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Theorem itg1mulc 21318
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 21302 . . 3  |-  ( S.1 `  ( RR  X.  {
0 } ) )  =  0
2 reex 9487 . . . . . 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 21290 . . . . . . 7  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  F : RR --> RR )
76adantr 465 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  F : RR --> RR )
8 i1fmulc.3 . . . . . 6  |-  ( ph  ->  A  e.  RR )
98adantr 465 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  e.  RR )
10 0red 9501 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  RR )
11 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  A  =  0 )
1211oveq1d 6218 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  ( 0  x.  x ) )
13 mul02lem2 9660 . . . . . . 7  |-  ( x  e.  RR  ->  (
0  x.  x )  =  0 )
1413adantl 466 . . . . . 6  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( 0  x.  x
)  =  0 )
1512, 14eqtrd 2495 . . . . 5  |-  ( ( ( ph  /\  A  =  0 )  /\  x  e.  RR )  ->  ( A  x.  x
)  =  0 )
163, 7, 9, 10, 15caofid2 6464 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  =  ( RR  X.  {
0 } ) )
1716fveq2d 5806 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( S.1 `  ( RR  X.  { 0 } ) ) )
18 simpr 461 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
1918oveq1d 6218 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  ( 0  x.  ( S.1 `  F ) ) )
20 itg1cl 21299 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( S.1 `  F )  e.  RR )
214, 20syl 16 . . . . . . 7  |-  ( ph  ->  ( S.1 `  F
)  e.  RR )
2221recnd 9526 . . . . . 6  |-  ( ph  ->  ( S.1 `  F
)  e.  CC )
2322mul02d 9681 . . . . 5  |-  ( ph  ->  ( 0  x.  ( S.1 `  F ) )  =  0 )
2423adantr 465 . . . 4  |-  ( (
ph  /\  A  = 
0 )  ->  (
0  x.  ( S.1 `  F ) )  =  0 )
2519, 24eqtrd 2495 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  ( A  x.  ( S.1 `  F ) )  =  0 )
261, 17, 253eqtr4a 2521 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
274, 8i1fmulc 21317 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  X.  { A } )  oF  x.  F )  e.  dom  S.1 )
2827adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( RR  X.  { A } )  oF  x.  F )  e. 
dom  S.1 )
29 i1ff 21290 . . . . . . . . . . . . 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 5676 . . . . . . . . . . . 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 3605 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  RR )
3433sselda 3467 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  RR )
3534recnd 9526 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  e.  CC )
368adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  RR )
3736recnd 9526 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  A  e.  CC )
3837adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
39 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
4035, 38, 39divcan2d 10223 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( A  x.  ( m  /  A
) )  =  m )
414, 8i1fmulclem 21316 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  RR )  ->  ( `' ( ( RR 
X.  { A }
)  oF  x.  F ) " {
m } )  =  ( `' F " { ( m  /  A ) } ) )
4234, 41syldan 470 . . . . . . . . 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 5806 . . . . . . . 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 2462 . . . . . . 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 6221 . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  RR )
4734, 46, 39redivcld 10273 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  RR )
4847recnd 9526 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  CC )
494ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  F  e.  dom  S.1 )
5046recnd 9526 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
51 eldifsni 4112 . . . . . . . . . . . 12  |-  ( m  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  m  =/=  0
)
5251adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  m  =/=  0 )
5335, 50, 52, 39divne0d 10237 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  =/=  0
)
54 eldifsn 4111 . . . . . . . . . 10  |-  ( ( m  /  A )  e.  ( RR  \  { 0 } )  <-> 
( ( m  /  A )  e.  RR  /\  ( m  /  A
)  =/=  0 ) )
5547, 53, 54sylanbrc 664 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( m  /  A )  e.  ( RR  \  { 0 } ) )
56 i1fima2sn 21294 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  ( m  /  A
)  e.  ( RR 
\  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5749, 55, 56syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  RR )
5857recnd 9526 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  m  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( vol `  ( `' F " { ( m  /  A ) } ) )  e.  CC )
