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Theorem itg2mulclem 21224
Description: Lemma for itg2mulc 21225. (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 11306 . . . . . . . 8  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
3 df-icc 11307 . . . . . . . 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 11126 . . . . . . . 8  |-  ( ( u  e.  RR*  /\ +oo  e.  RR* )  ->  (
u  < +oo  ->  u  <_ +oo ) )
62, 3, 4, 5ixxssixx 11314 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
7 fss 5567 . . . . . . 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 11037 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR+ )
1312adantr 465 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR+ )
1413rpred 11027 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR )
1510, 14i1fmulc 21181 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  oF  x.  f )  e.  dom  S.1 )
16 itg2ub 21211 . . . . . 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 1189 . . . . 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 21154 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  f : RR --> RR )
2019adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f : RR --> RR )
2120ffvelrnda 5843 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  RR )
22 0re 9386 . . . . . . . . . . . 12  |-  0  e.  RR
23 pnfxr 11092 . . . . . . . . . . . 12  |- +oo  e.  RR*
24 icossre 11376 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
2522, 23, 24mp2an 672 . . . . . . . . . . 11  |-  ( 0 [,) +oo )  C_  RR
26 fss 5567 . . . . . . . . . . 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 5843 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
3011rpred 11027 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3130ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR )
3211rpgt0d 11030 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
3332ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  0  <  A )
34 ledivmul 10205 . . . . . . . 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 1222 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( f `  y )  <_  ( A  x.  ( F `  y ) ) ) )
3621recnd 9412 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  CC )
3731recnd 9412 . . . . . . . . 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 11032 . . . . . . . . . 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 10111 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  /  A )  =  ( ( 1  /  A )  x.  ( f `  y
) ) )
4241breq1d 4302 . . . . . . 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 2731 . . . . 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 9373 . . . . . . 7  |-  RR  e.  _V
4645a1i 11 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  RR  e.  _V )
47 ovex 6116 . . . . . . 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 5744 . . . . . 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 4882 . . . . . . . 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 5744 . . . . . . . 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 6336 . . . . . 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 6337 . . . . 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 6116 . . . . . . 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 4882 . . . . . . . 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 6336 . . . . . 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 6337 . . . . 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 21182 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  oF  x.  f ) )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
66 itg1cl 21163 . . . . . . . . . 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 9412 . . . . . . . 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 9412 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  CC )
7168, 70, 39divrec2d 10111 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  /  A )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
7265, 71eqtr4d 2478 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  oF  x.  f ) )  =  ( ( S.1 `  f
)  /  A ) )
7372breq1d 4302 . . . . 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 10205 . . . . . 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 1222 . . . . 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 2799 . 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 11398 . . . . . 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 5597 . . . . . . 7  |-  ( A  e.  RR+  ->  ( RR 
X.  { A }
) : RR --> { A } )
8511, 84syl 16 . . . . . 6  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
86 rpre 10997 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
87 rpge0 11003 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  <_  A )
88 elrege0 11392 . . . . . . . . 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 4018 . . . . . 6  |-  ( ph  ->  { A }  C_  ( 0 [,) +oo ) )
92 fss 5567 . . . . . 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 3559 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
9683, 93, 1, 94, 94, 95off 6334 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F ) : RR --> ( 0 [,) +oo ) )
97 fss 5567 . . . 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 9414 . . . 4  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
10099rexrd 9433 . . 3  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
101 itg2leub 21212 . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972    C_ wss 3328   {csn 3877   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318    oRcofr 6319   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   +oocpnf 9415   RR*cxr 9417    < clt 9418    <_ cle 9419    / cdiv 9993   RR+crp 10991   [,)cico 11302   [,]cicc 11303   S.1citg1 21095   S.2citg2 21096
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xadd 11090  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-xmet 17810  df-met 17811  df-ovol 20948  df-vol 20949  df-mbf 21099  df-itg1 21100  df-itg2 21101
This theorem is referenced by:  itg2mulc  21225
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