bddmulibl - Metamath Proof Explorer

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Theorem bddmulibl 23605

Description: A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Assertion**
Ref	Expression
bddmulibl	MblFn $/\$ $/\$

Distinct variable groups:

Proof of Theorem bddmulibl Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbff 23394	. . . . . . 7 MblFn
2	1	ad2antrr 762	. . . . . 6 MblFn $/\$ $/\$ $/\$
3		ffn 6045	. . . . . 6
4	2, 3	syl 17	. . . . 5 MblFn $/\$ $/\$ $/\$
5		iblmbf 23534	. . . . . . . 8 MblFn
6	5	ad2antlr 763	. . . . . . 7 MblFn $/\$ $/\$ $/\$ MblFn
7		mbff 23394	. . . . . . 7 MblFn
8	6, 7	syl 17	. . . . . 6 MblFn $/\$ $/\$ $/\$
9		ffn 6045	. . . . . 6
10	8, 9	syl 17	. . . . 5 MblFn $/\$ $/\$ $/\$
11		mbfdm 23395	. . . . . 6 MblFn
12	11	ad2antrr 762	. . . . 5 MblFn $/\$ $/\$ $/\$
13		mbfdm 23395	. . . . . 6 MblFn
14	6, 13	syl 17	. . . . 5 MblFn $/\$ $/\$ $/\$
15		eqid 2622	. . . . 5
16		eqidd 2623	. . . . 5 MblFn $/\$ $/\$ $/\$ $/\$
17		eqidd 2623	. . . . 5 MblFn $/\$ $/\$ $/\$ $/\$
18	4, 10, 12, 14, 15, 16, 17	offval 6904	. . . 4 MblFn $/\$ $/\$ $/\$
19		ovexd 6680	. . . . 5 MblFn $/\$ $/\$ $/\$ $/\$
20		simpll 790	. . . . . . 7 MblFn $/\$ $/\$ $/\$ MblFn
21	20, 6	mbfmul 23493	. . . . . 6 MblFn $/\$ $/\$ $/\$ MblFn
22	18, 21	eqeltrrd 2702	. . . . 5 MblFn $/\$ $/\$ $/\$ MblFn
23	22, 19	mbfmptcl 23404	. . . . . . . 8 MblFn $/\$ $/\$ $/\$ $/\$
24		eqidd 2623	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
25		absf 14077	. . . . . . . . . 10
26	25	a1i 11	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$
27	26	feqmptd 6249	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
28		fveq2 6191	. . . . . . . 8
29	23, 24, 27, 28	fmptco 6396	. . . . . . 7 MblFn $/\$ $/\$ $/\$
30		eqid 2622	. . . . . . . . 9
31	23, 30	fmptd 6385	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
32		ax-resscn 9993	. . . . . . . . . . 11
33		ssid 3624	. . . . . . . . . . 11
34		cncfss 22702	. . . . . . . . . . 11 $/\$
35	32, 33, 34	mp2an 708	. . . . . . . . . 10
36		abscncf 22704	. . . . . . . . . 10
37	35, 36	sselii 3600	. . . . . . . . 9
38	37	a1i 11	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
39		cncombf 23425	. . . . . . . 8 MblFn $/\$ $/\$ MblFn
40	22, 31, 38, 39	syl3anc 1326	. . . . . . 7 MblFn $/\$ $/\$ $/\$ MblFn
41	29, 40	eqeltrrd 2702	. . . . . 6 MblFn $/\$ $/\$ $/\$ MblFn
42	23	abscld 14175	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
43	42	rexrd 10089	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
44	23	absge0d 14183	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
45		elxrge0 12281	. . . . . . . . . . 11 $/\$
46	43, 44, 45	sylanbrc 698	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$ $/\$
47		0e0iccpnf 12283	. . . . . . . . . . 11
48	47	a1i 11	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$ $/\$
49	46, 48	ifclda 4120	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$
50	49	adantr 481	. . . . . . . 8 MblFn $/\$ $/\$ $/\$ $/\$
51		eqid 2622	. . . . . . . 8
52	50, 51	fmptd 6385	. . . . . . 7 MblFn $/\$ $/\$ $/\$
53		reex 10027	. . . . . . . . . . . . . . 15
54	53	a1i 11	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$
55		simprl 794	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$
56	55	ad2antrr 762	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$ $/\$
57		elin 3796	. . . . . . . . . . . . . . . . . . . 20 $/\$
58	57	simprbi 480	. . . . . . . . . . . . . . . . . . 19
59		ffvelrn 6357	. . . . . . . . . . . . . . . . . . 19 $/\$
60	8, 58, 59	syl2an 494	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
61	60	abscld 14175	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$ $/\$
62	60	absge0d 14183	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$ $/\$
63		elrege0 12278	. . . . . . . . . . . . . . . . 17 $/\$
64	61, 62, 63	sylanbrc 698	. . . . . . . . . . . . . . . 16 MblFn $/\$ $/\$ $/\$ $/\$
65		0e0icopnf 12282	. . . . . . . . . . . . . . . . 17
66	65	a1i 11	. . . . . . . . . . . . . . . 16 MblFn $/\$ $/\$ $/\$ $/\$
67	64, 66	ifclda 4120	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$
68	67	ad2antrr 762	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$ $/\$
69		fconstmpt 5163	. . . . . . . . . . . . . . 15
70	69	a1i 11	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$
