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Theorem itg2mulc 19596
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 452 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,)  +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 452 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  e.  RR )
5 itg2mulc.4 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )
6 elrege0 10967 . . . . . . . 8  |-  ( A  e.  ( 0 [,) 
+oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
75, 6sylib 189 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 446 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 552 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 10574 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 204 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 19595 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) ) )
13 ge0mulcl 10970 . . . . . . . . 9  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
1413adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
15 fconst6g 5595 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) 
+oo )  ->  ( RR  X.  { A }
) : RR --> ( 0 [,)  +oo ) )
165, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
17 reex 9041 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3514 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6283 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
2120adantr 452 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
22 df-ico 10882 . . . . . . . . . 10  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
23 df-icc 10883 . . . . . . . . . 10  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
24 idd 22 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
25 xrltle 10702 . . . . . . . . . 10  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
2622, 23, 24, 25ixxssixx 10890 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
27 fss 5562 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
2928adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
308, 3remulcld 9076 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
3130adantr 452 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  e.  RR )
32 itg2lecl 19587 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  RR )
3411rpreccld 10618 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3521, 33, 34itg2mulclem 19595 . . . . 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 5742 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
37 0re 9051 . . . . . . . . . . . . . . 15  |-  0  e.  RR
38 pnfxr 10673 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
39 icossre 10951 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
4037, 38, 39mp2an 654 . . . . . . . . . . . . . 14  |-  ( 0 [,)  +oo )  C_  RR
41 ax-resscn 9007 . . . . . . . . . . . . . 14  |-  RR  C_  CC
4240, 41sstri 3321 . . . . . . . . . . . . 13  |-  ( 0 [,)  +oo )  C_  CC
43 fss 5562 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : RR
--> CC )
441, 42, 43sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
4544adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
4645ffvelrnda 5833 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4746mulid2d 9066 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4847mpteq2dva 4259 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
4936, 48eqtr4d 2443 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5017a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
51 1re 9050 . . . . . . . . 9  |-  1  e.  RR
5251a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
5350, 34, 11ofc12 6292 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( RR  X.  {
( ( 1  /  A )  x.  A
) } ) )
54 fconstmpt 4884 . . . . . . . . . 10  |-  ( RR 
X.  { ( ( 1  /  A )  x.  A ) } )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )
5553, 54syl6eq 2456 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
568recnd 9074 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
5756adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5811rpne0d 10613 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
5957, 58recid2d 9746 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
6059mpteq2dv 4260 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
6155, 60eqtrd 2440 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
6250, 52, 46, 61, 36offval2 6285 . . . . . . 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 10610 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
64 fconst6g 5595 . . . . . . . . 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 5595 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
6757, 66syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { A } ) : RR --> CC )
68 mulass 9038 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
6968adantl 453 . . . . . . . 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 6301 . . . . . . 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 2446 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  (
( RR  X.  { A } )  o F  x.  F ) ) )
7271fveq2d 5695 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( ( RR 
X.  { A }
)  o F  x.  F ) ) ) )
7333recnd 9074 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  CC )
7473, 57, 58divrec2d 9754 . . . . 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 4206 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  /  A ) )
764, 33, 11lemuldiv2d 10654 . . . 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 224 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) ) )
78 itg2cl 19581 . . . . . 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 9094 . . . . 5  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
81 xrletri3 10705 . . . . 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 643 . . . 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 452 . . 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 889 . 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 452 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
878adantr 452 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
8837a1i 11 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
89 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
9089oveq1d 6059 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  ( A  x.  x ) )
91 mul02 9204 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
9291adantl 453 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
9390, 92eqtr3d 2442 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
9485, 86, 87, 88, 93caofid2 6298 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
9594fveq2d 5695 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( S.2 `  ( RR  X.  { 0 } ) ) )
96 itg20 19586 . . . 4  |-  ( S.2 `  ( RR  X.  {
0 } ) )  =  0
9795, 96syl6eq 2456 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  0 )
983adantr 452 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
9998recnd 9074 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
10099mul02d 9224 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
101 simpr 448 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
102101oveq1d 6059 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
10397, 100, 1023eqtr2d 2446 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
1047simprd 450 . . 3  |-  ( ph  ->  0  <_  A )
105 leloe 9121 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10637, 8, 105sylancr 645 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
107104, 106mpbid 202 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10884, 103, 107mpjaodan 762 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    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2920    C_ wss 3284   {csn 3778   class class class wbr 4176    e. cmpt 4230    X. cxp 4839   -->wf 5413   ` cfv 5417  (class class class)co 6044    o Fcof 6266   CCcc 8948   RRcr 8949   0cc0 8950   1c1 8951    x. cmul 8955    +oocpnf 9077   RR*cxr 9079    < clt 9080    <_ cle 9081    / cdiv 9637   RR+crp 10572   [,)cico 10878   [,]cicc 10879   S.2citg2 19465
This theorem is referenced by:  iblmulc2  19679  itgmulc2lem1  19680  bddmulibl  19687  iblmulc2nc  26173  itgmulc2nclem1  26174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-disj 4147  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-ofr 6269  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xadd 10671  df-ioo 10880  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-xmet 16654  df-met 16655  df-ovol 19318  df-vol 19319  df-mbf 19469  df-itg1 19470  df-itg2 19471  df-0p 19519
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