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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.
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 11390 . . . . . . . 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 10991 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 212 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 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 )
)
1413adantl 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 ) )
165, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
17 reex 9371 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3557 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6332 . . . . . . 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 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 ) )
2622, 23, 24, 25ixxssixx 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 ) )
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 9412 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
3130adantr 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 )
3329, 31, 12, 32syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  RR )
3411rpreccld 11035 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3521, 33, 34itg2mulclem 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 ) ) ) )
362feqmptd 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 )
4037, 38, 39mp2an 672 . . . . . . . . . . . . . 14  |-  ( 0 [,) +oo )  C_  RR
41 ax-resscn 9337 . . . . . . . . . . . . . 14  |-  RR  C_  CC
4240, 41sstri 3363 . . . . . . . . . . . . 13  |-  ( 0 [,) +oo )  C_  CC
43 fss 5565 . . . . . . . . . . . . 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 5841 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4746mulid2d 9402 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4847mpteq2dva 4376 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4936, 48eqtr4d 2476 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5017a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
51 1red 9399 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
5250, 34, 11ofc12 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 ) )
5452, 53syl6eq 2489 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
558recnd 9410 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
5655adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5711rpne0d 11030 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5856, 57recid2d 10101 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
5958mpteq2dv 4377 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
6054, 59eqtrd 2473 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
6150, 51, 46, 60, 36offval2 6334 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( RR  X.  {
( 1  /  A
) } )  oF  x.  ( RR 
X.  { A }
) )  oF  x.  F )  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
6234rpcnd 11027 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
63 fconst6g 5597 . . . . . . . . 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 5597 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
6656, 65syl 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
) ) )
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 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 ) ) )
7049, 61, 693eqtr2d 2479 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  (
( RR  X.  { A } )  oF  x.  F ) ) )
7170fveq2d 5693 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  ( ( RR 
X.  { A }
)  oF  x.  F ) ) ) )
7233recnd 9410 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  e.  CC )
7372, 56, 57divrec2d 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 ) ) ) )
7435, 71, 733brtr4d 4320 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  oF  x.  F ) )  /  A ) )
754, 33, 11lemuldiv2d 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 ) ) )
7674, 75mpbird 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* )
7828, 77syl 16 . . . . 5  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  e.  RR* )
7930rexrd 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 ) ) ) ) )
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 913 . 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 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 )
9190adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
9289, 91eqtr3d 2475 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
9384, 85, 86, 87, 92caofid2 6349 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
9493fveq2d 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
9694, 95syl6eq 2489 . . 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 9410 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
9998mul02d 9565 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
100 simpr 461 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
101100oveq1d 6104 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
10296, 99, 1013eqtr2d 2479 . 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 9459 . . . 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 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
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