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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.
StepHypRef Expression
1 itg2mulc.2 . . . . 5  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
21adantr 465 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,) +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 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 ) )
75, 6sylib 196 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 459 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 568 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 11222 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 212 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 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 )
)
1413adantl 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 ) )
165, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
17 reex 9583 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3707 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6538 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
2120adantr 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 ) )
2622, 23, 24, 25ixxssixx 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 ) )
2820, 26, 27sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
2928adantr 465 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
308, 3remulcld 9624 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
3130adantr 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 )
3329, 31, 12, 32syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  RR )
3411rpreccld 11266 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3521, 33, 34itg2mulclem 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 ) ) ) )
362feqmptd 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 )
4037, 38, 39mp2an 672 . . . . . . . . . . . . . 14  |-  ( 0 [,) +oo )  C_  RR
41 ax-resscn 9549 . . . . . . . . . . . . . 14  |-  RR  C_  CC
4240, 41sstri 3513 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  CC
43 fss 5739 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  F : RR --> CC )
441, 42, 43sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
4544adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
4645ffvelrnda 6021 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4746mulid2d 9614 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4847mpteq2dva 4533 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4936, 48eqtr4d 2511 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5017a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
51 1red 9611 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
5250, 34, 11ofc12 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 ) )
5452, 53syl6eq 2524 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
558recnd 9622 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
5655adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5711rpne0d 11261 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5856, 57recid2d 10316 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
5958mpteq2dv 4534 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
6054, 59eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
6150, 51, 46, 60, 36offval2 6540 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( RR  X.  {
( 1  /  A
) } )  oF  x.  ( RR 
X.  { A }
) )  oF  x.  F )  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
6234rpcnd 11258 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
63 fconst6g 5774 . . . . . . . . 9  |-  ( ( 1  /  A )  e.  CC  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
6462, 63syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
65 fconst6g 5774 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
6656, 65syl 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
) ) )
6867adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) ) )
6950, 64, 66, 45, 68caofass 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 ) ) )
7049, 61, 693eqtr2d 2514 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  (
( RR  X.  { A } )  oF  x.  F ) ) )
7170fveq2d 5870 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( ( RR 
X.  { A }
)  oF  x.  F ) ) ) )
7233recnd 9622 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  CC )
7372, 56, 57divrec2d 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 ) ) ) )
7435, 71, 733brtr4d 4477 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  /  A ) )
754, 33, 11lemuldiv2d 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 ) ) )
7674, 75mpbird 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* )
7828, 77syl 16 . . . . 5  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  e.  RR* )
7930rexrd 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 ) ) ) ) )
8178, 79, 80syl2anc 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 ) ) ) ) )
8281adantr 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 ) ) ) ) )
8312, 76, 82mpbir2and 920 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F
) ) )
8417a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  RR  e.  _V )
8544adantr 465 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
868adantr 465 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
8737a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
88 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
8988oveq1d 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 )
9190adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
9289, 91eqtr3d 2510 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
9384, 85, 86, 87, 92caofid2 6555 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
9493fveq2d 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
9694, 95syl6eq 2524 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  0 )
973adantr 465 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
9897recnd 9622 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
9998mul02d 9777 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
100 simpr 461 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
101100oveq1d 6299 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
10296, 99, 1013eqtr2d 2514 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
1037simprd 463 . . 3  |-  ( ph  ->  0  <_  A )
104 leloe 9671 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10537, 8, 104sylancr 663 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
106103, 105mpbid 210 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10783, 102, 106mpjaodan 784 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 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
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