Theorem itg2mulc 21223

Description: The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
itg2mulc.2	|- ( ph -> F : RR --> ( 0 [,) +oo ) )
itg2mulc.3	|- ( ph -> ( S.2 ` F ) e. RR )
itg2mulc.4	|- ( ph -> A e. ( 0 [,) +oo ) )

Assertion

Ref	Expression
itg2mulc	|- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Proof of Theorem itg2mulc

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

Step	Hyp	Ref	Expression
1		itg2mulc.2	. . . . 5 |- ( ph -> F : RR --> ( 0 [,) +oo ) )
2	1	adantr 465	. . . 4 |- ( ( ph /\ 0 < A ) -> F : RR --> ( 0 [,) +oo ) )
3		itg2mulc.3	. . . . 5 |- ( ph -> ( S.2 ` F ) e. RR )
4	3	adantr 465	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) e. RR )
5		itg2mulc.4	. . . . . . . 8 |- ( ph -> A e. ( 0 [,) +oo ) )
6		elrege0 11390	. . . . . . . 8 |- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
7	5, 6	sylib 196	. . . . . . 7 |- ( ph -> ( A e. RR /\ 0 <_ A ) )
8	7	simpld 459	. . . . . 6 |- ( ph -> A e. RR )
9	8	anim1i 568	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( A e. RR /\ 0 < A ) )
10		elrp 10991	. . . . 5 |- ( A e. RR+ <-> ( A e. RR /\ 0 < A ) )
11	9, 10	sylibr 212	. . . 4 |- ( ( ph /\ 0 < A ) -> A e. RR+ )
12	2, 4, 11	itg2mulclem 21222	. . 3 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
13		ge0mulcl 11396	. . . . . . . . 9 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
14	13	adantl 466	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x x. y ) e. ( 0 [,) +oo ) )
15		fconst6g 5597	. . . . . . . . 9 |- ( A e. ( 0 [,) +oo ) -> ( RR X. { A } ) : RR --> ( 0 [,) +oo ) )
16	5, 15	syl 16	. . . . . . . 8 |- ( ph -> ( RR X. { A } ) : RR --> ( 0 [,) +oo ) )
17		reex 9371	. . . . . . . . 9 |- RR e. _V
18	17	a1i 11	. . . . . . . 8 |- ( ph -> RR e. _V )
19		inidm 3557	. . . . . . . 8 |- ( RR i^i RR ) = RR
20	14, 16, 1, 18, 18, 19	off 6332	. . . . . . 7 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
21	20	adantr 465	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) )
22		df-ico 11304	. . . . . . . . . 10 |- [,) = ( z e. RR* , w e. RR* |-> { v e. RR* | ( z <_ v /\ v < w ) } )
23		df-icc 11305	. . . . . . . . . 10 |- [,] = ( z e. RR* , w e. RR* |-> { v e. RR* | ( z <_ v /\ v <_ w ) } )
24		idd 24	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ u e. RR* ) -> ( 0 <_ u -> 0 <_ u ) )
25		xrltle 11124	. . . . . . . . . 10 |- ( ( u e. RR* /\ +oo e. RR* ) -> ( u < +oo -> u <_ +oo ) )
26	22, 23, 24, 25	ixxssixx 11312	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
27		fss 5565	. . . . . . . . 9 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
28	20, 26, 27	sylancl 662	. . . . . . . 8 |- ( ph -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
29	28	adantr 465	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) )
30	8, 3	remulcld 9412	. . . . . . . 8 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR )
31	30	adantr 465	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) e. RR )
32		itg2lecl 21214	. . . . . . 7 |- ( ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) /\ ( A x. ( S.2 ` F ) ) e. RR /\ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
33	29, 31, 12, 32	syl3anc 1218	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
34	11	rpreccld 11035	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. RR+ )
35	21, 33, 34	itg2mulclem 21222	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) <_ ( ( 1 / A ) x. ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) )
36	2	feqmptd 5742	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( F ` y ) ) )
37		0re 9384	. . . . . . . . . . . . . . 15 |- 0 e. RR
38		pnfxr 11090	. . . . . . . . . . . . . . 15 |- +oo e. RR*
39		icossre 11374	. . . . . . . . . . . . . . 15 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( 0 [,) +oo ) C_ RR )
40	37, 38, 39	mp2an 672	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) C_ RR
41		ax-resscn 9337	. . . . . . . . . . . . . 14 |- RR C_ CC
42	40, 41	sstri 3363	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ CC
43		fss 5565	. . . . . . . . . . . . 13 |- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ CC ) -> F : RR --> CC )
44	1, 42, 43	sylancl 662	. . . . . . . . . . . 12 |- ( ph -> F : RR --> CC )
45	44	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> F : RR --> CC )
46	45	ffvelrnda 5841	. . . . . . . . . 10 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( F ` y ) e. CC )
47	46	mulid2d 9402	. . . . . . . . 9 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( 1 x. ( F ` y ) ) = ( F ` y ) )
48	47	mpteq2dva 4376	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( 1 x. ( F ` y ) ) ) = ( y e. RR |-> ( F ` y ) ) )
49	36, 48	eqtr4d 2476	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( 1 x. ( F ` y ) ) ) )
50	17	a1i 11	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> RR e. _V )
51		1red 9399	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> 1 e. RR )
52	50, 34, 11	ofc12 6343	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( RR X. { ( ( 1 / A ) x. A ) } ) )
53		fconstmpt 4880	. . . . . . . . . 10 |- ( RR X. { ( ( 1 / A ) x. A ) } ) = ( y e. RR |-> ( ( 1 / A ) x. A ) )
54	52, 53	syl6eq 2489	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> ( ( 1 / A ) x. A ) ) )
55	8	recnd 9410	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
56	55	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A e. CC )
57	11	rpne0d 11030	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A =/= 0 )
58	56, 57	recid2d 10101	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( 1 / A ) x. A ) = 1 )
59	58	mpteq2dv 4377	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( ( 1 / A ) x. A ) ) = ( y e. RR |-> 1 ) )
