Theorem ioombl 22574

Description: An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
ioombl	|- ( A (,) B ) e. dom vol

Proof of Theorem ioombl

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

Step	Hyp	Ref	Expression
1		snunioo 11793	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
2	1	3expa 1215	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
3	2	adantrr 728	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
4		lbico1 11723	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> A e. ( A [,) B ) )
5	4	3expa 1215	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> A e. ( A [,) B ) )
6	5	adantrr 728	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> A e. ( A [,) B ) )
7	6	snssd 4130	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> { A } C_ ( A [,) B ) )
8		iccid 11715	. . . . . . . . . . 11 |- ( A e. RR* -> ( A [,] A ) = { A } )
9	8	ad2antrr 737	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( A [,] A ) = { A } )
10	9	ineq1d 3645	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( ( A [,] A ) i^i ( A (,) B ) ) = ( { A } i^i ( A (,) B ) ) )
11		simpll 765	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> A e. RR* )
12		simplr 767	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> B e. RR* )
13		df-icc 11676	. . . . . . . . . . 11 |- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
14		df-ioo 11673	. . . . . . . . . . 11 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
15		xrltnle 9732	. . . . . . . . . . 11 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w <-> -. w <_ A ) )
16	13, 14, 15	ixxdisj 11684	. . . . . . . . . 10 |- ( ( A e. RR* /\ A e. RR* /\ B e. RR* ) -> ( ( A [,] A ) i^i ( A (,) B ) ) = (/) )
17	11, 11, 12, 16	syl3anc 1276	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( ( A [,] A ) i^i ( A (,) B ) ) = (/) )
18	10, 17	eqtr3d 2498	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( { A } i^i ( A (,) B ) ) = (/) )
19		uneqdifeq 3868	. . . . . . . 8 |- ( ( { A } C_ ( A [,) B ) /\ ( { A } i^i ( A (,) B ) ) = (/) ) -> ( ( { A } u. ( A (,) B ) ) = ( A [,) B ) <-> ( ( A [,) B ) \ { A } ) = ( A (,) B ) ) )
20	7, 18, 19	syl2anc 671	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( ( { A } u. ( A (,) B ) ) = ( A [,) B ) <-> ( ( A [,) B ) \ { A } ) = ( A (,) B ) ) )
21	3, 20	mpbid 215	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( ( A [,) B ) \ { A } ) = ( A (,) B ) )
22		mnfxr 11448	. . . . . . . . . 10 |- -oo e. RR*
23	22	a1i 11	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> -oo e. RR* )
24		simprr 771	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> -oo < A )
25		simprl 769	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> A < B )
26		xrre2 11499	. . . . . . . . 9 |- ( ( ( -oo e. RR* /\ A e. RR* /\ B e. RR* ) /\ ( -oo < A /\ A < B ) ) -> A e. RR )
27	23, 11, 12, 24, 25, 26	syl32anc 1284	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> A e. RR )
28		icombl 22573	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) e. dom vol )
29	27, 12, 28	syl2anc 671	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( A [,) B ) e. dom vol )
30	27	snssd 4130	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> { A } C_ RR )
31		ovolsn 22503	. . . . . . . . 9 |- ( A e. RR -> ( vol* ` { A } ) = 0 )
32	27, 31	syl 17	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( vol* ` { A } ) = 0 )