71		eqidd 2623	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$
72	54, 56, 68, 70, 71	offval2 6914	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
73		ovif2 6738	. . . . . . . . . . . . . . 15
74	55	recnd 10068	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$
75	74	adantr 481	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$ $/\$
76	75	mul01d 10235	. . . . . . . . . . . . . . . 16 MblFn $/\$ $/\$ $/\$ $/\$
77	76	ifeq2d 4105	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$ $/\$
78	73, 77	syl5eq 2668	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$
79	78	mpteq2dv 4745	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
80	72, 79	eqtrd 2656	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
81	80	fveq2d 6195	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
82	67	adantr 481	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$
83		eqid 2622	. . . . . . . . . . . . . 14
84	82, 83	fmptd 6385	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$
85	84	adantr 481	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
86		inss2 3834	. . . . . . . . . . . . . . . . . 18
87	86	a1i 11	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$
88	22, 19	mbfdm2 23405	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$
89	8	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$ $/\$
90	8	feqmptd 6249	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$
91		simplr 792	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$
92	90, 91	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$
93	87, 88, 89, 92	iblss 23571	. . . . . . . . . . . . . . . 16 MblFn $/\$ $/\$ $/\$
94	60, 93	iblabs 23595	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$
95	61, 62	iblpos 23559	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$ MblFn $/\$
96	94, 95	mpbid 222	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ MblFn $/\$
97	96	simprd 479	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$
98	97	adantr 481	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
99		simplrl 800	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
100		neq0 3930	. . . . . . . . . . . . . . 15
101		0re 10040	. . . . . . . . . . . . . . . . . . 19
102	101	a1i 11	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
103	57	simplbi 476	. . . . . . . . . . . . . . . . . . . 20
104		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . 20 $/\$
105	2, 103, 104	syl2an 494	. . . . . . . . . . . . . . . . . . 19 MblFn $/\$ $/\$ $/\$ $/\$
106	105	abscld 14175	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
107		simplrl 800	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
108	105	absge0d 14183	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
109		simprr 796	. . . . . . . . . . . . . . . . . . 19 MblFn $/\$ $/\$ $/\$
110		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22
111	110	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21
112	111	breq1d 4663	. . . . . . . . . . . . . . . . . . . 20
113	112	rspccva 3308	. . . . . . . . . . . . . . . . . . 19 $/\$
114	109, 103, 113	syl2an 494	. . . . . . . . . . . . . . . . . 18 MblFn $/\$ $/\$ $/\$ $/\$
115	102, 106, 107, 108, 114	letrd 10194	. . . . . . . . . . . . . . . . 17 MblFn $/\$ $/\$ $/\$ $/\$
116	115	ex 450	. . . . . . . . . . . . . . . 16 MblFn $/\$ $/\$ $/\$
117	116	exlimdv 1861	. . . . . . . . . . . . . . 15 MblFn $/\$ $/\$ $/\$
118	100, 117	syl5bi 232	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$
119	118	imp 445	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
120		elrege0 12278	. . . . . . . . . . . . 13 $/\$
121	99, 119, 120	sylanbrc 698	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
122	85, 98, 121	itg2mulc 23514	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
123	81, 122	eqtr3d 2658	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$ $/\$
124	99, 98	remulcld 10070	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$ $/\$
125	123, 124	eqeltrd 2701	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$ $/\$
126	125	ex 450	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
127		noel 3919	. . . . . . . . . . . . . 14
128		eleq2 2690	. . . . . . . . . . . . . 14
129	127, 128	mtbiri 317	. . . . . . . . . . . . 13
130		iffalse 4095	. . . . . . . . . . . . 13
131	129, 130	syl 17	. . . . . . . . . . . 12
132	131	mpteq2dv 4745	. . . . . . . . . . 11