60	54, 59	eqtrd 2473	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> 1 ) )
61	50, 51, 46, 60, 36	offval2 6334	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) oF x. F ) = ( y e. RR |-> ( 1 x. ( F ` y ) ) ) )
62	34	rpcnd 11027	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. CC )
63		fconst6g 5597	. . . . . . . . 9 |- ( ( 1 / A ) e. CC -> ( RR X. { ( 1 / A ) } ) : RR --> CC )
64	62, 63	syl 16	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( RR X. { ( 1 / A ) } ) : RR --> CC )
65		fconst6g 5597	. . . . . . . . 9 |- ( A e. CC -> ( RR X. { A } ) : RR --> CC )
66	56, 65	syl 16	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( RR X. { A } ) : RR --> CC )
67		mulass 9368	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
68	67	adantl 466	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x x. y ) x. z ) = ( x x. ( y x. z ) ) )
69	50, 64, 66, 45, 68	caofass 6352	. . . . . . 7 |- ( ( ph /\ 0 < A ) -> ( ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) oF x. F ) = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
70	49, 61, 69	3eqtr2d 2479	. . . . . 6 |- ( ( ph /\ 0 < A ) -> F = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
71	70	fveq2d 5693	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) = ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) )
72	33	recnd 9410	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. CC )
73	72, 56, 57	divrec2d 10109	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) = ( ( 1 / A ) x. ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) )
74	35, 71, 73	3brtr4d 4320	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) )
75	4, 33, 11	lemuldiv2d 11071	. . . 4 |- ( ( ph /\ 0 < A ) -> ( ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <-> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) ) )
76	74, 75	mpbird 232	. . 3 |- ( ( ph /\ 0 < A ) -> ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) )
77		itg2cl 21208	. . . . . 6 |- ( ( ( RR X. { A } ) oF x. F ) : RR --> ( 0 [,] +oo ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* )
78	28, 77	syl 16	. . . . 5 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* )
79	30	rexrd 9431	. . . . 5 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR* )
80		xrletri3 11127	. . . . 5 |- ( ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR* /\ ( A x. ( S.2 ` F ) ) e. RR* ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
81	78, 79, 80	syl2anc 661	. . . 4 |- ( ph -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
82	81	adantr 465	. . 3 |- ( ( ph /\ 0 < A ) -> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) <-> ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) /\ ( A x. ( S.2 ` F ) ) <_ ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) ) ) )
83	12, 76, 82	mpbir2and 913	. 2 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
84	17	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> RR e. _V )
85	44	adantr 465	. . . . . 6 |- ( ( ph /\ 0 = A ) -> F : RR --> CC )
86	8	adantr 465	. . . . . 6 |- ( ( ph /\ 0 = A ) -> A e. RR )
87	37	a1i 11	. . . . . 6 |- ( ( ph /\ 0 = A ) -> 0 e. RR )
88		simplr 754	. . . . . . . 8 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> 0 = A )
89	88	oveq1d 6104	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = ( A x. x ) )
90		mul02 9545	. . . . . . . 8 |- ( x e. CC -> ( 0 x. x ) = 0 )
91	90	adantl 466	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = 0 )
92	89, 91	eqtr3d 2475	. . . . . 6 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( A x. x ) = 0 )
93	84, 85, 86, 87, 92	caofid2 6349	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
94	93	fveq2d 5693	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.2 ` ( RR X. { 0 } ) ) )
95		itg20 21213	. . . 4 |- ( S.2 ` ( RR X. { 0 } ) ) = 0
96	94, 95	syl6eq 2489	. . 3 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = 0 )
97	3	adantr 465	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. RR )
98	97	recnd 9410	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. CC )
99	98	mul02d 9565	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = 0 )
100		simpr 461	. . . 4 |- ( ( ph /\ 0 = A ) -> 0 = A )
101	100	oveq1d 6104	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = ( A x. ( S.2 ` F ) ) )
102	96, 99, 101	3eqtr2d 2479	. 2 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )
103	7	simprd 463	. . 3 |- ( ph -> 0 <_ A )
104		leloe 9459	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
105	37, 8, 104	sylancr 663	. . 3 |- ( ph -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
106	103, 105	mpbid 210	. 2 |- ( ph -> ( 0 < A \/ 0 = A ) )
107	83, 102, 106	mpjaodan 784	1 |- ( ph -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( A x. ( S.2 ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   _Vcvv 2970   C_ wss 3326   {csn 3875   class class class wbr 4290   e. cmpt 4348   X. cxp 4836   -->wf 5412   ` cfv 5416  (class class class)co 6089   oFcof 6316   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281   x. cmul 9285   +oocpnf 9413   RR*cxr 9415   < clt 9416   <_ cle 9417   / cdiv 9991   RR+crp 10989   [,)cico 11300   [,]cicc 11301   S.2citg2 21094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-ofr 6319  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-xadd 11088  df-ioo 11302  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-fl 11640  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-xmet 17808  df-met 17809  df-ovol 20946  df-vol 20947  df-mbf 21097  df-itg1 21098  df-itg2 21099  df-0p 21146

This theorem is referenced by:  iblmulc2  21306  itgmulc2lem1  21307  bddmulibl  21314  iblmulc2nc  28454  itgmulc2nclem1  28455