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Theorem itg2mulc 19206
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 451 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> ( 0 [,)  +oo ) )
3 itg2mulc.3 . . . . 5  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
43adantr 451 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  e.  RR )
5 itg2mulc.4 . . . . . . . 8  |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )
6 elrege0 10838 . . . . . . . 8  |-  ( A  e.  ( 0 [,) 
+oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
75, 6sylib 188 . . . . . . 7  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A )
)
87simpld 445 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98anim1i 551 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( A  e.  RR  /\  0  < 
A ) )
10 elrp 10448 . . . . 5  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
119, 10sylibr 203 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
122, 4, 11itg2mulclem 19205 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F
) ) )
13 ge0mulcl 10841 . . . . . . . . 9  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
1413adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
15 fconst6g 5513 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) 
+oo )  ->  ( RR  X.  { A }
) : RR --> ( 0 [,)  +oo ) )
165, 15syl 15 . . . . . . . 8  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
17 reex 8918 . . . . . . . . 9  |-  RR  e.  _V
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  RR  e.  _V )
19 inidm 3454 . . . . . . . 8  |-  ( RR 
i^i  RR )  =  RR
2014, 16, 1, 18, 18, 19off 6180 . . . . . . 7  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
2120adantr 451 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
22 df-ico 10754 . . . . . . . . . 10  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
23 df-icc 10755 . . . . . . . . . 10  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
24 idd 21 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
25 xrltle 10575 . . . . . . . . . 10  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
2622, 23, 24, 25ixxssixx 10762 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
27 fss 5480 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
2928adantr 451 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
308, 3remulcld 8953 . . . . . . . 8  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
3130adantr 451 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  e.  RR )
32 itg2lecl 19197 . . . . . . 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 1182 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  RR )
3411rpreccld 10492 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  RR+ )
3521, 33, 34itg2mulclem 19205 . . . . 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 5658 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
37 0re 8928 . . . . . . . . . . . . . . 15  |-  0  e.  RR
38 pnfxr 10547 . . . . . . . . . . . . . . 15  |-  +oo  e.  RR*
39 icossre 10822 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
4037, 38, 39mp2an 653 . . . . . . . . . . . . . 14  |-  ( 0 [,)  +oo )  C_  RR
41 ax-resscn 8884 . . . . . . . . . . . . . 14  |-  RR  C_  CC
4240, 41sstri 3264 . . . . . . . . . . . . 13  |-  ( 0 [,)  +oo )  C_  CC
43 fss 5480 . . . . . . . . . . . . 13  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  CC )  ->  F : RR
--> CC )
441, 42, 43sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  F : RR --> CC )
4544adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  F : RR
--> CC )
46 ffvelrn 5746 . . . . . . . . . . 11  |-  ( ( F : RR --> CC  /\  y  e.  RR )  ->  ( F `  y
)  e.  CC )
4745, 46sylan 457 . . . . . . . . . 10  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
4847mulid2d 8943 . . . . . . . . 9  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  (
1  x.  ( F `
 y ) )  =  ( F `  y ) )
4948mpteq2dva 4187 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( 1  x.  ( F `  y
) ) )  =  ( y  e.  RR  |->  ( F `  y ) ) )
5036, 49eqtr4d 2393 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( y  e.  RR  |->  ( 1  x.  ( F `  y )
) ) )
5117a1i 10 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  RR  e.  _V )
52 1re 8927 . . . . . . . . 9  |-  1  e.  RR
5352a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  y  e.  RR )  ->  1  e.  RR )
5451, 34, 11ofc12 6189 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( RR  X.  {
( ( 1  /  A )  x.  A
) } ) )
55 fconstmpt 4814 . . . . . . . . . 10  |-  ( RR 
X.  { ( ( 1  /  A )  x.  A ) } )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )
5654, 55syl6eq 2406 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A
) ) )
578recnd 8951 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
5857adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
5911rpne0d 10487 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
6058, 59recid2d 9622 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
1  /  A )  x.  A )  =  1 )
6160mpteq2dv 4188 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( y  e.  RR  |->  ( ( 1  /  A )  x.  A ) )  =  ( y  e.  RR  |->  1 ) )
6256, 61eqtrd 2390 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( RR  X.  { A } ) )  =  ( y  e.  RR  |->  1 ) )
6351, 53, 47, 62, 36offval2 6182 . . . . . . 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 )
) ) )
