Theorem ioombl1 23330

Description: An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)

Assertion

Ref	Expression
ioombl1	|- ( A e. RR* -> ( A (,) +oo ) e. dom vol )

Proof of Theorem ioombl1

Dummy variables f m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxr 11950	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		ioossre 12235	. . . . 5 |- ( A (,) +oo ) C_ RR
3	2	a1i 11	. . . 4 |- ( A e. RR -> ( A (,) +oo ) C_ RR )
4		elpwi 4168	. . . . . 6 |- ( x e. ~P RR -> x C_ RR )
5		simplrl 800	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) -> x C_ RR )
6		simplrr 801	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) -> ( vol* ` x ) e. RR )
7		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) -> y e. RR+ )
8		eqid 2622	. . . . . . . . . . . 12 |- seq 1 ( + , ( ( abs o. - ) o. f ) ) = seq 1 ( + , ( ( abs o. - ) o. f ) )
9	8	ovolgelb 23248	. . . . . . . . . . 11 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR /\ y e. RR+ ) -> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) )
10	5, 6, 7, 9	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) -> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) )
11		eqid 2622	. . . . . . . . . . 11 |- ( A (,) +oo ) = ( A (,) +oo )
12		simplll 798	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> A e. RR )
13	5	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> x C_ RR )
14	6	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> ( vol* ` x ) e. RR )
15		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> y e. RR+ )
16		eqid 2622	. . . . . . . . . . 11 |- seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) >. ) ) ) = seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) >. ) ) )
17		eqid 2622	. . . . . . . . . . 11 |- seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. ( 1st ` ( f ` m ) ) , if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) >. ) ) ) = seq 1 ( + , ( ( abs o. - ) o. ( m e. NN |-> <. ( 1st ` ( f ` m ) ) , if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) >. ) ) )
18		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
19		reex 10027	. . . . . . . . . . . . . . 15 |- RR e. _V
20	19, 19	xpex 6962	. . . . . . . . . . . . . 14 |- ( RR X. RR ) e. _V
21	20	inex2 4800	. . . . . . . . . . . . 13 |- ( <_ i^i ( RR X. RR ) ) e. _V
22		nnex 11026	. . . . . . . . . . . . 13 |- NN e. _V
23	21, 22	elmap 7886	. . . . . . . . . . . 12 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> f : NN --> ( <_ i^i ( RR X. RR ) ) )
24	18, 23	sylib 208	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> f : NN --> ( <_ i^i ( RR X. RR ) ) )
25		simprrl 804	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> x C_ U. ran ( (,) o. f ) )
26		simprrr 805	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) )
27		eqid 2622	. . . . . . . . . . 11 |- ( 1st ` ( f ` n ) ) = ( 1st ` ( f ` n ) )
28		eqid 2622	. . . . . . . . . . 11 |- ( 2nd ` ( f ` n ) ) = ( 2nd ` ( f ` n ) )
29		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( m = n -> ( f ` m ) = ( f ` n ) )
30	29	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( m = n -> ( 1st ` ( f ` m ) ) = ( 1st ` ( f ` n ) ) )
31	30	breq1d 4663	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( ( 1st ` ( f ` m ) ) <_ A <-> ( 1st ` ( f ` n ) ) <_ A ) )
32	31, 30	ifbieq2d 4111	. . . . . . . . . . . . . . 15 |- ( m = n -> if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) = if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) )
33	29	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( m = n -> ( 2nd ` ( f ` m ) ) = ( 2nd ` ( f ` n ) ) )
34	32, 33	breq12d 4666	. . . . . . . . . . . . . 14 |- ( m = n -> ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) <-> if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) ) )
35	34, 32, 33	ifbieq12d 4113	. . . . . . . . . . . . 13 |- ( m = n -> if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) = if ( if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) , if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) ) )
36	35, 33	opeq12d 4410	. . . . . . . . . . . 12 |- ( m = n -> <. if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) >. = <. if ( if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) , if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) >. )
37	36	cbvmptv 4750	. . . . . . . . . . 11 |- ( m e. NN |-> <. if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) >. ) = ( n e. NN |-> <. if ( if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) , if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) >. )
38	30, 35	opeq12d 4410	. . . . . . . . . . . 12 |- ( m = n -> <. ( 1st ` ( f ` m ) ) , if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) >. = <. ( 1st ` ( f ` n ) ) , if ( if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) , if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) ) >. )
