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Theorem itg2mulclem 19117
Description: Lemma for itg2mulc 19118. (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 10678 . . . . . . . 8  |-  [,)  =  ( z  e.  RR* ,  w  e.  RR*  |->  { v  e.  RR*  |  (
z  <_  v  /\  v  <  w ) } )
3 df-icc 10679 . . . . . . . 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 10499 . . . . . . . 8  |-  ( ( u  e.  RR*  /\  +oo  e.  RR* )  ->  (
u  <  +oo  ->  u  <_  +oo ) )
62, 3, 4, 5ixxssixx 10686 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
7 fss 5413 . . . . . . 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 10416 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR+ )
1312adantr 451 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR+ )
1413rpred 10406 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
1  /  A )  e.  RR )
1510, 14i1fmulc 19074 . . . . 5  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( RR  X.  {
( 1  /  A
) } )  o F  x.  f )  e.  dom  S.1 )
16 itg2ub 19104 . . . . . 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 19047 . . . . . . . . . 10  |-  ( f  e.  dom  S.1  ->  f : RR --> RR )
2019adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  f : RR --> RR )
21 ffvelrn 5679 . . . . . . . . 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 8854 . . . . . . . . . . . 12  |-  0  e.  RR
24 pnfxr 10471 . . . . . . . . . . . 12  |-  +oo  e.  RR*
25 icossre 10746 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
2623, 24, 25mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,)  +oo )  C_  RR
27 fss 5413 . . . . . . . . . . 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 5679 . . . . . . . . 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 10406 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
3332ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  A  e.  RR )
3411rpgt0d 10409 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
3534ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  0  <  A )
36 ledivmul 9645 . . . . . . . 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 8877 . . . . . . . . 9  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
f `  y )  e.  CC )
3933recnd 8877 . . . . . . . . 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 10411 . . . . . . . . . 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 9556 . . . . . . . 8  |-  ( ( ( ph  /\  f  e.  dom  S.1 )  /\  y  e.  RR )  ->  (
( f `  y
)  /  A )  =  ( ( 1  /  A )  x.  ( f `  y
) ) )
4443breq1d 4049 . . . . . . 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 2572 . . . . 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 8844 . . . . . . 7  |-  RR  e.  _V
4847a1i 10 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  RR  e.  _V )
49 ovex 5899 . . . . . . 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 5591 . . . . . 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 4748 . . . . . . . 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 5591 . . . . . . . 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 6111 . . . . . 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 6112 . . . . 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 5899 . . . . . . 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 4748 . . . . . . . 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 6111 . . . . . 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 6112 . . . . 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 19075 . . . . . . 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 19056 . . . . . . . . . 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 8877 . . . . . . . 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 8877 . . . . . . . 8  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  A  e.  CC )
7370, 72, 41divrec2d 9556 . . . . . . 7  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  (
( S.1 `  f )  /  A )  =  ( ( 1  /  A )  x.  ( S.1 `  f ) ) )
7467, 73eqtr4d 2331 . . . . . 6  |-  ( (
ph  /\  f  e.  dom  S.1 )  ->  ( S.1 `  ( ( RR 
X.  { ( 1  /  A ) } )  o F  x.  f ) )  =  ( ( S.1 `  f
)  /  A ) )
7574breq1d 4049 . . . . 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 9645 . . . . . 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 2639 . 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 10765 . . . . . 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 5444 . . . . . . 7  |-  ( A  e.  RR+  ->  ( RR 
X.  { A }
) : RR --> { A } )
8711, 86syl 15 . . . . . 6  |-  ( ph  ->  ( RR  X.  { A } ) : RR --> { A } )
88 rpre 10376 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  RR )
89 rpge0 10382 . . . . . . . . 9  |-  ( A  e.  RR+  ->  0  <_  A )
90 elrege0 10762 . . . . . . . . 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 3776 . . . . . 6  |-  ( ph  ->  { A }  C_  ( 0 [,)  +oo ) )
94 fss 5413 . . . . . 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 3391 . . . . 5  |-  ( RR 
i^i  RR )  =  RR
9885, 95, 1, 96, 96, 97off 6109 . . . 4  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F ) : RR --> ( 0 [,)  +oo ) )
99 fss 5413 . . . 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 8879 . . . 4  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR )
102101rexrd 8897 . . 3  |-  ( ph  ->  ( A  x.  ( S.2 `  F ) )  e.  RR* )
103 itg2leub 19105 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    o Rcofr 6093   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    / cdiv 9439   RR+crp 10370   [,)cico 10674   [,]cicc 10675   S.1citg1 18986   S.2citg2 18987
This theorem is referenced by:  itg2mulc  19118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-itg2 18993
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