Theorem i1fd 23448

Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fd.1	|- ( ph -> F : RR --> RR )
i1fd.2	|- ( ph -> ran F e. Fin )
i1fd.3	|- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( `' F " { x } ) e. dom vol )
i1fd.4	|- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { x } ) ) e. RR )

Assertion

Ref	Expression
i1fd	|- ( ph -> F e. dom S.1 )

Distinct variable groups:   x, F   ph, x

Proof of Theorem i1fd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fd.1	. . . . . . . . 9 |- ( ph -> F : RR --> RR )
2	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> F : RR --> RR )
3		ffun 6048	. . . . . . . 8 |- ( F : RR --> RR -> Fun F )
4		funcnvcnv 5956	. . . . . . . 8 |- ( Fun F -> Fun `' `' F )
5		imadif 5973	. . . . . . . 8 |- ( Fun `' `' F -> ( `' F " ( RR \ ( RR \ x ) ) ) = ( ( `' F " RR ) \ ( `' F " ( RR \ x ) ) ) )
6	2, 3, 4, 5	4syl 19	. . . . . . 7 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " ( RR \ ( RR \ x ) ) ) = ( ( `' F " RR ) \ ( `' F " ( RR \ x ) ) ) )
7		ioof 12271	. . . . . . . . . . . . 13 |- (,) : ( RR* X. RR* ) --> ~P RR
8		frn 6053	. . . . . . . . . . . . 13 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 |- ran (,) C_ ~P RR
10	9	sseli 3599	. . . . . . . . . . 11 |- ( x e. ran (,) -> x e. ~P RR )
11	10	elpwid 4170	. . . . . . . . . 10 |- ( x e. ran (,) -> x C_ RR )
12	11	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> x C_ RR )
13		dfss4 3858	. . . . . . . . 9 |- ( x C_ RR <-> ( RR \ ( RR \ x ) ) = x )
14	12, 13	sylib 208	. . . . . . . 8 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( RR \ ( RR \ x ) ) = x )
15	14	imaeq2d 5466	. . . . . . 7 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " ( RR \ ( RR \ x ) ) ) = ( `' F " x ) )
16	6, 15	eqtr3d 2658	. . . . . 6 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( ( `' F " RR ) \ ( `' F " ( RR \ x ) ) ) = ( `' F " x ) )
17		fimacnv 6347	. . . . . . . . 9 |- ( F : RR --> RR -> ( `' F " RR ) = RR )
18	2, 17	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " RR ) = RR )
19		rembl 23308	. . . . . . . 8 |- RR e. dom vol
20	18, 19	syl6eqel 2709	. . . . . . 7 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " RR ) e. dom vol )
21	1	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ -. 0 e. y ) -> F : RR --> RR )
22		inpreima 6342	. . . . . . . . . . . . . 14 |- ( Fun F -> ( `' F " ( y i^i ran F ) ) = ( ( `' F " y ) i^i ( `' F " ran F ) ) )
23		iunid 4575	. . . . . . . . . . . . . . . 16 |- U_ x e. ( y i^i ran F ) { x } = ( y i^i ran F )
24	23	imaeq2i 5464	. . . . . . . . . . . . . . 15 |- ( `' F " U_ x e. ( y i^i ran F ) { x } ) = ( `' F " ( y i^i ran F ) )
25		imaiun 6503	. . . . . . . . . . . . . . 15 |- ( `' F " U_ x e. ( y i^i ran F ) { x } ) = U_ x e. ( y i^i ran F ) ( `' F " { x } )
26	24, 25	eqtr3i 2646	. . . . . . . . . . . . . 14 |- ( `' F " ( y i^i ran F ) ) = U_ x e. ( y i^i ran F ) ( `' F " { x } )
