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Theorem itg2mulclem 22021
Description: Lemma for itg2mulc 22022. (Contributed by Mario Carneiro, 8-Jul-2014.)
Hypotheses
Ref Expression
itg2mulc.2  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
itg2mulc.3  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
itg2mulclem.4  |-  ( ph  ->  A  e.  RR+ )
Assertion
Ref Expression
itg2mulclem  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) ) )

Proof of Theorem itg2mulclem
Dummy variables  f  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itg2mulc.2 . . . . . . 7  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
2 df-ico 11547 . . . . . . . 8  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
3 df-icc 11548 . . . . . . . 8  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
4 idd 24 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
5 xrltle 11367 . . . . . . . 8  |-  ( ( u  e.  RR*  /\ +oo  e.  RR* )  ->  (
u  < +oo  ->  u  <_ +oo ) )
62, 3, 4, 5ixxssixx 11555 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
7 fss 5745 . . . . . . 7  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  F : RR --> ( 0 [,] +oo ) )
81, 6, 7sylancl 662 . . . . . 6  |-  ( ph  ->  F : RR --> ( 0 [,] +oo ) )
98adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] +oo ) )
10 simpr 461 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  e.  dom  S.1 )
11 itg2mulclem.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
1211rpreccld 11278 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR+ )
1312adantr 465 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR+ )
1413rpred 11268 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR )
1510, 14i1fmulc 21978 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  oF  x.  f )  e.  dom  S.1 )
16 itg2ub 22008 . . . . . 6  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  e.  dom  S.1  /\  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  oR  <_  F )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  oF  x.  f ) )  <_ 
( S.2 `  F ) )
17163expia 1198 . . . . 5  |-  ( ( F : RR --> ( 0 [,] +oo )  /\  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  e.  dom  S.1 )  ->  ( ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f )  oR  <_  F  ->  ( S.1 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f ) )  <_  ( S.2 `  F
) ) )
189, 15, 17syl2anc 661 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  oR  <_  F  ->  ( S.1 `  (
( RR  X.  {
( 1  /  A
) } )  oF  x.  f ) )  <_  ( S.2 `  F ) ) )
19 i1ff 21951 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  f : RR --> RR )
2019adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f : RR --> RR )
2120ffvelrnda 6032 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  RR )
22 0re 9608 . . . . . . . . . . . 12  |-  0  e.  RR
23 pnfxr 11333 . . . . . . . . . . . 12  |- +oo  e.  RR*
24 icossre 11617 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
2522, 23, 24mp2an 672 . . . . . . . . . . 11  |-  ( 0 [,) +oo )  C_  RR
26 fss 5745 . . . . . . . . . . 11  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  RR )  ->  F : RR --> RR )
271, 25, 26sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
2827adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> RR )
2928ffvelrnda 6032 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
3011rpred 11268 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3130ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR )
3211rpgt0d 11271 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
3332ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  0  <  A )
34 ledivmul 10430 . . . . . . . 8  |-  ( ( ( f `  y
)  e.  RR  /\  ( F `  y )  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( ( f `
 y )  /  A )  <_  ( F `  y )  <->  ( f `  y )  <_  ( A  x.  ( F `  y ) ) ) )
3521, 29, 31, 33, 34syl112anc 1232 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( f `  y )  <_  ( A  x.  ( F `  y ) ) ) )
3621recnd 9634 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  CC )
3731recnd 9634 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  CC )
3811adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR+ )
3938rpne0d 11273 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  =/=  0 )
4039adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  =/=  0 )
4136, 37, 40divrec2d 10336 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  /  A )  =  ( ( 1  /  A )  x.  ( f `  y
) ) )
4241breq1d 4463 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4335, 42bitr3d 255 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  <_  ( A  x.  ( F `  y
) )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4443ralbidva 2903 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( A. y  e.  RR  ( f `  y
)  <_  ( A  x.  ( F `  y
) )  <->  A. y  e.  RR  ( ( 1  /  A )  x.  ( f `  y
) )  <_  ( F `  y )
) )
45 reex 9595 . . . . . . 7  |-  RR  e.  _V
4645a1i 11 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  RR  e.  _V )
47 ovex 6320 . . . . . . 7  |-  ( A  x.  ( F `  y ) )  e. 
_V
4847a1i 11 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
_V )
4920feqmptd 5927 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  =  ( y  e.  RR  |->  ( f `  y ) ) )
5011ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR+ )
51 fconstmpt 5049 . . . . . . . 8  |-  ( RR 
X.  { A }
)  =  ( y  e.  RR  |->  A )
5251a1i 11 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( RR  X.  { A }
)  =  ( y  e.  RR  |->  A ) )
531feqmptd 5927 . . . . . . . 8  |-  ( ph  ->  F  =  ( y  e.  RR  |->  ( F `
 y ) ) )
5453adantr 465 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
5546, 50, 29, 52, 54offval2 6551 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( y  e.  RR  |->  ( A  x.  ( F `  y )
) ) )
5646, 21, 48, 49, 55ofrfval2 6552 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  oR  <_ 
( ( RR  X.  { A } )  oF  x.  F )  <->  A. y  e.  RR  ( f `  y
)  <_  ( A  x.  ( F `  y
) ) ) )
57 ovex 6320 . . . . . . 7  |-  ( ( 1  /  A )  x.  ( f `  y ) )  e. 