5938, 48, 58mulassd 9523 . . . . . 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 2497 . . . . 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 13301 . . . 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 21291 . . . . . . 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 3594 . . . . . 6  |-  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  C_  ran  ( ( RR  X.  { A } )  oF  x.  F )
65 ssfi 7647 . . . . . 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 662 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  e.  Fin )
6748, 58mulcld 9520 . . . . 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 13372 . . . 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 2498 . . 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 21297 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  F  e.  dom  S.1 )
73 itg1val 21297 . . . . . 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 3998 . . . . . . . . 9  |-  ( k  =  ( m  /  A )  ->  { k }  =  { ( m  /  A ) } )
7776imaeq2d 5280 . . . . . . . 8  |-  ( k  =  ( m  /  A )  ->  ( `' F " { k } )  =  ( `' F " { ( m  /  A ) } ) )
7877fveq2d 5806 . . . . . . 7  |-  ( k  =  ( m  /  A )  ->  ( vol `  ( `' F " { k } ) )  =  ( vol `  ( `' F " { ( m  /  A ) } ) ) )
7975, 78oveq12d 6221 . . . . . 6  |-  ( k  =  ( m  /  A )  ->  (
k  x.  ( vol `  ( `' F " { k } ) ) )  =  ( ( m  /  A
)  x.  ( vol `  ( `' F " { ( m  /  A ) } ) ) ) )
80 eqid 2454 . . . . . . 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 3589 . . . . . . . . 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 5670 . . . . . . . . . . . . . . . . . 18  |-  ( F : RR --> RR  ->  F  Fn  RR )
846, 83syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  RR )
85 eqidd 2455 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  RR )  ->  ( F `
 y )  =  ( F `  y
) )
8682, 8, 84, 85ofc1 6456 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  RR )  ->  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  =  ( A  x.  ( F `
 y ) ) )
8786adantlr 714 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  =  ( A  x.  ( F `  y )
) )
8887oveq1d 6218 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  =  ( ( A  x.  ( F `  y ) )  /  A ) )
896adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  A  =/=  0 )  ->  F : RR --> RR )
9089ffvelrnda 5955 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  RR )
9190recnd 9526 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  CC )
9237adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  e.  CC )
93 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  A  =/=  0 )
9491, 92, 93divcan3d 10226 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( A  x.  ( F `  y )
)  /  A )  =  ( F `  y ) )
9588, 94eqtrd 2495 . . . . . . . . . . . . 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 5952 . . . . . . . . . . . . . 14  |-  ( ( F  Fn  RR  /\  y  e.  RR )  ->  ( F `  y
)  e.  ran  F
)
9896, 97sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( F `  y )  e.  ran  F )
9995, 98eqeltrd 2542 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( ( RR 
X.  { A }
)  oF  x.  F ) `  y
)  /  A )  e.  ran  F )
10099ralrimiva 2830 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
)  e.  ran  F
)
101 ffn 5670 . . . . . . . . . . . . 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 6210 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  ->  (
n  /  A )  =  ( ( ( ( RR  X.  { A } )  oF  x.  F ) `  y )  /  A
) )
104103eleq1d 2523 . . . . . . . . . . . . 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 5958 . . . . . . . . . . . 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 2920 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )  -> 
( n  /  A
)  e.  ran  F
)
10981, 108sylan2 474 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ran  F )
11033sselda 3467 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  RR )
111110recnd 9526 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  e.  CC )
11237adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  e.  CC )
113 eldifsni 4112 . . . . . . . . . 10  |-  ( n  e.  ( ran  (
( RR  X.  { A } )  oF  x.  F )  \  { 0 } )  ->  n  =/=  0
)
114113adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  n  =/=  0 )
115 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  A  =/=  0 )
116111, 112, 114, 115divne0d 10237 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  =/=  0
)
117 eldifsn 4111 . . . . . . . 8  |-  ( ( n  /  A )  e.  ( ran  F  \  { 0 } )  <-> 
( ( n  /  A )  e.  ran  F  /\  ( n  /  A )  =/=  0
) )
118109, 116, 117sylanbrc 664 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F )  \  { 0 } ) )  ->  ( n  /  A )  e.  ( ran  F  \  {
0 } ) )
119 eldifi 3589 . . . . . . . . 9  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  e.  ran  F )
120 fnfvelrn 5952 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  (
( ( RR  X.  { A } )  oF  x.  F ) `
 y )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
12287, 121eqeltrrd 2543 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  0 )  /\  y  e.  RR )  ->  ( A  x.  ( F `  y ) )  e. 
