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Theorem itg2mulc 22705
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 467 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,) +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 467 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  e.  RR )
5 itg2mulc.4 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 0 [,) +oo ) )
6 elrege0 11738 . . . . . . . 8  |-  ( A  e.  ( 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
75, 6sylib 200 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 461 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 572 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 11304 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 216 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 22704 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) ) )
13 ge0mulcl 11745 . . . . . . . . 9  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
1413adantl 468 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) +oo ) )
15 fconst6g 5772 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) +oo )  ->  ( RR 
X.  { A }
) : RR --> ( 0 [,) +oo ) )
165, 15syl 17 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
17 reex 9630 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3641 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6546 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
2120adantr 467 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
22 icossicc 11721 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
23 fss 5737 . . . . . . . . 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 668 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
2524adantr 467 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
268, 3remulcld 9671 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
2726adantr 467 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  e.  RR )
28 itg2lecl 22696 . . . . . . 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 1268 . . . . . 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 22704 . . . . 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 5918 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
33 rge0ssre 11740 . . . . . . . . . . . . . 14  |-  ( 0 [,) +oo )  C_  RR
34 ax-resscn 9596 . . . . . . . . . . . . . 14  |-  RR  C_  CC
3533, 34sstri 3441 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  CC
36 fss 5737 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  F : RR --> CC )
371, 35, 36sylancl 668 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
3837adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
3938ffvelrnda 6022 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4039mulid2d 9661 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4140mpteq2dva 4489 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4232, 41eqtr4d 2488 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
4317a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
44 1red 9658 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
4543, 30, 11ofc12 6556 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( RR  X.  {
( ( 1  /  A )  x.  A
) } ) )
46 fconstmpt 4878 . . . . . . . . . 10  |-  ( RR 
X.  { ( ( 1  /  A )  x.  A ) } )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )
4745, 46syl6eq 2501 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
488recnd 9669 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
4948adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5011rpne0d 11346 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5149, 50recid2d 10379 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
5251mpteq2dv 4490 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
5347, 52eqtrd 2485 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
5443, 44, 39, 53, 32offval2 6548 . . . . . . 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 5772 . . . . . . . . 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 5772 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
5949, 58syl 17 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { A } ) : RR --> CC )
60 mulass 9627 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
6160adantl 468 . . . . . . . 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 6565 . . . . . . 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 2491 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  (
( RR  X.  { A } )  oF  x.  F ) ) )
6463fveq2d 5869 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( ( RR 
X.  { A }
)  oF  x.  F ) ) ) )
6529recnd 9669 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  CC )
6665, 49, 50divrec2d 10387 . . . . 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 4433 . . . 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 236 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) ) )
70 itg2cl 22690 . . . . . 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 9690 . . . . 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 667 . . . 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 467 . . 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 933 . 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 467 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
798adantr 467 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
80 0re 9643 . . . . . . 7  |-  0  e.  RR
8180a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
82 simplr 762 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
8382oveq1d 6305 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  ( A  x.  x ) )
84 mul02 9811 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
8584adantl 468 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
8683, 85eqtr3d 2487 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
8777, 78, 79, 81, 86caofid2 6562 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
8887fveq2d 5869 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( S.2 `  ( RR  X.  { 0 } ) ) )
89 itg20 22695 . . . 4  |-  ( S.2 `  ( RR  X.  {
0 } ) )  =  0
9088, 89syl6eq 2501 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  0 )
913adantr 467 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
9291recnd 9669 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
9392mul02d 9831 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
94 simpr 463 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
9594oveq1d 6305 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
9690, 93, 953eqtr2d 2491 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
977simprd 465 . . 3  |-  ( ph  ->  0  <_  A )
98 leloe 9720 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
9980, 8, 98sylancr 669 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10097, 99mpbid 214 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10176, 96, 100mpjaodan 795 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045    C_ wss 3404   {csn 3968   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   -->wf 5578   ` cfv 5582  (class class class)co 6290    oFcof 6529   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676    / cdiv 10269   RR+crp 11302   [,)cico 11637   [,]cicc 11638   S.2citg2 22574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-xmet 18963  df-met 18964  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-0p 22628
This theorem is referenced by:  iblmulc2  22788  itgmulc2lem1  22789  bddmulibl  22796  iblmulc2nc  32007  itgmulc2nclem1  32008
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