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Theorem itg2mulclem 19638
Description: Lemma for itg2mulc 19639. (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 } )  o F  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 10922 . . . . . . . 8  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
3 df-icc 10923 . . . . . . . 8  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
4 idd 22 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
5 xrltle 10742 . . . . . . . 8  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
62, 3, 4, 5ixxssixx 10930 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
7 fss 5599 . . . . . . 7  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  ( 0 [,]  +oo ) )  ->  F : RR --> ( 0 [,] 
+oo ) )
81, 6, 7sylancl 644 . . . . . 6  |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )
98adantr 452 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] 
+oo ) )
10 simpr 448 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  e.  dom  S.1 )
11 itg2mulclem.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
1211rpreccld 10658 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR+ )
1312adantr 452 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR+ )
1413rpred 10648 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR )
1510, 14i1fmulc 19595 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  e.  dom  S.1 )
16 itg2ub 19625 . . . . . 6  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  e.  dom  S.1  /\  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  f
)  o R  <_  F )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  <_ 
( S.2 `  F ) )
17163expia 1155 . . . . 5  |-  ( ( F : RR --> ( 0 [,]  +oo )  /\  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  e.  dom  S.1 )  ->  ( ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f )  o R  <_  F  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  <_ 
( S.2 `  F ) ) )
189, 15, 17syl2anc 643 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( RR  X.  { ( 1  /  A ) } )  o F  x.  f
)  o R  <_  F  ->  ( S.1 `  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f ) )  <_  ( S.2 `  F ) ) )
19 i1ff 19568 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  f : RR --> RR )
2019adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f : RR --> RR )
2120ffvelrnda 5870 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  RR )
22 0re 9091 . . . . . . . . . . . 12  |-  0  e.  RR
23 pnfxr 10713 . . . . . . . . . . . 12  |-  +oo  e.  RR*
24 icossre 10991 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2522, 23, 24mp2an 654 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
26 fss 5599 . . . . . . . . . . 11  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : RR
--> RR )
271, 25, 26sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
2827adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> RR )
2928ffvelrnda 5870 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
3011rpred 10648 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3130ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR )
3211rpgt0d 10651 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
3332ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  0  <  A )
34 ledivmul 9883 . . . . . . . 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 1188 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( f `  y )  <_  ( A  x.  ( F `  y ) ) ) )
3621recnd 9114 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  CC )
3731recnd 9114 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  CC )
3811adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR+ )
3938rpne0d 10653 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  =/=  0 )
4039adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  =/=  0 )
4136, 37, 40divrec2d 9794 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  /  A )  =  ( ( 1  /  A )  x.  ( f `  y
) ) )
4241breq1d 4222 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4335, 42bitr3d 247 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  <_  ( A  x.  ( F `  y
) )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4443ralbidva 2721 . . . . 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 9081 . . . . . . 7  |-  RR  e.  _V
4645a1i 11 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  RR  e.  _V )
47 ovex 6106 . . . . . . 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 5779 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  =  ( y  e.  RR  |->  ( f `  y ) ) )
5011ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR+ )
51 fconstmpt 4921 . . . . . . . 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 5779 . . . . . . . 8  |-  ( ph  ->  F  =  ( y  e.  RR  |->  ( F `
 y ) ) )
5453adantr 452 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
5546, 50, 29, 52, 54offval2 6322 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( y  e.  RR  |->  ( A  x.  ( F `  y )
) ) )
5646, 21, 48, 49, 55ofrfval2 6323 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  o R  <_ 
( ( RR  X.  { A } )  o F  x.  F )  <->  A. y  e.  RR  ( f `  y
)  <_  ( A  x.  ( F `  y
) ) ) )
57 ovex 6106 . . . . . . 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 707 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
1  /  A )  e.  RR+ )