33		nulmbl 22544	. . . . . . . 8 |- ( ( { A } C_ RR /\ ( vol* ` { A } ) = 0 ) -> { A } e. dom vol )
34	30, 32, 33	syl2anc 671	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> { A } e. dom vol )
35		difmbl 22552	. . . . . . 7 |- ( ( ( A [,) B ) e. dom vol /\ { A } e. dom vol ) -> ( ( A [,) B ) \ { A } ) e. dom vol )
36	29, 34, 35	syl2anc 671	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( ( A [,) B ) \ { A } ) e. dom vol )
37	21, 36	eqeltrrd 2541	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < B /\ -oo < A ) ) -> ( A (,) B ) e. dom vol )
38	37	expr 624	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( -oo < A -> ( A (,) B ) e. dom vol ) )
39		uncom 3590	. . . . . . . . 9 |- ( ( B [,) +oo ) u. ( -oo (,) B ) ) = ( ( -oo (,) B ) u. ( B [,) +oo ) )
40	22	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> -oo e. RR* )
41		simplr 767	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> B e. RR* )
42		pnfxr 11446	. . . . . . . . . . 11 |- +oo e. RR*
43	42	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> +oo e. RR* )
44		simpll 765	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> A e. RR* )
45		mnfle 11469	. . . . . . . . . . . 12 |- ( A e. RR* -> -oo <_ A )
46	45	ad2antrr 737	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> -oo <_ A )
47		simpr 467	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> A < B )
48	40, 44, 41, 46, 47	xrlelttrd 11491	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> -oo < B )
49		pnfge 11466	. . . . . . . . . . 11 |- ( B e. RR* -> B <_ +oo )
50	41, 49	syl 17	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> B <_ +oo )
51		df-ico 11675	. . . . . . . . . . 11 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
52		xrlenlt 9730	. . . . . . . . . . 11 |- ( ( B e. RR* /\ w e. RR* ) -> ( B <_ w <-> -. w < B ) )
53		xrltletr 11488	. . . . . . . . . . 11 |- ( ( w e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( w < B /\ B <_ +oo ) -> w < +oo ) )
54		xrltletr 11488	. . . . . . . . . . 11 |- ( ( -oo e. RR* /\ B e. RR* /\ w e. RR* ) -> ( ( -oo < B /\ B <_ w ) -> -oo < w ) )
55	14, 51, 52, 14, 53, 54	ixxun 11685	. . . . . . . . . 10 |- ( ( ( -oo e. RR* /\ B e. RR* /\ +oo e. RR* ) /\ ( -oo < B /\ B <_ +oo ) ) -> ( ( -oo (,) B ) u. ( B [,) +oo ) ) = ( -oo (,) +oo ) )
56	40, 41, 43, 48, 50, 55	syl32anc 1284	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( -oo (,) B ) u. ( B [,) +oo ) ) = ( -oo (,) +oo ) )
57	39, 56	syl5eq 2508	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( B [,) +oo ) u. ( -oo (,) B ) ) = ( -oo (,) +oo ) )
58		ioomax 11743	. . . . . . . 8 |- ( -oo (,) +oo ) = RR
59	57, 58	syl6eq 2512	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( B [,) +oo ) u. ( -oo (,) B ) ) = RR )
60		ssun1 3609	. . . . . . . . 9 |- ( B [,) +oo ) C_ ( ( B [,) +oo ) u. ( -oo (,) B ) )
61	60, 59	syl5sseq 3492	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B [,) +oo ) C_ RR )
62		incom 3637	. . . . . . . . 9 |- ( ( B [,) +oo ) i^i ( -oo (,) B ) ) = ( ( -oo (,) B ) i^i ( B [,) +oo ) )
63	14, 51, 52	ixxdisj 11684	. . . . . . . . . 10 |- ( ( -oo e. RR* /\ B e. RR* /\ +oo e. RR* ) -> ( ( -oo (,) B ) i^i ( B [,) +oo ) ) = (/) )
64	40, 41, 43, 63	syl3anc 1276	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( -oo (,) B ) i^i ( B [,) +oo ) ) = (/) )
