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Theorem itg1mulc 19549
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 19533 . . 3  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
2 reex 9037 . . . . . 6  |-  RR  e.  _V
32a1i 11 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  RR  e.  _V )
4 i1fmulc.2 . . . . . . 7  |-  ( ph  ->  F  e.  dom  S.1 )
5 i1ff 19521 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  F : RR --> RR )
76adantr 452 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
8 i1fmulc.3 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98adantr 452 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
10 0re 9047 . . . . . 6  |-  0  e.  RR
1110a1i 11 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
12 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1312oveq1d 6055 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
14 mul02lem2 9199 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1514adantl 453 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1613, 15eqtrd 2436 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
173, 7, 9, 11, 16caofid2 6294 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  =  ( RR  X.  {
0 } ) )
1817fveq2d 5691 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( S.1 `  ( RR  X.  { 0 } ) ) )
19 simpr 448 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
2019oveq1d 6055 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( 0  x.  ( S.1 `  F ) ) )
21 itg1cl 19530 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  e.  RR )
224, 21syl 16 . . . . . . 7  |-  ( ph  ->  ( S.1 `  F
)  e.  RR )
2322recnd 9070 . . . . . 6  |-  ( ph  ->  ( S.1 `  F
)  e.  CC )
2423mul02d 9220 . . . . 5  |-  ( ph  ->  ( 0  x.  ( S.1 `  F ) )  =  0 )
2524adantr 452 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
0  x.  ( S.1 `  F ) )  =  0 )
2620, 25eqtrd 2436 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  0 )
271, 18, 263eqtr4a 2462 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
284, 8i1fmulc 19548 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
2928adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1 )
30 i1ff 19521 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1  ->  ( ( RR  X.  { A }
)  o F  x.  F ) : RR --> RR )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F ) : RR --> RR )
32 frn 5556 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
3331, 32syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F ) 
C_  RR )
3433ssdifssd 3445 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  RR )
3534sselda 3308 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  e.  RR )
3635recnd 9070 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  e.  CC )
378adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  RR )
3837recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  CC )
3938adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
40 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
4136, 39, 40divcan2d 9748 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( A  x.  ( m  /  A
) )  =  m )
424, 8i1fmulclem 19547 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  o F  x.  F ) " {
m } )  =  ( `' F " { ( m  /  A ) } ) )
4335, 42syldan 457 . . . . . . . . 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 ) } ) )
4443fveq2d 5691 . . . . . . . 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 ) } ) ) )
4544eqcomd 2409 . . . . . . 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 } ) ) )
4641, 45oveq12d 6058 . . . . . 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 } ) ) ) )
478ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
4835, 47, 40redivcld 9798 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  RR )
4948recnd 9070 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  CC )
504ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
5147recnd 9070 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
52 eldifsni 3888 . . . . . . . . . . . 12  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  m  =/=  0
)
5352adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  m  =/=  0 )
5436, 51, 53, 40divne0d 9762 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  =/=  0
)
55 eldifsn 3887 . . . . . . . . . 10  |-  ( ( m  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( m  /  A )  e.  RR  /\  ( m  /  A
)  =/=  0 ) )
5648, 54, 55sylanbrc 646 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  ( RR  \  { 0 } ) )
57 i1fima2sn 19525 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  ( m  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5850, 56, 57syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5958recnd 9070 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  CC )
6039, 49, 59mulassd 9067 . . . . . 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 ) } ) ) ) ) )
6146, 60eqtr3d 2438 . . . . 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 ) } ) ) ) ) )
6261sumeq2dv 12452 . . . 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 ) } ) ) ) ) )
63 i1frn 19522 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  o F  x.  F )  e. 
dom  S.1  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e. 
Fin )
6429, 63syl 16 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  o F  x.  F )  e.  Fin )
65 difss 3434 . . . . . 6  |-  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  o F  x.  F )
66 ssfi 7288 . . . . . 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 )
6764, 65, 66sylancl 644 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  e.  Fin )
6849, 59mulcld 9064 . . . . 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, 38, 68fsummulc2 12522 . . . 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 ) } ) ) ) ) )
7062, 69eqtr4d 2439 . . 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 19528 . . . 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 } ) ) ) )
7229, 71syl 16 . . 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 452 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  F  e.  dom  S.1 )
74 itg1val 19528 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7573, 74syl 16 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
76 id 20 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  k  =  ( m  /  A ) )
77 sneq 3785 . . . . . . . . 9  |-  ( k  =  ( m  /  A )  ->  { k }  =  { ( m  /  A ) } )
7877imaeq2d 5162 . . . . . . . 8  |-  ( k  =  ( m  /  A )  ->  ( `' F " { k } )  =  ( `' F " { ( m  /  A ) } ) )
7978fveq2d 5691 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
8076, 79oveq12d 6058 . . . . . 6  |-  ( k  =  ( m  /  A )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( ( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
81 eqid 2404 . . . . . . 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 3429 . . . . . . . . 9  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  n  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) )
832a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  RR  e.  _V )
84 ffn 5550 . . . . . . . . . . . . . . . . . 18  |-  ( F : RR --> RR  ->  F  Fn  RR )
856, 84syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  RR )
86 eqidd 2405 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 y )  =  ( F `  y
) )
8783, 8, 85, 86ofc1 6286 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  =  ( A  x.  ( F `
 y ) ) )
8887adantlr 696 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 y )  =  ( A  x.  ( F `  y )
) )
8988oveq1d 6055 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  =  ( ( A  x.  ( F `  y ) )  /  A ) )
906adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  A  =/=  0 )  ->  F : RR --> RR )
9190ffvelrnda 5829 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
9291recnd 9070 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
9338adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  e.  CC )
94 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  =/=  0 )
9592, 93, 94divcan3d 9751 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( A  x.  ( F `  y )
)  /  A )  =  ( F `  y ) )
9689, 95eqtrd 2436 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  =  ( F `  y ) )
9790, 84syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  0 )  ->  F  Fn  RR )
98 fnfvelrn 5826 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
9997, 98sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  ran  F )
10096, 99eqeltrd 2478 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  o F  x.  F ) `  y
)  /  A )  e.  ran  F )
101100ralrimiva 2749 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  /  A
)  e.  ran  F
)
102 ffn 5550 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  o F  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
10331, 102syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  o F  x.  F )  Fn  RR )
104 oveq1 6047 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  ->  (
n  /  A )  =  ( ( ( ( RR  X.  { A } )  o F  x.  F ) `  y )  /  A
) )
105104eleq1d 2470 . . . . . . . . . . . . 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 ) )
106105ralrn 5832 . . . . . . . . . . . 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
) )
107103, 106syl 16 . . . . . . . . . . 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 ) )
108101, 107mpbird 224 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. n  e.  ran  ( ( RR 
X.  { A }
)  o F  x.  F ) ( n  /  A )  e. 
