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Theorem itg1mulc 22654
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 } )  oF  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 22638 . . 3  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
2 reex 9632 . . . . . 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 22626 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 17 . . . . . 6  |-  ( ph  ->  F : RR --> RR )
76adantr 467 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
8 i1fmulc.3 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98adantr 467 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
10 0red 9646 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
11 simplr 761 . . . . . . 7  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1211oveq1d 6318 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 9812 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 468 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2464 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
163, 7, 9, 10, 15caofid2 6574 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
1716fveq2d 5883 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( S.1 `  ( RR  X.  { 0 } ) ) )
18 simpr 463 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
1918oveq1d 6318 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( 0  x.  ( S.1 `  F ) ) )
20 itg1cl 22635 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  e.  RR )
214, 20syl 17 . . . . . . 7  |-  ( ph  ->  ( S.1 `  F
)  e.  RR )
2221recnd 9671 . . . . . 6  |-  ( ph  ->  ( S.1 `  F
)  e.  CC )
2322mul02d 9833 . . . . 5  |-  ( ph  ->  ( 0  x.  ( S.1 `  F ) )  =  0 )
2423adantr 467 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
0  x.  ( S.1 `  F ) )  =  0 )
2519, 24eqtrd 2464 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  0 )
261, 17, 253eqtr4a 2490 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
274, 8i1fmulc 22653 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F )  e.  dom  S.1 )
2827adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
29 i1ff 22626 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> RR )
3028, 29syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
31 frn 5750 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
3332ssdifssd 3604 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  RR )
3433sselda 3465 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  RR )
3534recnd 9671 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  CC )
368adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  RR )
3736recnd 9671 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  CC )
3837adantr 467 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
39 simplr 761 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
4035, 38, 39divcan2d 10387 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( A  x.  ( m  /  A
) )  =  m )
414, 8i1fmulclem 22652 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } )  =  ( `' F " { ( m  /  A ) } ) )
4234, 41syldan 473 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( `' ( ( RR  X.  { A } )  oF  x.  F )
" { m }
)  =  ( `' F " { ( m  /  A ) } ) )
4342fveq2d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { m } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
4443eqcomd 2431 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  =  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { m } ) ) )
4540, 44oveq12d 6321 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( ( A  x.  ( m  /  A ) )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  =  ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } ) ) ) )
468ad2antrr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
4734, 46, 39redivcld 10437 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  RR )
4847recnd 9671 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  CC )
494ad2antrr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
5046recnd 9671 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
51 eldifsni 4124 . . . . . . . . . . . 12  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  m  =/=  0
)
5251adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  =/=  0 )
5335, 50, 52, 39divne0d 10401 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  =/=  0
)
54 eldifsn 4123 . . . . . . . . . 10  |-  ( ( m  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( m  /  A )  e.  RR  /\  ( m  /  A
)  =/=  0 ) )
5547, 53, 54sylanbrc 669 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  ( RR  \  { 0 } ) )
56 i1fima2sn 22630 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  ( m  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5749, 55, 56syl2anc 666 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5857recnd 9671 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  CC )
5938, 48, 58mulassd 9668 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( ( A  x.  ( m  /  A ) )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  =  ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
6045, 59eqtr3d 2466 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } ) ) )  =  ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
6160sumeq2dv 13762 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { m } ) ) )  =  sum_ m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( A  x.  (
( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
62 i1frn 22627 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e. 
Fin )
6328, 62syl 17 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e.  Fin )
64 difss 3593 . . . . . 6  |-  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  oF  x.  F )
65 ssfi 7796 . . . . . 6  |-  ( ( ran  ( ( RR 
X.  { A }
)  oF  x.  F )  e.  Fin  /\  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
C_  ran  ( ( RR  X.  { A }
)  oF  x.  F ) )  -> 
( ran  ( ( RR  X.  { A }
)  oF  x.  F )  \  {
0 } )  e. 
