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Theorem itg2mulclem 19101
Description: Lemma for itg2mulc 19102. (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 10662 . . . . . . . 8  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
3 df-icc 10663 . . . . . . . 8  |-  [,]  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <_  w ) } )
4 idd 21 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  u  e.  RR* )  ->  (
0  <_  u  ->  0  <_  u ) )
5 xrltle 10483 . . . . . . . 8  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
62, 3, 4, 5ixxssixx 10670 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
7 fss 5397 . . . . . . 7  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  ( 0 [,]  +oo ) )  ->  F : RR --> ( 0 [,] 
+oo ) )
81, 6, 7sylancl 643 . . . . . 6  |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )
98adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> ( 0 [,] 
+oo ) )
10 simpr 447 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  e.  dom  S.1 )
11 itg2mulclem.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
1211rpreccld 10400 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR+ )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR+ )
1413rpred 10390 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR )
1510, 14i1fmulc 19058 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  e.  dom  S.1 )
16 itg2ub 19088 . . . . . 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 1153 . . . . 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 642 . . . 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 19031 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  f : RR --> RR )
2019adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f : RR --> RR )
21 ffvelrn 5663 . . . . . . . . 9  |-  ( ( f : RR --> RR  /\  y  e.  RR )  ->  ( f `  y
)  e.  RR )
2220, 21sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  RR )
23 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
24 pnfxr 10455 . . . . . . . . . . . 12  |-  +oo  e.  RR*
25 icossre 10730 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2623, 24, 25mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
27 fss 5397 . . . . . . . . . . 11  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  (
0 [,)  +oo )  C_  RR )  ->  F : RR
--> RR )
281, 26, 27sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
2928adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F : RR --> RR )
30 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : RR --> RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  RR )
3129, 30sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
3211rpred 10390 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3332ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR )
3411rpgt0d 10393 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
3534ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  0  <  A )
36 ledivmul 9629 . . . . . . . 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 ) ) ) )
3722, 31, 33, 35, 36syl112anc 1186 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( f `  y )  <_  ( A  x.  ( F `  y ) ) ) )
3822recnd 8861 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  CC )
3933recnd 8861 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  CC )
4011adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR+ )
4140rpne0d 10395 . . . . . . . . . 10  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  =/=  0 )
4241adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  =/=  0 )
4338, 39, 42divrec2d 9540 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  /  A )  =  ( ( 1  /  A )  x.  ( f `  y
) ) )
4443breq1d 4033 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( ( f `  y )  /  A
)  <_  ( F `  y )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4537, 44bitr3d 246 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  <_  ( A  x.  ( F `  y
) )  <->  ( (
1  /  A )  x.  ( f `  y ) )  <_ 
( F `  y
) ) )
4645ralbidva 2559 . . . . 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 )
) )
47 reex 8828 . . . . . . 7  |-  RR  e.  _V
4847a1i 10 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  RR  e.  _V )
49 ovex 5883 . . . . . . 7  |-  ( A  x.  ( F `  y ) )  e. 
_V
5049a1i 10 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
_V )
5120feqmptd 5575 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f  =  ( y  e.  RR  |->  ( f `  y ) ) )
5211ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR+ )
53 fconstmpt 4732 . . . . . . . 8  |-  ( RR 
X.  { A }
)  =  ( y  e.  RR  |->  A )
5453a1i 10 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( RR  X.  { A }
)  =  ( y  e.  RR  |->  A ) )
551feqmptd 5575 . . . . . . . 8  |-  ( ph  ->  F  =  ( y  e.  RR  |->  ( F `
 y ) ) )
5655adantr 451 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  F  =  ( y  e.  RR  |->  ( F `  y ) ) )
5748, 52, 31, 54, 56offval2 6095 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( y  e.  RR  |->  ( A  x.  ( F `  y )
) ) )
5848, 22, 50, 51, 57ofrfval2 6096 . . . . 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
) ) ) )
59 ovex 5883 . . . . . . 7  |-  ( ( 1  /  A )  x.  ( f `  y ) )  e. 