6434rpcnd 10484 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 1  /  A )  e.  CC )
65 fconst6g 5513 . . . . . . . . 9  |-  ( ( 1  /  A )  e.  CC  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
6664, 65syl 15 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { ( 1  /  A ) } ) : RR --> CC )
67 fconst6g 5513 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( RR  X.  { A }
) : RR --> CC )
6858, 67syl 15 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( RR  X.  { A } ) : RR --> CC )
69 mulass 8915 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( x  x.  y
)  x.  z )  =  ( x  x.  ( y  x.  z
) ) )
7069adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  0  <  A )  /\  (
x  e.  CC  /\  y  e.  CC  /\  z  e.  CC ) )  -> 
( ( x  x.  y )  x.  z
)  =  ( x  x.  ( y  x.  z ) ) )
7151, 66, 68, 45, 70caofass 6198 . . . . . . 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 ) ) )
7250, 63, 713eqtr2d 2396 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  F  =  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  (
( RR  X.  { A } )  o F  x.  F ) ) )
7372fveq2d 5612 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  =  ( S.2 `  ( ( RR  X.  { ( 1  /  A ) } )  o F  x.  ( ( RR 
X.  { A }
)  o F  x.  F ) ) ) )
7433recnd 8951 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  e.  CC )
7574, 58, 59divrec2d 9630 . . . . 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 ) ) ) )
7635, 73, 753brtr4d 4134 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  F )  <_  (
( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  /  A ) )
774, 33, 11lemuldiv2d 10528 . . . 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 ) ) )
7876, 77mpbird 223 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( S.2 `  F
) )  <_  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) ) )
79 itg2cl 19191 . . . . . 6  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,] 
+oo )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  e. 
RR* )
8028, 79syl 15 . . . . 5  |-  ( ph  ->  ( S.2 `  (
( RR  X.  { A } )  o F  x.  F ) )  e.  RR* )
8130rexrd 8971 . . . . 5  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
82 xrletri3 10578 . . . . 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 ) ) ) ) )
8380, 81, 82syl2anc 642 . . . 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 ) ) ) ) )
8483adantr 451 . . 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 ) ) ) ) )
8512, 78, 84mpbir2and 888 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( S.2 `  ( ( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F
) ) )
8617a1i 10 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  RR  e.  _V )
8744adantr 451 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  F : RR --> CC )
888adantr 451 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  A  e.  RR )
8937a1i 10 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  0  e.  RR )
90 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  0  =  A )
9190oveq1d 5960 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  ( A  x.  x ) )
92 mul02 9080 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
9392adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  (
0  x.  x )  =  0 )
9491, 93eqtr3d 2392 . . . . . 6  |-  ( ( ( ph  /\  0  =  A )  /\  x  e.  CC )  ->  ( A  x.  x )  =  0 )
9586, 87, 88, 89, 94caofid2 6195 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
9695fveq2d 5612 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( S.2 `  ( RR  X.  { 0 } ) ) )
97 itg20 19196 . . . 4  |-  ( S.2 `  ( RR  X.  {
0 } ) )  =  0
9896, 97syl6eq 2406 . . 3  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  0 )
993adantr 451 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  RR )
10099recnd 8951 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  F )  e.  CC )
101100mul02d 9100 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  0 )
102 simpr 447 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
103102oveq1d 5960 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  ( S.2 `  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
10498, 101, 1033eqtr2d 2396 . 2  |-  ( (
ph  /\  0  =  A )  ->  ( S.2 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
1057simprd 449 . . 3  |-  ( ph  ->  0  <_  A )
106 leloe 8998 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
10737, 8, 106sylancr 644 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
108105, 107mpbid 201 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
10985, 104, 108mpjaodan 761 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   {csn 3716   class class class wbr 4104    e. cmpt 4158    X. cxp 4769   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    +oocpnf 8954   RR*cxr 8956    < clt 8957    <_ cle 8958    / cdiv 9513   RR+crp 10446   [,)cico 10750   [,]cicc 10751   S.2citg2 19075
This theorem is referenced by:  iblmulc2  19289  itgmulc2lem1  19290  bddmulibl  19297  iblmulc2nc  25505  itgmulc2nclem1  25506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-disj 4075  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-ofr 6166  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-xadd 10545  df-ioo 10752  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-xmet 16475  df-met 16476  df-ovol 18928  df-vol 18929  df-mbf 19079  df-itg1 19080  df-itg2 19081  df-0p 19129
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