27		cnvimass 5485	. . . . . . . . . . . . . . . 16 |- ( `' F " y ) C_ dom F
28		cnvimarndm 5486	. . . . . . . . . . . . . . . 16 |- ( `' F " ran F ) = dom F
29	27, 28	sseqtr4i 3638	. . . . . . . . . . . . . . 15 |- ( `' F " y ) C_ ( `' F " ran F )
30		df-ss 3588	. . . . . . . . . . . . . . 15 |- ( ( `' F " y ) C_ ( `' F " ran F ) <-> ( ( `' F " y ) i^i ( `' F " ran F ) ) = ( `' F " y ) )
31	29, 30	mpbi 220	. . . . . . . . . . . . . 14 |- ( ( `' F " y ) i^i ( `' F " ran F ) ) = ( `' F " y )
32	22, 26, 31	3eqtr3g 2679	. . . . . . . . . . . . 13 |- ( Fun F -> U_ x e. ( y i^i ran F ) ( `' F " { x } ) = ( `' F " y ) )
33	21, 3, 32	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ -. 0 e. y ) -> U_ x e. ( y i^i ran F ) ( `' F " { x } ) = ( `' F " y ) )
34		i1fd.2	. . . . . . . . . . . . . . 15 |- ( ph -> ran F e. Fin )
35	34	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. 0 e. y ) -> ran F e. Fin )
36		inss2 3834	. . . . . . . . . . . . . 14 |- ( y i^i ran F ) C_ ran F
37		ssfi 8180	. . . . . . . . . . . . . 14 |- ( ( ran F e. Fin /\ ( y i^i ran F ) C_ ran F ) -> ( y i^i ran F ) e. Fin )
38	35, 36, 37	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ph /\ -. 0 e. y ) -> ( y i^i ran F ) e. Fin )
39		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ph )
40		inss1 3833	. . . . . . . . . . . . . . . . . . . . 21 |- ( y i^i ran F ) C_ y
41	40	sseli 3599	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 e. ( y i^i ran F ) -> 0 e. y )
42	41	con3i 150	. . . . . . . . . . . . . . . . . . 19 |- ( -. 0 e. y -> -. 0 e. ( y i^i ran F ) )
43	42	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ -. 0 e. y ) -> -. 0 e. ( y i^i ran F ) )
44		disjsn 4246	. . . . . . . . . . . . . . . . . 18 |- ( ( ( y i^i ran F ) i^i { 0 } ) = (/) <-> -. 0 e. ( y i^i ran F ) )
45	43, 44	sylibr 224	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. 0 e. y ) -> ( ( y i^i ran F ) i^i { 0 } ) = (/) )
46		reldisj 4020	. . . . . . . . . . . . . . . . . 18 |- ( ( y i^i ran F ) C_ ran F -> ( ( ( y i^i ran F ) i^i { 0 } ) = (/) <-> ( y i^i ran F ) C_ ( ran F \ { 0 } ) ) )
47	36, 46	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( ( ( y i^i ran F ) i^i { 0 } ) = (/) <-> ( y i^i ran F ) C_ ( ran F \ { 0 } ) )
48	45, 47	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ -. 0 e. y ) -> ( y i^i ran F ) C_ ( ran F \ { 0 } ) )
49	48	sselda 3603	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> x e. ( ran F \ { 0 } ) )
50		i1fd.3	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( `' F " { x } ) e. dom vol )
51	39, 49, 50	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( `' F " { x } ) e. dom vol )
52	51	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ( ph /\ -. 0 e. y ) -> A. x e. ( y i^i ran F ) ( `' F " { x } ) e. dom vol )