_V
6059a1i 10 . . . . . 6  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( 1  /  A
)  x.  ( f `
 y ) )  e.  _V )
6112ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
1  /  A )  e.  RR+ )
62 fconstmpt 4732 . . . . . . . 8  |-  ( RR 
X.  { ( 1  /  A ) } )  =  ( y  e.  RR  |->  ( 1  /  A ) )
6362a1i 10 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( RR  X.  { ( 1  /  A ) } )  =  ( y  e.  RR  |->  ( 1  /  A ) ) )
6448, 61, 22, 63, 51offval2 6095 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  =  ( y  e.  RR  |->  ( ( 1  /  A )  x.  ( f `  y
) ) ) )
6548, 60, 31, 64, 56ofrfval2 6096 . . . . 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 ) ) )
6646, 58, 653bitr4d 276 . . . 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 ) )
6710, 14itg1mulc 19059 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
68 itg1cl 19040 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  ( S.1 `  f )  e.  RR )
6968adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  RR )
7069recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  f )  e.  CC )
7132adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  RR )
7271recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  CC )
7370, 72, 41divrec2d 9540 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  /  A )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
7467, 73eqtr4d 2318 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  =  ( ( S.1 `  f
)  /  A ) )
7574breq1d 4033 . . . . 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 ) ) )
76 itg2mulc.3 . . . . . . 7  |-  ( ph  ->  ( S.2 `  F
)  e.  RR )
7776adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.2 `  F )  e.  RR )
7834adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  0  <  A )
79 ledivmul 9629 . . . . . 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 ) ) ) )
8069, 77, 71, 78, 79syl112anc 1186 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( ( S.1 `  f
)  /  A )  <_  ( S.2 `  F
)  <->  ( S.1 `  f
)  <_  ( A  x.  ( S.2 `  F
) ) ) )
8175, 80bitr2d 245 . . . 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 ) ) )
8218, 66, 813imtr4d 259 . . 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
) ) ) )
8382ralrimiva 2626 . 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 ) ) ) )
84 ge0mulcl 10749 . . . . . 6  |-  ( ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo ) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
8584adantl 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,)  +oo )  /\  y  e.  ( 0 [,)  +oo )
) )  ->  (
x  x.  y )  e.  ( 0 [,) 
+oo ) )
86 fconstg 5428 . . . . . . 7  |-  ( A  e.  RR+  ->  ( RR 
X.  { A }
) : RR --> { A } )
8711, 86syl 15 . . . . . 6  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
88 rpre 10360 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
89 rpge0 10366 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  <_  A )
90 elrege0 10746 . . . . . . . . 9  |-  ( A  e.  ( 0 [,) 
+oo )  <->  ( A  e.  RR  /\  0  <_  A ) )
9188, 89, 90sylanbrc 645 . . . . . . . 8  |-  ( A  e.  RR+  ->  A  e.  ( 0 [,)  +oo ) )
9211, 91syl 15 . . . . . . 7  |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )
9392snssd 3760 . . . . . 6  |-  ( ph  ->  { A }  C_  ( 0 [,)  +oo ) )
94 fss 5397 . . . . . 6  |-  ( ( ( RR  X.  { A } ) : RR --> { A }  /\  { A }  C_  ( 0 [,)  +oo ) )  -> 
( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
9587, 93, 94syl2anc 642 . . . . 5  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> ( 0 [,)  +oo ) )
9647a1i 10 . . . . 5  |-  ( ph  ->  RR  e.  _V )
97 inidm 3378 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
9885, 95, 1, 96, 96, 97off 6093 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
99 fss 5397 . . . 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 ) )
10098, 6, 99sylancl 643 . . 3  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,]  +oo ) )
10132, 76remulcld 8863 . . . 4  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
102101rexrd 8881 . . 3  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
103 itg2leub 19089 . . 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 ) ) ) ) )
104100, 102, 103syl2anc 642 . 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 ) ) ) ) )
10583, 104mpbird 223 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    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    o Rcofr 6077   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    / cdiv 9423   RR+crp 10354   [,)cico 10658   [,]cicc 10659   S.1citg1 18970   S.2citg2 18971
This theorem is referenced by:  itg2mulc  19102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976  df-itg2 18977
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