ran  F )
109108r19.21bi 2764 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )  -> 
( n  /  A
)  e.  ran  F
)
11082, 109sylan2 461 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ran  F )
11134sselda 3308 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
112111recnd 9070 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  e.  CC )
11338adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
114 eldifsni 3888 . . . . . . . . . 10  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  o F  x.  F )  \  { 0 } )  ->  n  =/=  0
)
115114adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  n  =/=  0 )
116 simplr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
117112, 113, 115, 116divne0d 9762 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  =/=  0
)
118 eldifsn 3887 . . . . . . . 8  |-  ( ( n  /  A )  e.  ( ran  F  \  { 0 } )  <-> 
( ( n  /  A )  e.  ran  F  /\  ( n  /  A )  =/=  0
) )
119110, 117, 118sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ( ran  F  \  {
0 } ) )
120 eldifi 3429 . . . . . . . . 9  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  e.  ran  F )
121 fnfvelrn 5826 . . . . . . . . . . . . . 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 ) )
122103, 121sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  o F  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
12388, 122eqeltrrd 2479 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
124123ralrimiva 2749 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) )
125 oveq2 6048 . . . . . . . . . . . . . 14  |-  ( k  =  ( F `  y )  ->  ( A  x.  k )  =  ( A  x.  ( F `  y ) ) )
126125eleq1d 2470 . . . . . . . . . . . . 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 ) ) )
127126ralrn 5832 . . . . . . . . . . . 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 ) ) )
12897, 127syl 16 . . . . . . . . . . 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 ) ) )
129124, 128mpbird 224 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. k  e.  ran  F ( A  x.  k )  e. 
ran  ( ( RR 
X.  { A }
)  o F  x.  F ) )
130129r19.21bi 2764 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ran  F )  -> 
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  o F  x.  F ) )
131120, 130sylan2 461 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  o F  x.  F ) )
13238adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  e.  CC )
133 frn 5556 . . . . . . . . . . . . 13  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
13490, 133syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  F 
C_  RR )
135134ssdifssd 3445 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  F  \  { 0 } )  C_  RR )
136135sselda 3308 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  RR )
137136recnd 9070 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  CC )
138 simplr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  =/=  0 )
139 eldifsni 3888 . . . . . . . . . 10  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
140139adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  =/=  0 )
141132, 137, 138, 140mulne0d 9630 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  =/=  0
)
142 eldifsn 3887 . . . . . . . 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
) )
143131, 141, 142sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ( ran  ( ( RR 
X.  { A }
)  o F  x.  F )  \  {
0 } ) )
144 simpl 444 . . . . . . . . . . . 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 } ) )
145 ssel2 3303 . . . . . . . . . . . 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 )
14634, 144, 145syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  RR )
147146recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  CC )
1488ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  RR )
149148recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  CC )
150136adantrl 697 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  RR )
151150recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  CC )
152 simplr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  o F  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  =/=  0 )
153147, 149, 151, 152divmuld 9768 . . . . . . . . 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 ) )
154153bicomd 193 . . . . . . . 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 ) )
155 eqcom 2406 . . . . . . . 8  |-  ( n  =  ( A  x.  k )  <->  ( A  x.  k )  =  n )
156 eqcom 2406 . . . . . . . 8  |-  ( k  =  ( n  /  A )  <->  ( n  /  A )  =  k )
157154, 155, 1563bitr4g 280 . . . . . . 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
) ) )
15881, 119, 143, 157f1o2d 6255 . . . . . 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 } ) )
159 oveq1 6047 . . . . . . . 8  |-  ( n  =  m  ->  (
n  /  A )  =  ( m  /  A ) )
160 ovex 6065 . . . . . . . 8  |-  ( m  /  A )  e. 
_V
161159, 81, 160fvmpt 5765 . . . . . . 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 ) )
162161adantl 453 . . . . . 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 ) )
163 i1fima2sn 19525 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
16473, 163sylan 458 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
165136, 164remulcld 9072 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
166165recnd 9070 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  CC )
16780, 67, 158, 162, 166fsumf1o 12472 . . . . 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 ) } ) ) ) )
16875, 167eqtrd 2436 . . . 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 ) } ) ) ) )
169168oveq2d 6056 . . 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 ) } ) ) ) ) )
17070, 72, 1693eqtr4d 2446 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
17127, 170pm2.61dane 2645 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951    / cdiv 9633   sum_csu 12434   volcvol 19313   S.1citg1 19460
This theorem is referenced by:  itg1sub  19554  itg2const  19585  itg2mulclem  19591  itg2monolem1  19595  itg2addnclem  26155
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466
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