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Theorem itg1mulc 19059
Description: The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
i1fmulc.2  |-  ( ph  ->  F  e.  dom  S.1 )
i1fmulc.3  |-  ( ph  ->  A  e.  RR )
Assertion
Ref Expression
itg1mulc  |-  ( ph  ->  ( S.1 `  (
( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )

Proof of Theorem itg1mulc
Dummy variables  k  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itg10 19043 . . 3  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
2 reex 8828 . . . . . 6  |-  RR  e.  _V
32a1i 10 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  RR  e.  _V )
4 i1fmulc.2 . . . . . . 7  |-  ( ph  ->  F  e.  dom  S.1 )
5 i1ff 19031 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  F : RR --> RR )
76adantr 451 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
8 i1fmulc.3 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98adantr 451 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
10 0re 8838 . . . . . 6  |-  0  e.  RR
1110a1i 10 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
12 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1312oveq1d 5873 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
14 mul02lem2 8989 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1514adantl 452 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1613, 15eqtrd 2315 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
173, 7, 9, 11, 16caofid2 6108 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
1817fveq2d 5529 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( S.1 `  ( RR  X.  { 0 } ) ) )
19 simpr 447 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
2019oveq1d 5873 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( 0  x.  ( S.1 `  F ) ) )
21 itg1cl 19040 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  e.  RR )
224, 21syl 15 . . . . . . 7  |-  ( ph  ->  ( S.1 `  F
)  e.  RR )
2322recnd 8861 . . . . . 6  |-  ( ph  ->  ( S.1 `  F
)  e.  CC )
2423mul02d 9010 . . . . 5  |-  ( ph  ->  ( 0  x.  ( S.1 `  F ) )  =  0 )
2524adantr 451 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
0  x.  ( S.1 `  F ) )  =  0 )
2620, 25eqtrd 2315 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  0 )
271, 18, 263eqtr4a 2341 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
28 difss 3303 . . . . . . . . . . 11  |-  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  o F  x.  F )
294, 8i1fmulc 19058 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
3029adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
31 i1ff 19031 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> RR )
3230, 31syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
33 frn 5395 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
3432, 33syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
3528, 34syl5ss 3190 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  RR )
3635sselda 3180 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  e.  RR )
3736recnd 8861 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  e.  CC )
388adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  RR )
3938recnd 8861 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  CC )
4039adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
41 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
4237, 40, 41divcan2d 9538 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( A  x.  ( m  /  A
) )  =  m )
434, 8i1fmulclem 19057 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
m } )  =  ( `' F " { ( m  /  A ) } ) )
4436, 43syldan 456 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  o F  x.  F )
" { m }
)  =  ( `' F " { ( m  /  A ) } ) )
4544fveq2d 5529 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { m } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
4645eqcomd 2288 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  =  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { m } ) ) )
4742, 46oveq12d 5876 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( ( A  x.  ( m  /  A ) )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  =  ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
m } ) ) ) )
488ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
4936, 48, 41redivcld 9588 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  RR )
5049recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  CC )
514ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
5248recnd 8861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
53 eldifsni 3750 . . . . . . . . . . . 12  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  m  =/=  0
)
5453adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  =/=  0 )
5537, 52, 54, 41divne0d 9552 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  =/=  0
)
56 eldifsn 3749 . . . . . . . . . 10  |-  ( ( m  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( m  /  A )  e.  RR  /\  ( m  /  A
)  =/=  0 ) )
5749, 55, 56sylanbrc 645 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  ( RR  \  { 0 } ) )
58 i1fima2sn 19035 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  ( m  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5951, 57, 58syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
6059recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  CC )
6140, 50, 60mulassd 8858 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( ( A  x.  ( m  /  A ) )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  =  ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
6247, 61eqtr3d 2317 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
m } ) ) )  =  ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
6362sumeq2dv 12176 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { m } ) ) )  =  sum_ m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( A  x.  (
( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
64 i1frn 19032 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e. 
Fin )
6530, 64syl 15 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e.  Fin )
66 ssfi 7083 . . . . . 6  |-  ( ( ran  ( ( RR 
X.  { A }
)  o F  x.  F )  e.  Fin  /\  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
C_  ran  ( ( RR  X.  { A }
)  o F  x.  F ) )  -> 
( ran  ( ( RR  X.  { A }
)  o F  x.  F )  \  {
0 } )  e. 