ran  ( ( RR 
X.  { A }
)  oF  x.  F ) )
123122ralrimiva 2830 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  A. y  e.  RR  ( A  x.  ( F `  y ) )  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
124 oveq2 6211 . . . . . . . . . . . . . 14  |-  ( k  =  ( F `  y )  ->  ( A  x.  k )  =  ( A  x.  ( F `  y ) ) )
125124eleq1d 2523 . . . . . . . . . . . . 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 5958 . . . . . . . . . . . 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 2920 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ran  F )  -> 
( A  x.  k
)  e.  ran  (
( RR  X.  { A } )  oF  x.  F ) )
130119, 129sylan2 474 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ran  ( ( RR  X.  { A } )  oF  x.  F ) )
13137adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  e.  CC )
132 frn 5676 . . . . . . . . . . . . 13  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
13389, 132syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  0 )  ->  ran  F 
C_  RR )
134133ssdifssd 3605 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  ( ran  F  \  { 0 } )  C_  RR )
135134sselda 3467 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  RR )
136135recnd 9526 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  e.  CC )
137 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  A  =/=  0 )
138 eldifsni 4112 . . . . . . . . . 10  |-  ( k  e.  ( ran  F  \  { 0 } )  ->  k  =/=  0
)
139138adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  k  =/=  0 )
140131, 136, 137, 139mulne0d 10102 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  =/=  0
)
141 eldifsn 4111 . . . . . . . 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 664 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( A  x.  k )  e.  ( ran  ( ( RR 
X.  { A }
)  oF  x.  F )  \  {
0 } ) )
143 simpl 457 . . . . . . . . . . . 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 3462 . . . . . . . . . . . 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 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  RR )
146145recnd 9526 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  n  e.  CC )
1478ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  RR )
148147recnd 9526 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  A  e.  CC )
149135adantrl 715 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  RR )
150149recnd 9526 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  (
n  e.  ( ran  ( ( RR  X.  { A } )  oF  x.  F ) 
\  { 0 } )  /\  k  e.  ( ran  F  \  { 0 } ) ) )  ->  k  e.  CC )
151 simplr 754 . . . . . . . . . 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 10243 . . . . . . . . 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 2463 . . . . . . . 8  |-  ( n  =  ( A  x.  k )  <->  ( A  x.  k )  =  n )
155 eqcom 2463 . . . . . . . 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 6425 . . . . . 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 6210 . . . . . . . 8  |-  ( n  =  m  ->  (
n  /  A )  =  ( m  /  A ) )
159 ovex 6228 . . . . . . . 8  |-  ( m  /  A )  e. 
_V
160158, 80, 159fvmpt 5886 . . . . . . 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 466 . . . . . 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 21294 . . . . . . . . 9  |-  ( ( F  e.  dom  S.1  /\  k  e.  ( ran 
F  \  { 0 } ) )  -> 
( vol `  ( `' F " { k } ) )  e.  RR )
16372, 162sylan 471 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( vol `  ( `' F " { k } ) )  e.  RR )
164135, 163remulcld 9528 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  RR )
165164recnd 9526 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  ( ran  F  \  { 0 } ) )  ->  ( k  x.  ( vol `  ( `' F " { k } ) ) )  e.  CC )
16679, 66, 157, 161, 165fsumf1o 13321 . . . . 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 2495 . . . 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 6219 . . 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 2505 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  ( S.1 `  ( ( RR 
X.  { A }
)  oF  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
17026, 169pm2.61dane 2770 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    \ cdif 3436    C_ wss 3439   {csn 3988    |-> cmpt 4461    X. cxp 4949   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   Fincfn 7423   CCcc 9394   RRcr 9395   0cc0 9396    x. cmul 9401    / cdiv 10107   sum_csu 13284   volcvol 21082   S.1citg1 21231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-xadd 11204  df-ioo 11418  df-ico 11420  df-icc 11421  df-fz 11558  df-fzo 11669  df-fl 11762  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-xmet 17938  df-met 17939  df-ovol 21083  df-vol 21084  df-mbf 21235  df-itg1 21236
This theorem is referenced by:  itg1sub  21323  itg2const  21354  itg2mulclem  21360  itg2monolem1  21364  itg2addnclem  28611
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