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Theorem itg1mulc 21024
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 21008 . . 3  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
2 reex 9361 . . . . . 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 20996 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  F : RR --> RR )
76adantr 462 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
8 i1fmulc.3 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98adantr 462 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
10 0red 9375 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
11 simplr 747 . . . . . . 7  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1211oveq1d 6095 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 9534 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 463 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2465 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
163, 7, 9, 10, 15caofid2 6340 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
1716fveq2d 5683 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( S.1 `  ( RR  X.  { 0 } ) ) )
18 simpr 458 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
1918oveq1d 6095 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( 0  x.  ( S.1 `  F ) ) )
20 itg1cl 21005 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  e.  RR )
214, 20syl 16 . . . . . . 7  |-  ( ph  ->  ( S.1 `  F
)  e.  RR )
2221recnd 9400 . . . . . 6  |-  ( ph  ->  ( S.1 `  F
)  e.  CC )
2322mul02d 9555 . . . . 5  |-  ( ph  ->  ( 0  x.  ( S.1 `  F ) )  =  0 )
2423adantr 462 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
0  x.  ( S.1 `  F ) )  =  0 )
2519, 24eqtrd 2465 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  0 )
261, 17, 253eqtr4a 2491 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
274, 8i1fmulc 21023 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F )  e.  dom  S.1 )
2827adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
29 i1ff 20996 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1  ->  ( ( RR  X.  { A }
)  oF  x.  F ) : RR --> RR )
3028, 29syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F ) : RR --> RR )
31 frn 5553 . . . . . . . . . . . 12  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F ) 
C_  RR )
3332ssdifssd 3482 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  RR )
3433sselda 3344 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  RR )
3534recnd 9400 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  CC )
368adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  RR )
3736recnd 9400 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  CC )
3837adantr 462 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
39 simplr 747 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
4035, 38, 39divcan2d 10097 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( A  x.  ( m  /  A
) )  =  m )
414, 8i1fmulclem 21022 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } )  =  ( `' F " { ( m  /  A ) } ) )
4234, 41syldan 467 . . . . . . . . 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 5683 . . . . . . . 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 2438 . . . . . . 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 6098 . . . . . 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 718 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
4734, 46, 39redivcld 10147 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  RR )
4847recnd 9400 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  CC )
494ad2antrr 718 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
5046recnd 9400 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
51 eldifsni 3989 . . . . . . . . . . . 12  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  m  =/=  0
)
5251adantl 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  =/=  0 )
5335, 50, 52, 39divne0d 10111 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  =/=  0
)
54 eldifsn 3988 . . . . . . . . . 10  |-  ( ( m  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( m  /  A )  e.  RR  /\  ( m  /  A
)  =/=  0 ) )
5547, 53, 54sylanbrc 657 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  ( RR  \  { 0 } ) )
56 i1fima2sn 21000 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  ( m  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5749, 55, 56syl2anc 654 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5857recnd 9400 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  CC )
5938, 48, 58mulassd 9397 . . . . . 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 2467 . . . . 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 13164 . . . 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 20997 . . . . . . 7  |-  ( ( ( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e. 
Fin )
6328, 62syl 16 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  ( ( RR  X.  { A } )  oF  x.  F )  e.  Fin )
64 difss 3471 . . . . . 6  |-  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  oF  x.  F )
65 ssfi 7521 . . . . . 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 655 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  e.  Fin )
6748, 58mulcld 9394 . . . . 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 13234 . . . 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 2468 . . 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 21003 . . . 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 16 . . 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 462 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  F  e.  dom  S.1 )
73 itg1val 21003 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  =  sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
7472, 73syl 16 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  F )  = 
sum_ k  e.  ( ran  F  \  {
0 } ) ( k  x.  ( vol `  ( `' F " { k } ) ) ) )
75 id 22 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  k  =  ( m  /  A ) )
76 sneq 3875 . . . . . . . . 9  |-  ( k  =  ( m  /  A )  ->  { k }  =  { ( m  /  A ) } )
7776imaeq2d 5157 . . . . . . . 8  |-  ( k  =  ( m  /  A )  ->  ( `' F " { k } )  =  ( `' F " { ( m  /  A ) } ) )
7877fveq2d 5683 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
7975, 78oveq12d 6098 . . . . . 6  |-  ( k  =  ( m  /  A )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( ( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
80 eqid 2433 . . . . . . 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 3466 . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . 18  |-  ( F : RR --> RR  ->  F  Fn  RR )
846, 83syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  RR )
85 eqidd 2434 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 y )  =  ( F `  y
) )
8682, 8, 84, 85ofc1 6332 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  =  ( A  x.  ( F `
 y ) ) )
8786adantlr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  =  ( A  x.  ( F `  y )
) )
8887oveq1d 6095 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  =  ( ( A  x.  ( F `  y ) )  /  A ) )
896adantr 462 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  A  =/=  0 )  ->  F : RR --> RR )
9089ffvelrnda 5831 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
9190recnd 9400 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
9237adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  e.  CC )
93 simplr 747 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  =/=  0 )
9491, 92, 93divcan3d 10100 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( A  x.  ( F `  y )
)  /  A )  =  ( F `  y ) )
9588, 94eqtrd 2465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  =  ( F `  y ) )
9689, 83syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  0 )  ->  F  Fn  RR )
97 fnfvelrn 5828 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
9896, 97sylan 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  ran  F )
9995, 98eqeltrd 2507 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  e.  ran  F )
10099ralrimiva 2789 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
)  e.  ran  F
)
101 ffn 5547 . . . . . . . . . . . . 13  |-  ( ( ( RR  X.  { A } )  oF  x.  F ) : RR --> RR  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
10230, 101syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  Fn  RR )
103 oveq1 6087 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  ->  (
n  /  A )  =  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
) )
104103eleq1d 2499 . . . . . . . . . . . . 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 5834 . . . . . . . . . . . 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 16 . . . . . . . . . . 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 232 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. n  e.  ran  ( ( RR 
X.  { A }
)  oF  x.  F ) ( n  /  A )  e. 