Fin )
6765, 28, 66sylancl 643 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  e.  Fin )
6850, 60mulcld 8855 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( (
m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  e.  CC )
6967, 39, 68fsummulc2 12246 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A  x.  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } ) ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
7063, 69eqtr4d 2318 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { m } ) ) )  =  ( A  x.  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
71 itg1val 19038 . . . 4  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1  ->  ( S.1 `  ( ( RR  X.  { A } )  o F  x.  F ) )  =  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  o F  x.  F ) " { m } ) ) ) )
7230, 71syl 15 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
m } ) ) ) )
734adantr 451 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  F  e.  dom  S.1 )
74 itg1val 19038 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7573, 74syl 15 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
76 id 19 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  k  =  ( m  /  A ) )
77 sneq 3651 . . . . . . . . 9  |-  ( k  =  ( m  /  A )  ->  { k }  =  { ( m  /  A ) } )
7877imaeq2d 5012 . . . . . . . 8  |-  ( k  =  ( m  /  A )  ->  ( `' F " { k } )  =  ( `' F " { ( m  /  A ) } ) )
7978fveq2d 5529 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
8076, 79oveq12d 5876 . . . . . 6  |-  ( k  =  ( m  /  A )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( ( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
81 eqid 2283 . . . . . . 7  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
|->  ( n  /  A
) )  =  ( n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) )
82 eldifi 3298 . . . . . . . . 9  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  n  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) )
832a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  RR  e.  _V )
84 ffn 5389 . . . . . . . . . . . . . . . . . 18  |-  ( F : RR --> RR  ->  F  Fn  RR )
856, 84syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  RR )
86 eqidd 2284 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 y )  =  ( F `  y
) )
8783, 8, 85, 86ofc1 6100 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  =  ( A  x.  ( F `
 y ) ) )
8887adantlr 695 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 y )  =  ( A  x.  ( F `  y )
) )
8988oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  =  ( ( A  x.  ( F `  y ) )  /  A ) )
906adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  A  =/=  0 )  ->  F : RR --> RR )
91 ffvelrn 5663 . . . . . . . . . . . . . . . . 17  |-  ( ( F : RR --> RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  RR )
9290, 91sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
9392recnd 8861 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
9439adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  e.  CC )
95 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  =/=  0 )
9693, 94, 95divcan3d 9541 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( A  x.  ( F `  y )
)  /  A )  =  ( F `  y ) )
9789, 96eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  =  ( F `  y ) )
9890, 84syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  0 )  ->  F  Fn  RR )
99 fnfvelrn 5662 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
10098, 99sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  ran  F )
10197, 100eqeltrd 2357 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  e.  ran  F )
102101ralrimiva 2626 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  /  A
)  e.  ran  F
)
103 ffn 5389 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
10432, 103syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
105 oveq1 5865 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  ->  (
n  /  A )  =  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  /  A
) )
106105eleq1d 2349 . . . . . . . . . . . . 13  |-  ( n  =  ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  ->  (
( n  /  A
)  e.  ran  F  <->  ( ( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  e.  ran  F ) )
107106ralrn 5668 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  Fn  RR  ->  ( A. n  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) ( n  /  A )  e.  ran  F  <->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  /  A
)  e.  ran  F
) )
108104, 107syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A. n  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) ( n  /  A
)  e.  ran  F  <->  A. y  e.  RR  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  e.  ran  F ) )
109102, 108mpbird 223 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. n  e.  ran  ( ( RR 
X.  { A }
)  o F  x.  F ) ( n  /  A )  e. 