53		finiunmbl 23312	. . . . . . . . . . . . 13 |- ( ( ( y i^i ran F ) e. Fin /\ A. x e. ( y i^i ran F ) ( `' F " { x } ) e. dom vol ) -> U_ x e. ( y i^i ran F ) ( `' F " { x } ) e. dom vol )
54	38, 52, 53	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ -. 0 e. y ) -> U_ x e. ( y i^i ran F ) ( `' F " { x } ) e. dom vol )
55	33, 54	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ( ph /\ -. 0 e. y ) -> ( `' F " y ) e. dom vol )
56	55	ex 450	. . . . . . . . . 10 |- ( ph -> ( -. 0 e. y -> ( `' F " y ) e. dom vol ) )
57	56	alrimiv 1855	. . . . . . . . 9 |- ( ph -> A. y ( -. 0 e. y -> ( `' F " y ) e. dom vol ) )
58	57	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> A. y ( -. 0 e. y -> ( `' F " y ) e. dom vol ) )
59		elndif 3734	. . . . . . . . 9 |- ( 0 e. x -> -. 0 e. ( RR \ x ) )
60	59	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> -. 0 e. ( RR \ x ) )
61		reex 10027	. . . . . . . . . 10 |- RR e. _V
62		difexg 4808	. . . . . . . . . 10 |- ( RR e. _V -> ( RR \ x ) e. _V )
63	61, 62	ax-mp 5	. . . . . . . . 9 |- ( RR \ x ) e. _V
64		eleq2 2690	. . . . . . . . . . 11 |- ( y = ( RR \ x ) -> ( 0 e. y <-> 0 e. ( RR \ x ) ) )
65	64	notbid 308	. . . . . . . . . 10 |- ( y = ( RR \ x ) -> ( -. 0 e. y <-> -. 0 e. ( RR \ x ) ) )
66		imaeq2 5462	. . . . . . . . . . 11 |- ( y = ( RR \ x ) -> ( `' F " y ) = ( `' F " ( RR \ x ) ) )
67	66	eleq1d 2686	. . . . . . . . . 10 |- ( y = ( RR \ x ) -> ( ( `' F " y ) e. dom vol <-> ( `' F " ( RR \ x ) ) e. dom vol ) )
68	65, 67	imbi12d 334	. . . . . . . . 9 |- ( y = ( RR \ x ) -> ( ( -. 0 e. y -> ( `' F " y ) e. dom vol ) <-> ( -. 0 e. ( RR \ x ) -> ( `' F " ( RR \ x ) ) e. dom vol ) ) )
69	63, 68	spcv 3299	. . . . . . . 8 |- ( A. y ( -. 0 e. y -> ( `' F " y ) e. dom vol ) -> ( -. 0 e. ( RR \ x ) -> ( `' F " ( RR \ x ) ) e. dom vol ) )
70	58, 60, 69	sylc 65	. . . . . . 7 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " ( RR \ x ) ) e. dom vol )
71		difmbl 23311	. . . . . . 7 |- ( ( ( `' F " RR ) e. dom vol /\ ( `' F " ( RR \ x ) ) e. dom vol ) -> ( ( `' F " RR ) \ ( `' F " ( RR \ x ) ) ) e. dom vol )
72	20, 70, 71	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( ( `' F " RR ) \ ( `' F " ( RR \ x ) ) ) e. dom vol )
73	16, 72	eqeltrrd 2702	. . . . 5 |- ( ( ( ph /\ x e. ran (,) ) /\ 0 e. x ) -> ( `' F " x ) e. dom vol )
74		eleq2 2690	. . . . . . . . . . 11 |- ( y = x -> ( 0 e. y <-> 0 e. x ) )
75	74	notbid 308	. . . . . . . . . 10 |- ( y = x -> ( -. 0 e. y <-> -. 0 e. x ) )
76		imaeq2 5462	. . . . . . . . . . 11 |- ( y = x -> ( `' F " y ) = ( `' F " x ) )
77	76	eleq1d 2686	. . . . . . . . . 10 |- ( y = x -> ( ( `' F " y ) e. dom vol <-> ( `' F " x ) e. dom vol ) )
78	75, 77	imbi12d 334	. . . . . . . . 9 |- ( y = x -> ( ( -. 0 e. y -> ( `' F " y ) e. dom vol ) <-> ( -. 0 e. x -> ( `' F " x ) e. dom vol ) ) )
79	78	spv 2260	. . . . . . . 8 |- ( A. y ( -. 0 e. y -> ( `' F " y ) e. dom vol ) -> ( -. 0 e. x -> ( `' F " x ) e. dom vol ) )