65	62, 64	syl5eq 2508	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( B [,) +oo ) i^i ( -oo (,) B ) ) = (/) )
66		uneqdifeq 3868	. . . . . . . 8 |- ( ( ( B [,) +oo ) C_ RR /\ ( ( B [,) +oo ) i^i ( -oo (,) B ) ) = (/) ) -> ( ( ( B [,) +oo ) u. ( -oo (,) B ) ) = RR <-> ( RR \ ( B [,) +oo ) ) = ( -oo (,) B ) ) )
67	61, 65, 66	syl2anc 671	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( ( B [,) +oo ) u. ( -oo (,) B ) ) = RR <-> ( RR \ ( B [,) +oo ) ) = ( -oo (,) B ) ) )
68	59, 67	mpbid 215	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( RR \ ( B [,) +oo ) ) = ( -oo (,) B ) )
69		rembl 22549	. . . . . . 7 |- RR e. dom vol
70		xrleloe 11477	. . . . . . . . . . 11 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( B <_ +oo <-> ( B < +oo \/ B = +oo ) ) )
71	41, 42, 70	sylancl 673	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B <_ +oo <-> ( B < +oo \/ B = +oo ) ) )
72	50, 71	mpbid 215	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B < +oo \/ B = +oo ) )
73		xrre2 11499	. . . . . . . . . . . 12 |- ( ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) /\ ( A < B /\ B < +oo ) ) -> B e. RR )
74	73	expr 624	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* /\ +oo e. RR* ) /\ A < B ) -> ( B < +oo -> B e. RR ) )
75	42, 74	mp3anl3 1369	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B < +oo -> B e. RR ) )
76	75	orim1d 855	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( ( B < +oo \/ B = +oo ) -> ( B e. RR \/ B = +oo ) ) )
77	72, 76	mpd 15	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B e. RR \/ B = +oo ) )
78		icombl1 22572	. . . . . . . . 9 |- ( B e. RR -> ( B [,) +oo ) e. dom vol )
79		oveq1 6327	. . . . . . . . . . 11 |- ( B = +oo -> ( B [,) +oo ) = ( +oo [,) +oo ) )
80		pnfge 11466	. . . . . . . . . . . . 13 |- ( +oo e. RR* -> +oo <_ +oo )
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 |- +oo <_ +oo
82		ico0 11716	. . . . . . . . . . . . 13 |- ( ( +oo e. RR* /\ +oo e. RR* ) -> ( ( +oo [,) +oo ) = (/) <-> +oo <_ +oo ) )
83	42, 42, 82	mp2an 683	. . . . . . . . . . . 12 |- ( ( +oo [,) +oo ) = (/) <-> +oo <_ +oo )
84	81, 83	mpbir 214	. . . . . . . . . . 11 |- ( +oo [,) +oo ) = (/)
85	79, 84	syl6eq 2512	. . . . . . . . . 10 |- ( B = +oo -> ( B [,) +oo ) = (/) )
86		0mbl 22548	. . . . . . . . . 10 |- (/) e. dom vol
87	85, 86	syl6eqel 2548	. . . . . . . . 9 |- ( B = +oo -> ( B [,) +oo ) e. dom vol )
88	78, 87	jaoi 385	. . . . . . . 8 |- ( ( B e. RR \/ B = +oo ) -> ( B [,) +oo ) e. dom vol )
89	77, 88	syl 17	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( B [,) +oo ) e. dom vol )
90		difmbl 22552	. . . . . . 7 |- ( ( RR e. dom vol /\ ( B [,) +oo ) e. dom vol ) -> ( RR \ ( B [,) +oo ) ) e. dom vol )
91	69, 89, 90	sylancr 674	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( RR \ ( B [,) +oo ) ) e. dom vol )
92	68, 91	eqeltrrd 2541	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( -oo (,) B ) e. dom vol )
93		oveq1 6327	. . . . . 6 |- ( -oo = A -> ( -oo (,) B ) = ( A (,) B ) )
94	93	eleq1d 2524	. . . . 5 |- ( -oo = A -> ( ( -oo (,) B ) e. dom vol <-> ( A (,) B ) e. dom vol ) )
95	92, 94	syl5ibcom 228	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( -oo = A -> ( A (,) B ) e. dom vol ) )
96		xrleloe 11477	. . . . . 6 |- ( ( -oo e. RR* /\ A e. RR* ) -> ( -oo <_ A <-> ( -oo < A \/ -oo = A ) ) )