Fin )
6663, 64, 65sylancl 667 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  e.  Fin )
6748, 58mulcld 9665 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( (
m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) )  e.  CC )
6866, 37, 67fsummulc2 13838 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A  x.  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } ) ( A  x.  ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
6961, 68eqtr4d 2467 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { m } ) ) )  =  ( A  x.  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
70 itg1val 22633 . . . 4  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1  ->  ( S.1 `  ( ( RR  X.  { A } )  oF  x.  F ) )  =  sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR  X.  { A } )  oF  x.  F ) " { m } ) ) ) )
7128, 70syl 17 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } ) ( m  x.  ( vol `  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } ) ) ) )
724adantr 467 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  F  e.  dom  S.1 )
73 itg1val 22633 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7472, 73syl 17 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
75 id 23 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  k  =  ( m  /  A ) )
76 sneq 4007 . . . . . . . . 9  |-  ( k  =  ( m  /  A )  ->  { k }  =  { ( m  /  A ) } )
7776imaeq2d 5185 . . . . . . . 8  |-  ( k  =  ( m  /  A )  ->  ( `' F " { k } )  =  ( `' F " { ( m  /  A ) } ) )
7877fveq2d 5883 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
7975, 78oveq12d 6321 . . . . . 6  |-  ( k  =  ( m  /  A )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( ( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
80 eqid 2423 . . . . . . 7  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
|->  ( n  /  A
) )  =  ( n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) )
81 eldifi 3588 . . . . . . . . 9  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  n  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) )
822a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  RR  e.  _V )
83 ffn 5744 . . . . . . . . . . . . . . . . . 18  |-  ( F : RR --> RR  ->  F  Fn  RR )
846, 83syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  RR )
85 eqidd 2424 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 y )  =  ( F `  y
) )
8682, 8, 84, 85ofc1 6566 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  =  ( A  x.  ( F `
 y ) ) )
8786adantlr 720 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  =  ( A  x.  ( F `  y )
) )
8887oveq1d 6318 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  =  ( ( A  x.  ( F `  y ) )  /  A ) )
896adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  A  =/=  0 )  ->  F : RR --> RR )
9089ffvelrnda 6035 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
9190recnd 9671 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
9237adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  e.  CC )
93 simplr 761 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  =/=  0 )
9491, 92, 93divcan3d 10390 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( A  x.  ( F `  y )
)  /  A )  =  ( F `  y ) )
9588, 94eqtrd 2464 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  =  ( F `  y ) )
9689, 83syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  0 )  ->  F  Fn  RR )
97 fnfvelrn 6032 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
9896, 97sylan 474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  ran  F )
9995, 98eqeltrd 2511 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  e.  ran  F )
10099ralrimiva 2840 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
)  e.  ran  F
)
101 ffn 5744 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
10230, 101syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
103 oveq1 6310 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  ->  (
n  /  A )  =  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
) )
104103eleq1d 2492 . . . . . . . . . . . . 13  |-  ( n  =  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  ->  (
( n  /  A
)  e.  ran  F  <->  ( ( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  e.  ran  F ) )
105104ralrn 6038 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  Fn  RR  ->  ( A. n  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) ( n  /  A )  e.  ran  F  <->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
)  e.  ran  F
) )
106102, 105syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A. n  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) ( n  /  A
)  e.  ran  F  <->  A. y  e.  RR  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  e.  ran  F ) )
107100, 106mpbird 236 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. n  e.  ran  ( ( RR 
X.  { A }
)  oF  x.  F ) ( n  /  A )  e. 
ran  F )
108107r19.21bi 2795 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )  -> 
( n  /  A
)  e.  ran  F
)
10981, 108sylan2 477 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ran  F )
11033sselda 3465 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
111110recnd 9671 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  CC )
11237adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
113 eldifsni 4124 . . . . . . . . . 10  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  n  =/=  0
)
114113adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  =/=  0 )
115 simplr 761 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
116111, 112, 114, 115divne0d 10401 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  =/=  0
)
117 eldifsn 4123 . . . . . . . 8  |-  ( ( n  /  A )  e.  ( ran  F  \  { 0 } )  <-> 
( ( n  /  A )  e.  ran  F  /\  ( n  /  A )  =/=  0
) )
118109, 116, 117sylanbrc 669 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ( ran  F  \  {
0 } ) )
119 eldifi 3588 . . . . . . . . 9  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  e.  ran  F )
120 fnfvelrn 6032 . . . . . . . . . . . . . 14  |-  ( ( ( ( RR  X.  { A } )  oF  x.  F )  Fn  RR  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
121102, 120sylan 474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
12287, 121eqeltrrd 2512 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