80	57, 79	syl 17	. . . . . . 7 |- ( ph -> ( -. 0 e. x -> ( `' F " x ) e. dom vol ) )
81	80	imp 445	. . . . . 6 |- ( ( ph /\ -. 0 e. x ) -> ( `' F " x ) e. dom vol )
82	81	adantlr 751	. . . . 5 |- ( ( ( ph /\ x e. ran (,) ) /\ -. 0 e. x ) -> ( `' F " x ) e. dom vol )
83	73, 82	pm2.61dan 832	. . . 4 |- ( ( ph /\ x e. ran (,) ) -> ( `' F " x ) e. dom vol )
84	83	ralrimiva 2966	. . 3 |- ( ph -> A. x e. ran (,) ( `' F " x ) e. dom vol )
85		ismbf 23397	. . . 4 |- ( F : RR --> RR -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) )
86	1, 85	syl 17	. . 3 |- ( ph -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) )
87	84, 86	mpbird 247	. 2 |- ( ph -> F e. MblFn )
88		mblvol 23298	. . . . . . . 8 |- ( ( `' F " y ) e. dom vol -> ( vol ` ( `' F " y ) ) = ( vol* ` ( `' F " y ) ) )
89	55, 88	syl 17	. . . . . . 7 |- ( ( ph /\ -. 0 e. y ) -> ( vol ` ( `' F " y ) ) = ( vol* ` ( `' F " y ) ) )
90		mblss 23299	. . . . . . . . 9 |- ( ( `' F " y ) e. dom vol -> ( `' F " y ) C_ RR )
91	55, 90	syl 17	. . . . . . . 8 |- ( ( ph /\ -. 0 e. y ) -> ( `' F " y ) C_ RR )
92		mblvol 23298	. . . . . . . . . . 11 |- ( ( `' F " { x } ) e. dom vol -> ( vol ` ( `' F " { x } ) ) = ( vol* ` ( `' F " { x } ) ) )
93	51, 92	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( vol ` ( `' F " { x } ) ) = ( vol* ` ( `' F " { x } ) ) )
94		i1fd.4	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ran F \ { 0 } ) ) -> ( vol ` ( `' F " { x } ) ) e. RR )
95	39, 49, 94	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( vol ` ( `' F " { x } ) ) e. RR )
96	93, 95	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( vol* ` ( `' F " { x } ) ) e. RR )
97	38, 96	fsumrecl 14465	. . . . . . . 8 |- ( ( ph /\ -. 0 e. y ) -> sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) e. RR )
98	33	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ -. 0 e. y ) -> ( vol* ` U_ x e. ( y i^i ran F ) ( `' F " { x } ) ) = ( vol* ` ( `' F " y ) ) )
99		mblss 23299	. . . . . . . . . . . . 13 |- ( ( `' F " { x } ) e. dom vol -> ( `' F " { x } ) C_ RR )
100	51, 99	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( `' F " { x } ) C_ RR )
101	100, 96	jca 554	. . . . . . . . . . 11 |- ( ( ( ph /\ -. 0 e. y ) /\ x e. ( y i^i ran F ) ) -> ( ( `' F " { x } ) C_ RR /\ ( vol* ` ( `' F " { x } ) ) e. RR ) )
102	101	ralrimiva 2966	. . . . . . . . . 10 |- ( ( ph /\ -. 0 e. y ) -> A. x e. ( y i^i ran F ) ( ( `' F " { x } ) C_ RR /\ ( vol* ` ( `' F " { x } ) ) e. RR ) )
103		ovolfiniun 23269	. . . . . . . . . 10 |- ( ( ( y i^i ran F ) e. Fin /\ A. x e. ( y i^i ran F ) ( ( `' F " { x } ) C_ RR /\ ( vol* ` ( `' F " { x } ) ) e. RR ) ) -> ( vol* ` U_ x e. ( y i^i ran F ) ( `' F " { x } ) ) <_ sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) )
104	38, 102, 103	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ -. 0 e. y ) -> ( vol* ` U_ x e. ( y i^i ran F ) ( `' F " { x } ) ) <_ sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) )