133		fconstmpt 5163	. . . . . . . . . . 11
134	132, 133	syl6eqr 2674	. . . . . . . . . 10
135	134	fveq2d 6195	. . . . . . . . 9
136		itg20 23504	. . . . . . . . . 10
137	136, 101	eqeltri 2697	. . . . . . . . 9
138	135, 137	syl6eqel 2709	. . . . . . . 8
139	126, 138	pm2.61d2 172	. . . . . . 7 MblFn $/\$ $/\$ $/\$
140	107, 61	remulcld 10070	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
141	140	rexrd 10089	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
142	107, 61, 115, 62	mulge0d 10604	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
143		elxrge0 12281	. . . . . . . . . . . 12 $/\$
144	141, 142, 143	sylanbrc 698	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
145	144, 48	ifclda 4120	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$
146	145	adantr 481	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$ $/\$
147		eqid 2622	. . . . . . . . 9
148	146, 147	fmptd 6385	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
149	105, 60	absmuld 14193	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
150		abscl 14018	. . . . . . . . . . . . . . . 16
151		absge0 14027	. . . . . . . . . . . . . . . 16
152	150, 151	jca 554	. . . . . . . . . . . . . . 15 $/\$
153	60, 152	syl 17	. . . . . . . . . . . . . 14 MblFn $/\$ $/\$ $/\$ $/\$ $/\$
154		lemul1a 10877	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
155	106, 107, 153, 114, 154	syl31anc 1329	. . . . . . . . . . . . 13 MblFn $/\$ $/\$ $/\$ $/\$
156	149, 155	eqbrtrd 4675	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
157		iftrue 4092	. . . . . . . . . . . . 13
158	157	adantl 482	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
159		iftrue 4092	. . . . . . . . . . . . 13
160	159	adantl 482	. . . . . . . . . . . 12 MblFn $/\$ $/\$ $/\$ $/\$
161	156, 158, 160	3brtr4d 4685	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
162		0le0 11110	. . . . . . . . . . . . . 14
163	162	a1i 11	. . . . . . . . . . . . 13
164		iffalse 4095	. . . . . . . . . . . . 13
165	163, 164, 130	3brtr4d 4685	. . . . . . . . . . . 12
166	165	adantl 482	. . . . . . . . . . 11 MblFn $/\$ $/\$ $/\$ $/\$
167	161, 166	pm2.61dan 832	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$
168	167	ralrimivw 2967	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$
169	53	a1i 11	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$
170		eqidd 2623	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$
171		eqidd 2623	. . . . . . . . . 10 MblFn $/\$ $/\$ $/\$
172	169, 50, 146, 170, 171	ofrfval2 6915	. . . . . . . . 9 MblFn $/\$ $/\$ $/\$
173	168, 172	mpbird 247	. . . . . . . 8 MblFn $/\$ $/\$ $/\$
174		itg2le 23506	. . . . . . . 8 $/\$ $/\$
175	52, 148, 173, 174	syl3anc 1326	. . . . . . 7 MblFn $/\$ $/\$ $/\$
176		itg2lecl 23505	. . . . . . 7 $/\$ $/\$
177	52, 139, 175, 176	syl3anc 1326	. . . . . 6 MblFn $/\$ $/\$ $/\$
178	42, 44	iblpos 23559	. . . . . 6 MblFn $/\$ $/\$ $/\$ MblFn $/\$
179	41, 177, 178	mpbir2and 957	. . . . 5 MblFn $/\$ $/\$ $/\$
180	19, 22, 179	iblabsr 23596	. . . 4 MblFn $/\$ $/\$ $/\$
181	18, 180	eqeltrd 2701	. . 3 MblFn $/\$ $/\$ $/\$
182	181	rexlimdvaa 3032	. 2 MblFn $/\$
183	182	3impia 1261	1 MblFn $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wex 1704

wcel 1990

wral 2912

wrex 2913

cvv 3200

cin 3573

wss 3574

c0 3915

cif 4086

csn 4177 class class class wbr 4653

cmpt 4729

cxp 5112

cdm 5114

ccom 5118

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cof 6895

cofr 6896

cc 9934

cr 9935

cc0 9936

cmul 9941

cpnf 10071

cxr 10073

cle 10075

cico 12177

cicc 12178

cabs 13974

ccncf 22679

cvol 23232 MblFncmbf 23383

citg2 23385

cibl 23386

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 ax-addf 10015 ax-mulf 10016

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-iin 4523 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-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 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-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 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-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-0p 23437

This theorem is referenced by: bddibl 23606 itgsubstlem 23811 fourierdlem16 40340 fourierdlem21 40345 fourierdlem22 40346 fourierdlem83 40406

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