39	38	cbvmptv 4750	. . . . . . . . . . 11 |- ( m e. NN |-> <. ( 1st ` ( f ` m ) ) , if ( if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) <_ ( 2nd ` ( f ` m ) ) , if ( ( 1st ` ( f ` m ) ) <_ A , A , ( 1st ` ( f ` m ) ) ) , ( 2nd ` ( f ` m ) ) ) >. ) = ( n e. NN |-> <. ( 1st ` ( f ` n ) ) , if ( if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) <_ ( 2nd ` ( f ` n ) ) , if ( ( 1st ` ( f ` n ) ) <_ A , A , ( 1st ` ( f ` n ) ) ) , ( 2nd ` ( f ` n ) ) ) >. )
40	11, 12, 13, 14, 15, 8, 16, 17, 24, 25, 26, 27, 28, 37, 39	ioombl1lem4 23329	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) /\ ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ ( x C_ U. ran ( (,) o. f ) /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <_ ( ( vol* ` x ) + y ) ) ) ) -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( ( vol* ` x ) + y ) )
41	10, 40	rexlimddv 3035	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) /\ y e. RR+ ) -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( ( vol* ` x ) + y ) )
42	41	ralrimiva 2966	. . . . . . . 8 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> A. y e. RR+ ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( ( vol* ` x ) + y ) )
43		inss1 3833	. . . . . . . . . . . 12 |- ( x i^i ( A (,) +oo ) ) C_ x
44		ovolsscl 23254	. . . . . . . . . . . 12 |- ( ( ( x i^i ( A (,) +oo ) ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( A (,) +oo ) ) ) e. RR )
45	43, 44	mp3an1 1411	. . . . . . . . . . 11 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( A (,) +oo ) ) ) e. RR )
46	45	adantl 482	. . . . . . . . . 10 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( vol* ` ( x i^i ( A (,) +oo ) ) ) e. RR )
47		difss 3737	. . . . . . . . . . . 12 |- ( x \ ( A (,) +oo ) ) C_ x
48		ovolsscl 23254	. . . . . . . . . . . 12 |- ( ( ( x \ ( A (,) +oo ) ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ ( A (,) +oo ) ) ) e. RR )
49	47, 48	mp3an1 1411	. . . . . . . . . . 11 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ ( A (,) +oo ) ) ) e. RR )
50	49	adantl 482	. . . . . . . . . 10 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( vol* ` ( x \ ( A (,) +oo ) ) ) e. RR )
51	46, 50	readdcld 10069	. . . . . . . . 9 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) e. RR )
52		simprr 796	. . . . . . . . 9 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( vol* ` x ) e. RR )
53		alrple 12037	. . . . . . . . 9 |- ( ( ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) e. RR /\ ( vol* ` x ) e. RR ) -> ( ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) <-> A. y e. RR+ ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( ( vol* ` x ) + y ) ) )
54	51, 52, 53	syl2anc 693	. . . . . . . 8 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) <-> A. y e. RR+ ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( ( vol* ` x ) + y ) ) )
55	42, 54	mpbird 247	. . . . . . 7 |- ( ( A e. RR /\ ( x C_ RR /\ ( vol* ` x ) e. RR ) ) -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) )
56	55	expr 643	. . . . . 6 |- ( ( A e. RR /\ x C_ RR ) -> ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) ) )
57	4, 56	sylan2 491	. . . . 5 |- ( ( A e. RR /\ x e. ~P RR ) -> ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) ) )
58	57	ralrimiva 2966	. . . 4 |- ( A e. RR -> A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) ) )
59		ismbl2 23295	. . . 4 |- ( ( A (,) +oo ) e. dom vol <-> ( ( A (,) +oo ) C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i ( A (,) +oo ) ) ) + ( vol* ` ( x \ ( A (,) +oo ) ) ) ) <_ ( vol* ` x ) ) ) )
60	3, 58, 59	sylanbrc 698	. . 3 |- ( A e. RR -> ( A (,) +oo ) e. dom vol )
61		oveq1 6657	. . . . 5 |- ( A = +oo -> ( A (,) +oo ) = ( +oo (,) +oo ) )
62		iooid 12203	. . . . 5 |- ( +oo (,) +oo ) = (/)
63	61, 62	syl6eq 2672	. . . 4 |- ( A = +oo -> ( A (,) +oo ) = (/) )
64		0mbl 23307	. . . 4 |- (/) e. dom vol
65	63, 64	syl6eqel 2709	. . 3 |- ( A = +oo -> ( A (,) +oo ) e. dom vol )
66		oveq1 6657	. . . . 5 |- ( A = -oo -> ( A (,) +oo ) = ( -oo (,) +oo ) )
67		ioomax 12248	. . . . 5 |- ( -oo (,) +oo ) = RR
68	66, 67	syl6eq 2672	. . . 4 |- ( A = -oo -> ( A (,) +oo ) = RR )
69		rembl 23308	. . . 4 |- RR e. dom vol
70	68, 69	syl6eqel 2709	. . 3 |- ( A = -oo -> ( A (,) +oo ) e. dom vol )
71	60, 65, 70	3jaoi 1391	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> ( A (,) +oo ) e. dom vol )
72	1, 71	sylbi 207	1 |- ( A e. RR* -> ( A (,) +oo ) e. dom vol )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   supcsup 8346   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   RR+crp 11832   (,)cioo 12175   seqcseq 12801   abscabs 13974   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-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-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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-xadd 11947  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-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234

This theorem is referenced by:  icombl1  23331