ran  F )
110109r19.21bi 2641 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )  -> 
( n  /  A
)  e.  ran  F
)
11182, 110sylan2 460 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ran  F )
11235sselda 3180 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
113112recnd 8861 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  e.  CC )
11439adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
115 eldifsni 3750 . . . . . . . . . 10  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  n  =/=  0
)
116115adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  =/=  0 )
117 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
118113, 114, 116, 117divne0d 9552 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  =/=  0
)
119 eldifsn 3749 . . . . . . . 8  |-  ( ( n  /  A )  e.  ( ran  F  \  { 0 } )  <-> 
( ( n  /  A )  e.  ran  F  /\  ( n  /  A )  =/=  0
) )
120111, 118, 119sylanbrc 645 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ( ran  F  \  {
0 } ) )
121 eldifi 3298 . . . . . . . . 9  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  e.  ran  F )
122 fnfvelrn 5662 . . . . . . . . . . . . . 14  |-  ( ( ( ( RR  X.  { A } )  o F  x.  F )  Fn  RR  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
123104, 122sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
12488, 123eqeltrrd 2358 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
125124ralrimiva 2626 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) )
126 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( k  =  ( F `  y )  ->  ( A  x.  k )  =  ( A  x.  ( F `  y ) ) )
127126eleq1d 2349 . . . . . . . . . . . . 13  |-  ( k  =  ( F `  y )  ->  (
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  o F  x.  F )  <->  ( A  x.  ( F `  y
) )  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) ) )
128127ralrn 5668 . . . . . . . . . . . 12  |-  ( F  Fn  RR  ->  ( A. k  e.  ran  F ( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  o F  x.  F )  <->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) ) )
12998, 128syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A. k  e.  ran  F ( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  o F  x.  F )  <->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) ) )
130125, 129mpbird 223 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. k  e.  ran  F ( A  x.  k )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
131130r19.21bi 2641 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ran  F )  -> 
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) )
132121, 131sylan2 460 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) )
13339adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  e.  CC )
134 difss 3303 . . . . . . . . . . . 12  |-  ( ran 
F  \  { 0 } )  C_  ran  F
135 frn 5395 . . . . . . . . . . . . 13  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
13690, 135syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  F 
C_  RR )
137134, 136syl5ss 3190 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  F  \  { 0 } )  C_  RR )
138137sselda 3180 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  RR )
139138recnd 8861 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  CC )
140 simplr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  =/=  0 )
141 eldifsni 3750 . . . . . . . . . 10  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
142141adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  =/=  0 )
143133, 139, 140, 142mulne0d 9420 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  =/=  0
)
144 eldifsn 3749 . . . . . . . 8  |-  ( ( A  x.  k )  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  <-> 
( ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  o F  x.  F )  /\  ( A  x.  k )  =/=  0
) )
145132, 143, 144sylanbrc 645 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ( ran  ( ( RR 
X.  { A }
)  o F  x.  F )  \  {
0 } ) )
146 simpl 443 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  n  e.  ( ran  ( ( RR 
X.  { A }
)  o F  x.  F )  \  {
0 } ) )
147 ssel2 3175 . . . . . . . . . . . 12  |-  ( ( ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
C_  RR  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
14835, 146, 147syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  RR )
149148recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  CC )
1508ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  RR )
151150recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  CC )
152138adantrl 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  RR )
153152recnd 8861 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  CC )
154 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  =/=  0 )
155149, 151, 153, 154divmuld 9558 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
( n  /  A
)  =  k  <->  ( A  x.  k )  =  n ) )
156155bicomd 192 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
( A  x.  k
)  =  n  <->  ( n  /  A )  =  k ) )
157 eqcom 2285 . . . . . . . 8  |-  ( n  =  ( A  x.  k )  <->  ( A  x.  k )  =  n )
158 eqcom 2285 . . . . . . . 8  |-  ( k  =  ( n  /  A )  <->  ( n  /  A )  =  k )
159156, 157, 1583bitr4g 279 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
n  =  ( A  x.  k )  <->  k  =  ( n  /  A
) ) )
16081, 120, 145, 159f1o2d 6069 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) ) : ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) -1-1-onto-> ( ran  F  \  {
0 } ) )
161 oveq1 5865 . . . . . . . 8  |-  ( n  =  m  ->  (
n  /  A )  =  ( m  /  A ) )
162 ovex 5883 . . . . . . . 8  |-  ( m  /  A )  e. 
_V
163161, 81, 162fvmpt 5602 . . . . . . 7  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  ( ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) 
|->  ( n  /  A
) ) `  m
)  =  ( m  /  A ) )
164163adantl 452 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) ) `  m )  =  ( m  /  A ) )
165 i1fima2sn 19035 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
16673, 165sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
167138, 166remulcld 8863 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
168167recnd 8861 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  CC )
16980, 67, 160, 164, 168fsumf1o 12196 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
17075, 169eqtrd 2315 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
171170oveq2d 5874 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( A  x.  sum_ m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
17270, 72, 1713eqtr4d 2325 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
17327, 172pm2.61dane 2524 1  |-  ( ph  ->  ( S.1 `  (
( RR  X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.1 `  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    \ cdif 3149    C_ wss 3152   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    / cdiv 9423   sum_csu 12158   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  itg1sub  19064  itg2const  19095  itg2mulclem  19101  itg2monolem1  19105
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-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
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