Theorem itg2mulc 21917

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 11627	. . . . . . . 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 11222	. . . . 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 21916	. . 3 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) <_ ( A x. ( S.2 ` F ) ) )
13		ge0mulcl 11633	. . . . . . . . 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 5774	. . . . . . . . 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 9583	. . . . . . . . 9 |- RR e. _V
18	17	a1i 11	. . . . . . . 8 |- ( ph -> RR e. _V )
19		inidm 3707	. . . . . . . 8 |- ( RR i^i RR ) = RR
20	14, 16, 1, 18, 18, 19	off 6538	. . . . . . 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 11535	. . . . . . . . . 10 |- [,) = ( z e. RR* , w e. RR* |-> { v e. RR* | ( z <_ v /\ v < w ) } )
23		df-icc 11536	. . . . . . . . . 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 11355	. . . . . . . . . 10 |- ( ( u e. RR* /\ +oo e. RR* ) -> ( u < +oo -> u <_ +oo ) )
26	22, 23, 24, 25	ixxssixx 11543	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
27		fss 5739	. . . . . . . . 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 9624	. . . . . . . 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 21908	. . . . . . 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 1228	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. RR )
34	11	rpreccld 11266	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. RR+ )
35	21, 33, 34	itg2mulclem 21916	. . . . 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 5920	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> F = ( y e. RR |-> ( F ` y ) ) )
37		0re 9596	. . . . . . . . . . . . . . 15 |- 0 e. RR
38		pnfxr 11321	. . . . . . . . . . . . . . 15 |- +oo e. RR*
39		icossre 11605	. . . . . . . . . . . . . . 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 9549	. . . . . . . . . . . . . 14 |- RR C_ CC
42	40, 41	sstri 3513	. . . . . . . . . . . . 13 |- ( 0 [,) +oo ) C_ CC
43		fss 5739	. . . . . . . . . . . . 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 6021	. . . . . . . . . 10 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( F ` y ) e. CC )
47	46	mulid2d 9614	. . . . . . . . 9 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> ( 1 x. ( F ` y ) ) = ( F ` y ) )
48	47	mpteq2dva 4533	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( 1 x. ( F ` y ) ) ) = ( y e. RR |-> ( F ` y ) ) )
49	36, 48	eqtr4d 2511	. . . . . . 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 9611	. . . . . . . 8 |- ( ( ( ph /\ 0 < A ) /\ y e. RR ) -> 1 e. RR )
52	50, 34, 11	ofc12 6549	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( RR X. { ( ( 1 / A ) x. A ) } ) )
53		fconstmpt 5043	. . . . . . . . . 10 |- ( RR X. { ( ( 1 / A ) x. A ) } ) = ( y e. RR |-> ( ( 1 / A ) x. A ) )
54	52, 53	syl6eq 2524	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> ( ( 1 / A ) x. A ) ) )
55	8	recnd 9622	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
56	55	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A e. CC )
57	11	rpne0d 11261	. . . . . . . . . . 11 |- ( ( ph /\ 0 < A ) -> A =/= 0 )
58	56, 57	recid2d 10316	. . . . . . . . . 10 |- ( ( ph /\ 0 < A ) -> ( ( 1 / A ) x. A ) = 1 )
59	58	mpteq2dv 4534	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( y e. RR |-> ( ( 1 / A ) x. A ) ) = ( y e. RR |-> 1 ) )
60	54, 59	eqtrd 2508	. . . . . . . 8 |- ( ( ph /\ 0 < A ) -> ( ( RR X. { ( 1 / A ) } ) oF x. ( RR X. { A } ) ) = ( y e. RR |-> 1 ) )
61	50, 51, 46, 60, 36	offval2 6540	. . . . . . 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 11258	. . . . . . . . 9 |- ( ( ph /\ 0 < A ) -> ( 1 / A ) e. CC )
63		fconst6g 5774	. . . . . . . . 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 5774	. . . . . . . . 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 9580	. . . . . . . . 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 6558	. . . . . . 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 2514	. . . . . 6 |- ( ( ph /\ 0 < A ) -> F = ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) )
71	70	fveq2d 5870	. . . . 5 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) = ( S.2 ` ( ( RR X. { ( 1 / A ) } ) oF x. ( ( RR X. { A } ) oF x. F ) ) ) )
72	33	recnd 9622	. . . . . 6 |- ( ( ph /\ 0 < A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) e. CC )
73	72, 56, 57	divrec2d 10324	. . . . 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 4477	. . . 4 |- ( ( ph /\ 0 < A ) -> ( S.2 ` F ) <_ ( ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) / A ) )
75	4, 33, 11	lemuldiv2d 11302	. . . 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 21902	. . . . . 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 9643	. . . . 5 |- ( ph -> ( A x. ( S.2 ` F ) ) e. RR* )
80		xrletri3 11358	. . . . 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 920	. 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 6299	. . . . . . 7 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( 0 x. x ) = ( A x. x ) )
90		mul02 9757	. . . . . . . 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 2510	. . . . . 6 |- ( ( ( ph /\ 0 = A ) /\ x e. CC ) -> ( A x. x ) = 0 )
93	84, 85, 86, 87, 92	caofid2 6555	. . . . 5 |- ( ( ph /\ 0 = A ) -> ( ( RR X. { A } ) oF x. F ) = ( RR X. { 0 } ) )
94	93	fveq2d 5870	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` ( ( RR X. { A } ) oF x. F ) ) = ( S.2 ` ( RR X. { 0 } ) ) )
95		itg20 21907	. . . 4 |- ( S.2 ` ( RR X. { 0 } ) ) = 0
96	94, 95	syl6eq 2524	. . 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 9622	. . . 4 |- ( ( ph /\ 0 = A ) -> ( S.2 ` F ) e. CC )
99	98	mul02d 9777	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = 0 )
100		simpr 461	. . . 4 |- ( ( ph /\ 0 = A ) -> 0 = A )
101	100	oveq1d 6299	. . 3 |- ( ( ph /\ 0 = A ) -> ( 0 x. ( S.2 ` F ) ) = ( A x. ( S.2 ` F ) ) )
102	96, 99, 101	3eqtr2d 2514	. 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 9671	. . . 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 973   = wceq 1379   e. wcel 1767   _Vcvv 3113   C_ wss 3476   {csn 4027   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284   oFcof 6522   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   x. cmul 9497   +oocpnf 9625   RR*cxr 9627   < clt 9628   <_ cle 9629   / cdiv 10206   RR+crp 11220   [,)cico 11531   [,]cicc 11532   S.2citg2 21788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-xadd 11319  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-xmet 18211  df-met 18212  df-ovol 21639  df-vol 21640  df-mbf 21791  df-itg1 21792  df-itg2 21793  df-0p 21840

This theorem is referenced by:  iblmulc2  22000  itgmulc2lem1  22001  bddmulibl  22008  iblmulc2nc  29685  itgmulc2nclem1  29686