123122ralrimiva 2840 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
124 oveq2 6311 . . . . . . . . . . . . . 14  |-  ( k  =  ( F `  y )  ->  ( A  x.  k )  =  ( A  x.  ( F `  y ) ) )
125124eleq1d 2492 . . . . . . . . . . . . 13  |-  ( k  =  ( F `  y )  ->  (
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F )  <->  ( A  x.  ( F `  y
) )  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) ) )
126125ralrn 6038 . . . . . . . . . . . 12  |-  ( F  Fn  RR  ->  ( A. k  e.  ran  F ( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F )  <->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) ) )
12796, 126syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A. k  e.  ran  F ( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F )  <->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) ) )
128123, 127mpbird 236 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. k  e.  ran  F ( A  x.  k )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
129128r19.21bi 2795 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ran  F )  -> 
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
130119, 129sylan2 477 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) )
13137adantr 467 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  e.  CC )
132 frn 5750 . . . . . . . . . . . . 13  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
13389, 132syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  F 
C_  RR )
134133ssdifssd 3604 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  F  \  { 0 } )  C_  RR )
135134sselda 3465 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  RR )
136135recnd 9671 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  CC )
137 simplr 761 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  =/=  0 )
138 eldifsni 4124 . . . . . . . . . 10  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
139138adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  =/=  0 )
140131, 136, 137, 139mulne0d 10266 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  =/=  0
)
141 eldifsn 4123 . . . . . . . 8  |-  ( ( A  x.  k )  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  <-> 
( ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  oF  x.  F )  /\  ( A  x.  k )  =/=  0
) )
142130, 140, 141sylanbrc 669 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ( ran  ( ( RR 
X.  { A }
)  oF  x.  F )  \  {
0 } ) )
143 simpl 459 . . . . . . . . . . . 12  |-  ( ( n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  n  e.  ( ran  ( ( RR 
X.  { A }
)  oF  x.  F )  \  {
0 } ) )
144 ssel2 3460 . . . . . . . . . . . 12  |-  ( ( ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
C_  RR  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
14533, 143, 144syl2an 480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  RR )
146145recnd 9671 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  CC )
1478ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  RR )
148147recnd 9671 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  CC )
149135adantrl 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  RR )
150149recnd 9671 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  CC )
151 simplr 761 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  =/=  0 )
152146, 148, 150, 151divmuld 10407 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
( n  /  A
)  =  k  <->  ( A  x.  k )  =  n ) )
153152bicomd 205 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
( A  x.  k
)  =  n  <->  ( n  /  A )  =  k ) )
154 eqcom 2432 . . . . . . . 8  |-  ( n  =  ( A  x.  k )  <->  ( A  x.  k )  =  n )
155 eqcom 2432 . . . . . . . 8  |-  ( k  =  ( n  /  A )  <->  ( n  /  A )  =  k )
156153, 154, 1553bitr4g 292 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  (
n  =  ( A  x.  k )  <->  k  =  ( n  /  A
) ) )
15780, 118, 142, 156f1o2d 6533 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) ) : ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) -1-1-onto-> ( ran  F  \  {
0 } ) )
158 oveq1 6310 . . . . . . . 8  |-  ( n  =  m  ->  (
n  /  A )  =  ( m  /  A ) )
159 ovex 6331 . . . . . . . 8  |-  ( m  /  A )  e. 
_V
160158, 80, 159fvmpt 5962 . . . . . . 7  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  ( ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) 
|->  ( n  /  A
) ) `  m
)  =  ( m  /  A ) )
161160adantl 468 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  |->  ( n  /  A ) ) `  m )  =  ( m  /  A ) )
162 i1fima2sn 22630 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
16372, 162sylan 474 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
164135, 163remulcld 9673 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
165164recnd 9671 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  CC )
16679, 66, 157, 161, 165fsumf1o 13782 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  sum_ k  e.  ( ran  F  \  { 0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) )  =  sum_ m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
16774, 166eqtrd 2464 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
168167oveq2d 6319 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( A  x.  sum_ m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) ( ( m  /  A )  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) ) )
16969, 71, 1683eqtr4d 2474 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
17026, 169pm2.61dane 2743 1  |-  ( ph  ->  ( S.1 `  (
( RR  X.  { A } )  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   _Vcvv 3082    \ cdif 3434    C_ wss 3437   {csn 3997    |-> cmpt 4480    X. cxp 4849   `'ccnv 4850   dom cdm 4851   ran crn 4852   "cima 4854    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    oFcof 6541   Fincfn 7575   CCcc 9539   RRcr 9540   0cc0 9541    x. cmul 9546    / cdiv 10271   sum_csu 13745   volcvol 22407   S.1citg1 22565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-q 11267  df-rp 11305  df-xadd 11412  df-ioo 11641  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-sum 13746  df-xmet 18956  df-met 18957  df-ovol 22408  df-vol 22410  df-mbf 22569  df-itg1 22570
This theorem is referenced by:  itg1sub  22659  itg2const  22690  itg2mulclem  22696  itg2monolem1  22700  itg2addnclem  31913
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