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Theorem itg2mulc 19641
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 } )  o F  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 453 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,)  +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 453 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  e.  RR )
5 itg2mulc.4 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )
6 elrege0 11009 . . . . . . . 8  |-  ( A  e.  ( 0 [,) 
+oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
75, 6sylib 190 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 447 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 553 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 10616 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 205 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 19640 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) ) )
13 ge0mulcl 11012 . . . . . . . . 9  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
1413adantl 454 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
15 fconst6g 5634 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) 
+oo )  ->  ( RR  X.  { A }
) : RR --> ( 0 [,)  +oo ) )
165, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
17 reex 9083 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3552 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6322 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
2120adantr 453 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
22 df-ico 10924 . . . . . . . . . 10  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
23 df-icc 10925 . . . . . . . . . 10  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
24 idd 23 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
25 xrltle 10744 . . . . . . . . . 10  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
2622, 23, 24, 25ixxssixx 10932 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
27 fss 5601 . . . . . . . . 9  |-  ( ( ( ( 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 ) )
2820, 26, 27sylancl 645 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
2928adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
308, 3remulcld 9118 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
3130adantr 453 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  e.  RR )
32 itg2lecl 19632 . . . . . . 7  |-  ( ( ( ( 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
) ) )  -> 
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  RR )
3329, 31, 12, 32syl3anc 1185 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  RR )
3411rpreccld 10660 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3521, 33, 34itg2mulclem 19640 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  (
( RR  X.  { A } )  o F  x.  F ) ) )  <_  ( (
1  /  A )  x.  ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) ) ) )
362feqmptd 5781 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
37 0re 9093 . . . . . . . . . . . . . . 15  |-  0  e.  RR
38 pnfxr 10715 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
39 icossre 10993 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
4037, 38, 39mp2an 655 . . . . . . . . . . . . . 14  |-  ( 0 [,)  +oo )  C_  RR
41 ax-resscn 9049 . . . . . . . . . . . . . 14  |-  RR  C_  CC
4240, 41sstri 3359 . . . . . . . . . . . . 13  |-  ( 0 [,)  +oo )  C_  CC
43 fss 5601 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : RR
--> CC )
441, 42, 43sylancl 645 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
4544adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
4645ffvelrnda 5872 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4746mulid2d 9108 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4847mpteq2dva 4297 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4936, 48eqtr4d 2473 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5017a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
51 1re 9092 . . . . . . . . 9  |-  1  e.  RR
5251a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
5350, 34, 11ofc12 6331 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( RR  X.  {
( ( 1  /  A )  x.  A
) } ) )
54 fconstmpt 4923 . . . . . . . . . 10  |-  ( RR 
X.  { ( ( 1  /  A )  x.  A ) } )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )
5553, 54syl6eq 2486 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
568recnd 9116 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
5756adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5811rpne0d 10655 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5957, 58recid2d 9788 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
6059mpteq2dv 4298 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
6155, 60eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
6250, 52, 46, 61, 36offval2 6324 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( RR  X.  {
( 1  /  A
) } )  o F  x.  ( RR 
X.  { A }
) )  o F  x.  F )  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
6334rpcnd 10652 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
64 fconst6g 5634 . . . . . . . . 9  |-  ( ( 1  /  A )  e.  CC  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
6563, 64syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
66 fconst6g 5634 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
6757, 66syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { A } ) : RR --> CC )
68 mulass 9080 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
6968adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) ) )
7050, 65, 67, 45, 69caofass 6340 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( RR  X.  {
( 1  /  A
) } )  o F  x.  ( RR 
X.  { A }
) )  o F  x.  F )  =  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  (
( RR  X.  { A } )  o F  x.  F ) ) )
7149, 62, 703eqtr2d 2476 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  (
( RR  X.  { A } )  o F  x.  F ) ) )
7271fveq2d 5734 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( ( RR 
X.  { A }
)  o F  x.  F ) ) ) )
7333recnd 9116 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  CC )
7473, 57, 58divrec2d 9796 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  /  A )  =  ( ( 1  /  A
)  x.  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) ) ) )
7535, 72, 743brtr4d 4244 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  /  A ) )
764, 33, 11lemuldiv2d 10696 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  ( S.2 `  F ) )  <_ 
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <-> 
( S.2 `  F )  <_  ( ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  /  A ) ) )
7775, 76mpbird 225 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) ) )
78 itg2cl 19626 . . . . . 6  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,] 
+oo )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  e. 
RR* )
7928, 78syl 16 . . . . 5  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  e.  RR* )
8030rexrd 9136 . . . . 5  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
81 xrletri3 10747 . . . . 5  |-  ( ( ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  e.  RR*  /\  ( A  x.  ( S.2 `  F ) )  e. 
RR* )  ->  (
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) )  <->  ( ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) )  /\  ( A  x.  ( S.2 `  F ) )  <_ 
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) ) ) ) )
8279, 80, 81syl2anc 644 . . . 4  |-  ( ph  ->  ( ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) )  <->  ( ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) )  /\  ( A  x.  ( S.2 `  F ) )  <_ 
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) ) ) ) )
8382adantr 453 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) )  <-> 
( ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) )  /\  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) ) ) ) )
8412, 77, 83mpbir2and 890 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F
) ) )
8517a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  RR  e.  _V )
8644adantr 453 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
878adantr 453 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
8837a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
89 simplr 733 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
9089oveq1d 6098 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  ( A  x.  x ) )
91 mul02 9246 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
9291adantl 454 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
9390, 92eqtr3d 2472 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
9485, 86, 87, 88, 93caofid2 6337 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
9594fveq2d 5734 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( S.2 `  ( RR  X.  { 0 } ) ) )
96 itg20 19631 . . . 4  |-  ( S.2 `  ( RR  X.  {
0 } ) )  =  0
9795, 96syl6eq 2486 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  0 )
983adantr 453 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
9998recnd 9116 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
10099mul02d 9266 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
101 simpr 449 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
102101oveq1d 6098 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
10397, 100, 1023eqtr2d 2476 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
1047simprd 451 . . 3  |-  ( ph  ->  0  <_  A )
105 leloe 9163 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10637, 8, 105sylancr 646 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
107104, 106mpbid 203 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10884, 103, 107mpjaodan 763 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {csn 3816   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123    / cdiv 9679   RR+crp 10614   [,)cico 10920   [,]cicc 10921   S.2citg2 19510
This theorem is referenced by:  iblmulc2  19724  itgmulc2lem1  19725  bddmulibl  19732  iblmulc2nc  26272  itgmulc2nclem1  26273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-ofr 6308  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515  df-itg2 19516  df-0p 19564
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