97	22, 44, 96	sylancr 674	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( -oo <_ A <-> ( -oo < A \/ -oo = A ) ) )
98	46, 97	mpbid 215	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( -oo < A \/ -oo = A ) )
99	38, 95, 98	mpjaod 387	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A < B ) -> ( A (,) B ) e. dom vol )
100		ioo0 11695	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> B <_ A ) )
101		xrlenlt 9730	. . . . . . 7 |- ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
102	101	ancoms 459	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
103	100, 102	bitrd 261	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) = (/) <-> -. A < B ) )
104	103	biimpar 492	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A < B ) -> ( A (,) B ) = (/) )
105	104, 86	syl6eqel 2548	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. A < B ) -> ( A (,) B ) e. dom vol )
106	99, 105	pm2.61dan 805	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. dom vol )
107		ndmioo 11697	. . 3 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) = (/) )
108	107, 86	syl6eqel 2548	. 2 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. dom vol )
109	106, 108	pm2.61i 169	1 |- ( A (,) B ) e. dom vol

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   \ cdif 3413   u. cun 3414   i^i cin 3415   C_ wss 3416   (/)c0 3743   {csn 3980   class class class wbr 4418   dom cdm 4856   ` cfv 5605  (class class class)co 6320   RRcr 9569   0cc0 9570   +oocpnf 9703   -oocmnf 9704   RR*cxr 9705   < clt 9706   <_ cle 9707   (,)cioo 11669   [,)cico 11671   [,]cicc 11672   vol*covol 22468   volcvol 22470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-of 6563  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-2o 7214  df-oadd 7217  df-er 7394  df-map 7505  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-inf 7988  df-oi 8056  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-q 11299  df-rp 11337  df-xadd 11444  df-ioo 11673  df-ico 11675  df-icc 11676  df-fz 11820  df-fzo 11953  df-fl 12066  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-clim 13607  df-rlim 13608  df-sum 13808  df-xmet 19018  df-met 19019  df-ovol 22471  df-vol 22473

This theorem is referenced by:  iccmbl  22575  ovolioo  22577  ioovolcl  22578  uniioovol  22592  uniioombllem4  22600  uniioombllem5  22601  opnmblALT  22617  mbfid  22648  ditgcl  22869  ditgswap  22870  ditgsplitlem  22871  ftc1lem1  23043  ftc1lem2  23044  ftc1a  23045  ftc1lem4  23047  ftc2  23052  ftc2ditglem  23053  itgsubstlem  23056  itg2gt0cn  32043  ftc1cnnclem  32061  ftc1anclem7  32069  ftc1anclem8  32070  ftc1anc  32071  ftc2nc  32072  areacirc  32083  iocmbl  36143  cnioobibld  36144  itgpowd  36145  lhe4.4ex1a  36723  volioo  37911  itgsin0pilem1  37912  iblioosinexp  37915  itgsinexplem1  37916  itgsinexp  37917  itgcoscmulx  37932  volioc  37935  itgsincmulx  37937  iblcncfioo  37941  itgiccshift  37943  itgperiod  37944  itgsbtaddcnst  37945  volico  37947  volioof  37951  wallispilem2  38029  dirkeritg  38065  fourierdlem16  38086  fourierdlem21  38091  fourierdlem22  38092  fourierdlem39  38110  fourierdlem73  38144  fourierdlem83  38154  fourierdlem103  38174  fourierdlem104  38175  fourierdlem111  38182  fourierdlem112  38183  sqwvfoura  38193  sqwvfourb  38194  etransclem18  38218  etransclem23  38223  ovolval4lem1  38578  ovolval5lem1  38581