9 |- ( ph -> ( m e. NN |-> ( vol* ` ( T ` m ) ) ) = ( ( ZZ>= ` 1 ) X. { 0 } ) )
84	83	seqeq3d 12809	. . . . . . . 8 |- ( ph -> seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) = seq 1 ( + , ( ( ZZ>= ` 1 ) X. { 0 } ) ) )
85		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
86		serclim0 14308	. . . . . . . . 9 |- ( 1 e. ZZ -> seq 1 ( + , ( ( ZZ>= ` 1 ) X. { 0 } ) ) ~~> 0 )
87	85, 86	ax-mp 5	. . . . . . . 8 |- seq 1 ( + , ( ( ZZ>= ` 1 ) X. { 0 } ) ) ~~> 0
88	84, 87	syl6eqbr 4692	. . . . . . 7 |- ( ph -> seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) ~~> 0 )
89		seqex 12803	. . . . . . . 8 |- seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) e. _V
90		c0ex 10034	. . . . . . . 8 |- 0 e. _V
91	89, 90	breldm 5329	. . . . . . 7 |- ( seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) ~~> 0 -> seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) e. dom ~~> )
92	88, 91	syl 17	. . . . . 6 |- ( ph -> seq 1 ( + , ( m e. NN |-> ( vol* ` ( T ` m ) ) ) ) e. dom ~~> )
93	72, 73, 75, 77, 92	ovoliun2 23274	. . . . 5 |- ( ph -> ( vol* ` U_ m e. NN ( T ` m ) ) <_ sum_ m e. NN ( vol* ` ( T ` m ) ) )
94	76	sumeq2dv 14433	. . . . . 6 |- ( ph -> sum_ m e. NN ( vol* ` ( T ` m ) ) = sum_ m e. NN 0 )
95	80	eqimssi 3659	. . . . . . . 8 |- NN C_ ( ZZ>= ` 1 )
96	95	orci 405	. . . . . . 7 |- ( NN C_ ( ZZ>= ` 1 ) \/ NN e. Fin )
97		sumz 14453	. . . . . . 7 |- ( ( NN C_ ( ZZ>= ` 1 ) \/ NN e. Fin ) -> sum_ m e. NN 0 = 0 )
98	96, 97	ax-mp 5	. . . . . 6 |- sum_ m e. NN 0 = 0
99	94, 98	syl6eq 2672	. . . . 5 |- ( ph -> sum_ m e. NN ( vol* ` ( T ` m ) ) = 0 )
100	93, 99	breqtrd 4679	. . . 4 |- ( ph -> ( vol* ` U_ m e. NN ( T ` m ) ) <_ 0 )
101		ovolge0 23249	. . . . 5 |- ( U_ m e. NN ( T ` m ) C_ RR -> 0 <_ ( vol* ` U_ m e. NN ( T ` m ) ) )
102	69, 101	syl 17	. . . 4 |- ( ph -> 0 <_ ( vol* ` U_ m e. NN ( T ` m ) ) )
103		ovolcl 23246	. . . . . 6 |- ( U_ m e. NN ( T ` m ) C_ RR -> ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* )
104	69, 103	syl 17	. . . . 5 |- ( ph -> ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* )
105		0xr 10086	. . . . 5 |- 0 e. RR*
106		xrletri3 11985	. . . . 5 |- ( ( ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* /\ 0 e. RR* ) -> ( ( vol* ` U_ m e. NN ( T ` m ) ) = 0 <-> ( ( vol* ` U_ m e. NN ( T ` m ) ) <_ 0 /\ 0 <_ ( vol* ` U_ m e. NN ( T ` m ) ) ) ) )
107	104, 105, 106	sylancl 694	. . . 4 |- ( ph -> ( ( vol* ` U_ m e. NN ( T ` m ) ) = 0 <-> ( ( vol* ` U_ m e. NN ( T ` m ) ) <_ 0 /\ 0 <_ ( vol* ` U_ m e. NN ( T ` m ) ) ) ) )
108	100, 102, 107	mpbir2and 957	. . 3 |- ( ph -> ( vol* ` U_ m e. NN ( T ` m ) ) = 0 )
109	71, 108	breqtrd 4679	. 2 |- ( ph -> ( vol* ` ( 0 [,] 1 ) ) <_ 0 )
110	12, 109	mto 188	1 |- -. ph

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Er wer 7739   /.cqs 7741   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   [,]cicc 12178   seqcseq 12801   ~~> cli 14215   sum_csu 14416   vol*covol 23231   volcvol 23232

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-inf2 8538  ax-cc 9257  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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  vitali  23382