105	98, 104	eqbrtrrd 4677	. . . . . . . 8 |- ( ( ph /\ -. 0 e. y ) -> ( vol* ` ( `' F " y ) ) <_ sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) )
106		ovollecl 23251	. . . . . . . 8 |- ( ( ( `' F " y ) C_ RR /\ sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) e. RR /\ ( vol* ` ( `' F " y ) ) <_ sum_ x e. ( y i^i ran F ) ( vol* ` ( `' F " { x } ) ) ) -> ( vol* ` ( `' F " y ) ) e. RR )
107	91, 97, 105, 106	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ -. 0 e. y ) -> ( vol* ` ( `' F " y ) ) e. RR )
108	89, 107	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ -. 0 e. y ) -> ( vol ` ( `' F " y ) ) e. RR )
109	108	ex 450	. . . . 5 |- ( ph -> ( -. 0 e. y -> ( vol ` ( `' F " y ) ) e. RR ) )
110	109	alrimiv 1855	. . . 4 |- ( ph -> A. y ( -. 0 e. y -> ( vol ` ( `' F " y ) ) e. RR ) )
111		neldifsn 4321	. . . 4 |- -. 0 e. ( RR \ { 0 } )
112		difexg 4808	. . . . . 6 |- ( RR e. _V -> ( RR \ { 0 } ) e. _V )
113	61, 112	ax-mp 5	. . . . 5 |- ( RR \ { 0 } ) e. _V
114		eleq2 2690	. . . . . . 7 |- ( y = ( RR \ { 0 } ) -> ( 0 e. y <-> 0 e. ( RR \ { 0 } ) ) )
115	114	notbid 308	. . . . . 6 |- ( y = ( RR \ { 0 } ) -> ( -. 0 e. y <-> -. 0 e. ( RR \ { 0 } ) ) )
116		imaeq2 5462	. . . . . . . 8 |- ( y = ( RR \ { 0 } ) -> ( `' F " y ) = ( `' F " ( RR \ { 0 } ) ) )
117	116	fveq2d 6195	. . . . . . 7 |- ( y = ( RR \ { 0 } ) -> ( vol ` ( `' F " y ) ) = ( vol ` ( `' F " ( RR \ { 0 } ) ) ) )
118	117	eleq1d 2686	. . . . . 6 |- ( y = ( RR \ { 0 } ) -> ( ( vol ` ( `' F " y ) ) e. RR <-> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) )
119	115, 118	imbi12d 334	. . . . 5 |- ( y = ( RR \ { 0 } ) -> ( ( -. 0 e. y -> ( vol ` ( `' F " y ) ) e. RR ) <-> ( -. 0 e. ( RR \ { 0 } ) -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )
120	113, 119	spcv 3299	. . . 4 |- ( A. y ( -. 0 e. y -> ( vol ` ( `' F " y ) ) e. RR ) -> ( -. 0 e. ( RR \ { 0 } ) -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) )
121	110, 111, 120	mpisyl 21	. . 3 |- ( ph -> ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR )
122	1, 34, 121	3jca 1242	. 2 |- ( ph -> ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) )
123		isi1f 23441	. 2 |- ( F e. dom S.1 <-> ( F e. MblFn /\ ( F : RR --> RR /\ ran F e. Fin /\ ( vol ` ( `' F " ( RR \ { 0 } ) ) ) e. RR ) ) )
124	87, 122, 123	sylanbrc 698	1 |- ( ph -> F e. dom S.1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888   Fincfn 7955   RRcr 9935   0cc0 9936   RR*cxr 10073   <_ cle 10075   (,)cioo 12175   sum_csu 14416   vol*covol 23231   volcvol 23232  MblFncmbf 23383   S.1citg1 23384

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-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389

This theorem is referenced by:  i1f0  23454  i1f1  23457  i1fadd  23462  i1fmul  23463  i1fmulc  23470  i1fres  23472  mbfi1fseqlem4  23485  itg2addnclem2  33462  ftc1anclem3  33487