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Theorem itg2mulc 22590
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  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itg2mulc.2 . . . . 5  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
21adantr 466 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,) +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 466 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  e.  RR )
5 itg2mulc.4 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 0 [,) +oo ) )
6 elrege0 11737 . . . . . . . 8  |-  ( A  e.  ( 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
75, 6sylib 199 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 460 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 570 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 11304 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 215 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 22589 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) ) )
13 ge0mulcl 11743 . . . . . . . . 9  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
1413adantl 467 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) +oo ) )
15 fconst6g 5789 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) +oo )  ->  ( RR 
X.  { A }
) : RR --> ( 0 [,) +oo ) )
165, 15syl 17 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
17 reex 9629 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3677 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6560 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
2120adantr 466 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
22 icossicc 11721 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
23 fss 5754 . . . . . . . . 9  |-  ( ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
2420, 22, 23sylancl 666 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
2524adantr 466 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
268, 3remulcld 9670 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
2726adantr 466 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  e.  RR )
28 itg2lecl 22581 . . . . . . 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 )
2925, 27, 12, 28syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  RR )
3011rpreccld 11351 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3121, 29, 30itg2mulclem 22589 . . . . 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 ) ) ) )
322feqmptd 5934 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
33 rge0ssre 11738 . . . . . . . . . . . . . 14  |-  ( 0 [,) +oo )  C_  RR
34 ax-resscn 9595 . . . . . . . . . . . . . 14  |-  RR  C_  CC
3533, 34sstri 3479 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  CC
36 fss 5754 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  F : RR --> CC )
371, 35, 36sylancl 666 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
3837adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
3938ffvelrnda 6037 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4039mulid2d 9660 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4140mpteq2dva 4512 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4232, 41eqtr4d 2473 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
4317a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
44 1red 9657 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
4543, 30, 11ofc12 6570 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( RR  X.  {
( ( 1  /  A )  x.  A
) } ) )
46 fconstmpt 4898 . . . . . . . . . 10  |-  ( RR 
X.  { ( ( 1  /  A )  x.  A ) } )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )
4745, 46syl6eq 2486 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
488recnd 9668 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
4948adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5011rpne0d 11346 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5149, 50recid2d 10378 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
5251mpteq2dv 4513 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
5347, 52eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
5443, 44, 39, 53, 32offval2 6562 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( RR  X.  {
( 1  /  A
) } )  oF  x.  ( RR 
X.  { A }
) )  oF  x.  F )  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5530rpcnd 11343 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
56 fconst6g 5789 . . . . . . . . 9  |-  ( ( 1  /  A )  e.  CC  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
5755, 56syl 17 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
58 fconst6g 5789 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
5949, 58syl 17 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { A } ) : RR --> CC )
60 mulass 9626 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
6160adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) ) )
6243, 57, 59, 38, 61caofass 6579 . . . . . . 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 ) ) )
6342, 54, 623eqtr2d 2476 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  (
( RR  X.  { A } )  oF  x.  F ) ) )
6463fveq2d 5885 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( ( RR 
X.  { A }
)  oF  x.  F ) ) ) )
6529recnd 9668 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  CC )
6665, 49, 50divrec2d 10386 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  /  A )  =  ( ( 1  /  A
)  x.  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) ) ) )
6731, 64, 663brtr4d 4456 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  /  A ) )
684, 29, 11lemuldiv2d 11388 . . . 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 ) ) )
6967, 68mpbird 235 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) ) )
70 itg2cl 22575 . . . . . 6  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  RR* )
7124, 70syl 17 . . . . 5  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  e.  RR* )
7226rexrd 9689 . . . . 5  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
73 xrletri3 11451 . . . . 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 ) ) ) ) )
7471, 72, 73syl2anc 665 . . . 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 ) ) ) ) )
7574adantr 466 . . 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 ) ) ) ) )
7612, 69, 75mpbir2and 930 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F
) ) )
7717a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  RR  e.  _V )
7837adantr 466 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
798adantr 466 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
80 0re 9642 . . . . . . 7  |-  0  e.  RR
8180a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
82 simplr 760 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
8382oveq1d 6320 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  ( A  x.  x ) )
84 mul02 9810 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
8584adantl 467 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
8683, 85eqtr3d 2472 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
8777, 78, 79, 81, 86caofid2 6576 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
8887fveq2d 5885 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( S.2 `  ( RR  X.  { 0 } ) ) )
89 itg20 22580 . . . 4  |-  ( S.2 `  ( RR  X.  {
0 } ) )  =  0
9088, 89syl6eq 2486 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  0 )
913adantr 466 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
9291recnd 9668 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
9392mul02d 9830 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
94 simpr 462 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
9594oveq1d 6320 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
9690, 93, 953eqtr2d 2476 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
977simprd 464 . . 3  |-  ( ph  ->  0  <_  A )
98 leloe 9719 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9980, 8, 98sylancr 667 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10097, 99mpbid 213 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10176, 96, 100mpjaodan 793 1  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   {csn 4002   class class class wbr 4426    |-> cmpt 4484    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    / cdiv 10268   RR+crp 11302   [,)cico 11637   [,]cicc 11638   S.2citg2 22459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-xmet 18902  df-met 18903  df-ovol 22301  df-vol 22303  df-mbf 22462  df-itg1 22463  df-itg2 22464  df-0p 22513
This theorem is referenced by:  iblmulc2  22673  itgmulc2lem1  22674  bddmulibl  22681  iblmulc2nc  31722  itgmulc2nclem1  31723
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