60 fconstmpt 4921 . . . . . . . 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 6322 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  ( f `  y
) ) ) )
6346, 58, 29, 62, 54ofrfval2 6323 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( RR  X.  { ( 1  /  A ) } )  o F  x.  f
)  o R  <_  F 
<-> 
A. y  e.  RR  ( ( 1  /  A )  x.  (
f `  y )
)  <_  ( F `  y ) ) )
6444, 56, 633bitr4d 277 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  o R  <_ 
( ( RR  X.  { A } )  o F  x.  F )  <-> 
( ( RR  X.  { ( 1  /  A ) } )  o F  x.  f
)  o R  <_  F ) )
6510, 14itg1mulc 19596 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
66 itg1cl 19577 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  ( S.1 `  f )  e.  RR )
6766adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  RR )
6867recnd 9114 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  CC )
6930adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR )
7069recnd 9114 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  CC )
7168, 70, 39divrec2d 9794 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  /  A )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
7265, 71eqtr4d 2471 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  =  ( ( S.1 `  f
)  /  A ) )
7372breq1d 4222 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  f ) )  <_  ( S.2 `  F
)  <->  ( ( S.1 `  f )  /  A
)  <_  ( S.2 `  F ) ) )
74 itg2mulc.3 . . . . . . 7  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
7574adantr 452 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.2 `  F )  e.  RR )
7632adantr 452 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  0  <  A )
77 ledivmul 9883 . . . . . 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 1188 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( S.1 `  f
)  /  A )  <_  ( S.2 `  F
)  <->  ( S.1 `  f
)  <_  ( A  x.  ( S.2 `  F
) ) ) )
7973, 78bitr2d 246 . . . 4  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  <_  ( A  x.  ( S.2 `  F ) )  <->  ( S.1 `  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f ) )  <_  ( S.2 `  F ) ) )
8018, 64, 793imtr4d 260 . . 3  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
f  o R  <_ 
( ( RR  X.  { A } )  o F  x.  F )  ->  ( S.1 `  f
)  <_  ( A  x.  ( S.2 `  F
) ) ) )
8180ralrimiva 2789 . 2  |-  ( ph  ->  A. f  e.  dom  S.1 ( f  o R  <_  ( ( RR 
X.  { A }
)  o F  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) )
82 ge0mulcl 11010 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
8382adantl 453 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
84 fconstg 5630 . . . . . . 7  |-  ( A  e.  RR+  ->  ( RR 
X.  { A }
) : RR --> { A } )
8511, 84syl 16 . . . . . 6  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
86 rpre 10618 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
87 rpge0 10624 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  <_  A )
88 elrege0 11007 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) 
+oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
8986, 87, 88sylanbrc 646 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  ( 0 [,)  +oo ) )
9011, 89syl 16 . . . . . . 7  |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )
9190snssd 3943 . . . . . 6  |-  ( ph  ->  { A }  C_  ( 0 [,)  +oo ) )
92 fss 5599 . . . . . 6  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  ( 0 [,)  +oo ) )  -> 
( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
9385, 91, 92syl2anc 643 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
9445a1i 11 . . . . 5  |-  ( ph  ->  RR  e.  _V )
95 inidm 3550 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
9683, 93, 1, 94, 94, 95off 6320 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
97 fss 5599 . . . 4  |-  ( ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  ( 0 [,]  +oo ) )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,] 
+oo ) )
9896, 6, 97sylancl 644 . . 3  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
9930, 74remulcld 9116 . . . 4  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
10099rexrd 9134 . . 3  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
101 itg2leub 19626 . . 3  |-  ( ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo )  /\  ( A  x.  ( S.2 `  F ) )  e. 
RR* )  ->  (
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) )  <->  A. f  e.  dom  S.1 ( f  o R  <_  ( ( RR 
X.  { A }
)  o F  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) ) )
10298, 100, 101syl2anc 643 . 2  |-  ( ph  ->  ( ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) )  <->  A. f  e.  dom  S.1 ( f  o R  <_  ( ( RR 
X.  { A }
)  o F  x.  F )  ->  ( S.1 `  f )  <_ 
( A  x.  ( S.2 `  F ) ) ) ) )
10381, 102mpbird 224 1  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    C_ wss 3320   {csn 3814   class class class wbr 4212    e. cmpt 4266    X. cxp 4876   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303    o Rcofr 6304   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121    / cdiv 9677   RR+crp 10612   [,)cico 10918   [,]cicc 10919   S.1citg1 19507   S.2citg2 19508
This theorem is referenced by:  itg2mulc  19639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xadd 10711  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-xmet 16695  df-met 16696  df-ovol 19361  df-vol 19362  df-mbf 19512  df-itg1 19513  df-itg2 19514
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