_V
5857a1i 11 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( 1  /  A
)  x.  ( f `
 y ) )  e.  _V )
5912ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
1  /  A )  e.  RR+ )
60 fconstmpt 5049 . . . . . . . 8  |-  ( RR 
X.  { ( 1  /  A ) } )  =  ( y  e.  RR  |->  ( 1  /  A ) )
6160a1i 11 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( RR  X.  { ( 1  /  A ) } )  =  ( y  e.  RR  |->  ( 1  /  A ) ) )
6246, 59, 21, 61, 49offval2 6551 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  oF  x.  f )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  ( f `  y
) ) ) )
6346, 58, 29, 62, 54ofrfval2 6552 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  oR  <_  F 
<-> 
A. y  e.  RR  ( ( 1  /  A )  x.  (
f `  y )
)  <_  ( F `  y ) ) )
6444, 56, 633bitr4d 285 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  oR  <_ 
( ( RR  X.  { A } )  oF  x.  F )  <-> 
( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f
)  oR  <_  F ) )
6510, 14itg1mulc 21979 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  oF  x.  f ) )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
66 itg1cl 21960 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  ( S.1 `  f )  e.  RR )
6766adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  RR )
6867recnd 9634 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  CC )
6930adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR )
7069recnd 9634 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  CC )
7168, 70, 39divrec2d 10336 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  /  A )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
7265, 71eqtr4d 2511 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  oF  x.  f ) )  =  ( ( S.1 `  f
)  /  A ) )
7372breq1d 4463 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  ( ( RR  X.  { ( 1  /  A ) } )  oF  x.  f ) )  <_  ( S.2 `  F
)  <->  ( ( S.1 `  f )  /  A
)  <_  ( S.2 `  F ) ) )
74 itg2mulc.3 . . . . . . 7  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
7574adantr 465 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.2 `  F )  e.  RR )
7632adantr 465 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  0  <  A )
77 ledivmul 10430 . . . . . 6  |-  ( ( ( S.1 `  f
)  e.  RR  /\  ( S.2 `  F )  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( ( S.1 `  f )  /  A
)  <_  ( S.2 `  F )  <->  ( S.1 `  f )  <_  ( A  x.  ( S.2 `  F ) ) ) )
7867, 75, 69, 76, 77syl112anc 1232 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( S.1 `  f
)  /  A )  <_  ( S.2 `  F
)  <->  ( S.1 `  f
)  <_  ( A  x.  ( S.2 `  F
) ) ) )
7973, 78bitr2d 254 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  <_  ( A  x.  ( S.2 `  F ) )  <->  ( S.1 `  (
( RR  X.  {
( 1  /  A
) } )  oF  x.  f ) )  <_  ( S.2 `  F ) ) )
8018, 64, 793imtr4d 268 . . 3  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  oR  <_ 
( ( RR  X.  { A } )  oF  x.  F )  ->  ( S.1 `  f
)  <_  ( A  x.  ( S.2 `  F
) ) ) )
8180ralrimiva 2881 . 2  |-  ( ph  ->  A. f  e.  dom  S.1 ( f  oR  <_  ( ( RR 
X.  { A }
)  oF  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) )
82 ge0mulcl 11645 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
8382adantl 466 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) +oo ) )
84 fconstg 5778 . . . . . . 7  |-  ( A  e.  RR+  ->  ( RR 
X.  { A }
) : RR --> { A } )
8511, 84syl 16 . . . . . 6  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
86 rpre 11238 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
87 rpge0 11244 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  <_  A )
88 elrege0 11639 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) +oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
8986, 87, 88sylanbrc 664 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  ( 0 [,) +oo ) )
9011, 89syl 16 . . . . . . 7  |-  ( ph  ->  A  e.  ( 0 [,) +oo ) )
9190snssd 4178 . . . . . 6  |-  ( ph  ->  { A }  C_  ( 0 [,) +oo ) )
92 fss 5745 . . . . . 6  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  ( 0 [,) +oo ) )  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
9385, 91, 92syl2anc 661 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,) +oo ) )
9445a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
95 inidm 3712 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
9683, 93, 1, 94, 94, 95off 6549 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
97 fss 5745 . . . 4  |-  ( ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
9896, 6, 97sylancl 662 . . 3  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,] +oo ) )
9930, 74remulcld 9636 . . . 4  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
10099rexrd 9655 . . 3  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
101 itg2leub 22009 . . 3  |-  ( ( ( ( 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 ) )  <->  A. f  e.  dom  S.1 ( f  oR  <_  ( ( RR 
X.  { A }
)  oF  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) ) )
10298, 100, 101syl2anc 661 . 2  |-  ( ph  ->  ( ( S.2 `  (
( RR  X.  { A } )  oF  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) )  <->  A. f  e.  dom  S.1 ( f  oR  <_  ( ( RR 
X.  { A }
)  oF  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) ) )
10381, 102mpbird 232 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    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    C_ wss 3481   {csn 4033   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533    oRcofr 6534   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641    / cdiv 10218   RR+crp 11232   [,)cico 11543   [,]cicc 11544   S.1citg1 21892   S.2citg2 21893
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xadd 11331  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-xmet 18282  df-met 18283  df-ovol 21744  df-vol 21745  df-mbf 21896  df-itg1 21897  df-itg2 21898
This theorem is referenced by:  itg2mulc  22022
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