ran  F )
108107r19.21bi 2804 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )  -> 
( n  /  A
)  e.  ran  F
)
10981, 108sylan2 471 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ran  F )
11033sselda 3344 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
111110recnd 9400 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  CC )
11237adantr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
113 eldifsni 3989 . . . . . . . . . 10  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  n  =/=  0
)
114113adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  =/=  0 )
115 simplr 747 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
116111, 112, 114, 115divne0d 10111 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  =/=  0
)
117 eldifsn 3988 . . . . . . . 8  |-  ( ( n  /  A )  e.  ( ran  F  \  { 0 } )  <-> 
( ( n  /  A )  e.  ran  F  /\  ( n  /  A )  =/=  0
) )
118109, 116, 117sylanbrc 657 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ( ran  F  \  {
0 } ) )
119 eldifi 3466 . . . . . . . . 9  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  e.  ran  F )
120 fnfvelrn 5828 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
12287, 121eqeltrrd 2508 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
123122ralrimiva 2789 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
124 oveq2 6088 . . . . . . . . . . . . . 14  |-  ( k  =  ( F `  y )  ->  ( A  x.  k )  =  ( A  x.  ( F `  y ) ) )
125124eleq1d 2499 . . . . . . . . . . . . 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 5834 . . . . . . . . . . . 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 16 . . . . . . . . . . 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 232 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A. k  e.  ran  F ( A  x.  k )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
129128r19.21bi 2804 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ran  F )  -> 
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
130119, 129sylan2 471 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) )
13137adantr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  e.  CC )
132 frn 5553 . . . . . . . . . . . . 13  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
13389, 132syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  F 
C_  RR )
134133ssdifssd 3482 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  F  \  { 0 } )  C_  RR )
135134sselda 3344 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  RR )
136135recnd 9400 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  CC )
137 simplr 747 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  =/=  0 )
138 eldifsni 3989 . . . . . . . . . 10  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
139138adantl 463 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  =/=  0 )
140131, 136, 137, 139mulne0d 9976 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  =/=  0
)
141 eldifsn 3988 . . . . . . . 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 657 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ( ran  ( ( RR 
X.  { A }
)  oF  x.  F )  \  {
0 } ) )
143 simpl 454 . . . . . . . . . . . 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 3339 . . . . . . . . . . . 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 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  RR )
146145recnd 9400 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  CC )
1478ad2antrr 718 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  RR )
148147recnd 9400 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  CC )
149135adantrl 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  RR )
150149recnd 9400 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  CC )
151 simplr 747 . . . . . . . . . 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 10117 . . . . . . . . 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 201 . . . . . . . 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 2435 . . . . . . . 8  |-  ( n  =  ( A  x.  k )  <->  ( A  x.  k )  =  n )
155 eqcom 2435 . . . . . . . 8  |-  ( k  =  ( n  /  A )  <->  ( n  /  A )  =  k )
156153, 154, 1553bitr4g 288 . . . . . . 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 6301 . . . . . 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 6087 . . . . . . . 8  |-  ( n  =  m  ->  (
n  /  A )  =  ( m  /  A ) )
159 ovex 6105 . . . . . . . 8  |-  ( m  /  A )  e. 
_V
160158, 80, 159fvmpt 5762 . . . . . . 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 463 . . . . . 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 21000 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
16372, 162sylan 468 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
164135, 163remulcld 9402 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
165164recnd 9400 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  CC )
16679, 66, 157, 161, 165fsumf1o 13184 . . . . 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 2465 . . . 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 6096 . . 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 2475 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
17026, 169pm2.61dane 2679 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 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   _Vcvv 2962    \ cdif 3313    C_ wss 3316   {csn 3865    e. cmpt 4338    X. cxp 4825   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    oFcof 6307   Fincfn 7298   CCcc 9268   RRcr 9269   0cc0 9270    x. cmul 9275    / cdiv 9981   sum_csu 13147   volcvol 20789   S.1citg1 20937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xadd 11078  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-xmet 17654  df-met 17655  df-ovol 20790  df-vol 20791  df-mbf 20941  df-itg1 20942
This theorem is referenced by:  itg1sub  21029  itg2const  21060  itg2mulclem  21066  itg2monolem1  21070  itg2addnclem  28287
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