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Theorem i1fmul 19051
Description: The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
Assertion
Ref Expression
i1fmul  |-  ( ph  ->  ( F  o F  x.  G )  e. 
dom  S.1 )

Proof of Theorem i1fmul
Dummy variables  y 
z  w  v  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 remulcl 8822 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
21adantl 452 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
3 i1fadd.1 . . . 4  |-  ( ph  ->  F  e.  dom  S.1 )
4 i1ff 19031 . . . 4  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 15 . . 3  |-  ( ph  ->  F : RR --> RR )
6 i1fadd.2 . . . 4  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 19031 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 15 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 8828 . . . 4  |-  RR  e.  _V
109a1i 10 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3378 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6093 . 2  |-  ( ph  ->  ( F  o F  x.  G ) : RR --> RR )
13 i1frn 19032 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 15 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 19032 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 15 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7128 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 642 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2283 . . . . . 6  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  x.  v
) )  =  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) )
20 ovex 5883 . . . . . 6  |-  ( u  x.  v )  e. 
_V
2119, 20fnmpt2i 6193 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  x.  v
) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5457 . . . . 5  |-  ( ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) )  Fn  ( ran  F  X.  ran  G
)  <->  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) : ( ran 
F  X.  ran  G
) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) )
2321, 22mpbi 199 . . . 4  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  x.  v
) ) : ( ran  F  X.  ran  G ) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) )
24 fofi 7142 . . . 4  |-  ( ( ( ran  F  X.  ran  G )  e.  Fin  /\  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) ) : ( ran  F  X.  ran  G ) -onto-> ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) )  ->  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) )  e.  Fin )
2518, 23, 24sylancl 643 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) )  e.  Fin )
26 eqid 2283 . . . . . . . . 9  |-  ( x  x.  y )  =  ( x  x.  y
)
27 rspceov 5893 . . . . . . . . 9  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G  /\  ( x  x.  y )  =  ( x  x.  y ) )  ->  E. u  e.  ran  F E. v  e.  ran  G ( x  x.  y )  =  ( u  x.  v
) )
2826, 27mp3an3 1266 . . . . . . . 8  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  E. u  e.  ran  F E. v  e.  ran  G ( x  x.  y )  =  ( u  x.  v
) )
29 ovex 5883 . . . . . . . . 9  |-  ( x  x.  y )  e. 
_V
30 eqeq1 2289 . . . . . . . . . 10  |-  ( w  =  ( x  x.  y )  ->  (
w  =  ( u  x.  v )  <->  ( x  x.  y )  =  ( u  x.  v ) ) )
31302rexbidv 2586 . . . . . . . . 9  |-  ( w  =  ( x  x.  y )  ->  ( E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v )  <->  E. u  e.  ran  F E. v  e.  ran  G ( x  x.  y )  =  ( u  x.  v
) ) )
3229, 31elab 2914 . . . . . . . 8  |-  ( ( x  x.  y )  e.  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) }  <->  E. u  e.  ran  F E. v  e.  ran  G ( x  x.  y
)  =  ( u  x.  v ) )
3328, 32sylibr 203 . . . . . . 7  |-  ( ( x  e.  ran  F  /\  y  e.  ran  G )  ->  ( x  x.  y )  e.  {
w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v
) } )
3433adantl 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  F  /\  y  e.  ran  G ) )  ->  ( x  x.  y )  e.  {
w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v
) } )
35 ffn 5389 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5396 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 188 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5389 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 15 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5396 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 188 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6093 . . . . 5  |-  ( ph  ->  ( F  o F  x.  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) } )
44 frn 5395 . . . . 5  |-  ( ( F  o F  x.  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) }  ->  ran  ( F  o F  x.  G )  C_  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v
) } )
4543, 44syl 15 . . . 4  |-  ( ph  ->  ran  ( F  o F  x.  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) } )
4619rnmpt2 5954 . . . 4  |-  ran  (
u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) )  =  {
w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v
) }
4745, 46syl6sseqr 3225 . . 3  |-  ( ph  ->  ran  ( F  o F  x.  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) )
48 ssfi 7083 . . 3  |-  ( ( ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) )  e.  Fin  /\  ran  ( F  o F  x.  G )  C_  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) ) )  ->  ran  ( F  o F  x.  G )  e. 
Fin )
4925, 47, 48syl2anc 642 . 2  |-  ( ph  ->  ran  ( F  o F  x.  G )  e.  Fin )
50 frn 5395 . . . . . . . 8  |-  ( ( F  o F  x.  G ) : RR --> RR  ->  ran  ( F  o F  x.  G
)  C_  RR )
5112, 50syl 15 . . . . . . 7  |-  ( ph  ->  ran  ( F  o F  x.  G )  C_  RR )
52 ax-resscn 8794 . . . . . . 7  |-  RR  C_  CC
5351, 52syl6ss 3191 . . . . . 6  |-  ( ph  ->  ran  ( F  o F  x.  G )  C_  CC )
54 ssdif 3311 . . . . . 6  |-  ( ran  ( F  o F  x.  G )  C_  CC  ->  ( ran  ( F  o F  x.  G
)  \  { 0 } )  C_  ( CC  \  { 0 } ) )
5553, 54syl 15 . . . . 5  |-  ( ph  ->  ( ran  ( F  o F  x.  G
)  \  { 0 } )  C_  ( CC  \  { 0 } ) )
5655sselda 3180 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  y  e.  ( CC  \  { 0 } ) )
573, 6i1fmullem 19049 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  { 0 } ) )  -> 
( `' ( F  o F  x.  G
) " { y } )  =  U_ z  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )
5856, 57syldan 456 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( `' ( F  o F  x.  G ) " {
y } )  = 
U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )
59 difss 3303 . . . . . 6  |-  ( ran 
G  \  { 0 } )  C_  ran  G
60 ssfi 7083 . . . . . 6  |-  ( ( ran  G  e.  Fin  /\  ( ran  G  \  { 0 } ) 
C_  ran  G )  ->  ( ran  G  \  { 0 } )  e.  Fin )
6116, 59, 60sylancl 643 . . . . 5  |-  ( ph  ->  ( ran  G  \  { 0 } )  e.  Fin )
62 i1fima 19033 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  z ) } )  e.  dom  vol )
633, 62syl 15 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( y  / 
z ) } )  e.  dom  vol )
64 i1fima 19033 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
656, 64syl 15 . . . . . . 7  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
66 inmbl 18899 . . . . . . 7  |-  ( ( ( `' F " { ( y  / 
z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
6763, 65, 66syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
6867ralrimivw 2627 . . . . 5  |-  ( ph  ->  A. z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
69 finiunmbl 18901 . . . . 5  |-  ( ( ( ran  G  \  { 0 } )  e.  Fin  /\  A. z  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )  ->  U_ z  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
7061, 68, 69syl2anc 642 . . . 4  |-  ( ph  ->  U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
7170adantr 451 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  U_ z  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
7258, 71eqeltrd 2357 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( `' ( F  o F  x.  G ) " {
y } )  e. 
dom  vol )
73 mblvol 18889 . . . 4  |-  ( ( `' ( F  o F  x.  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  o F  x.  G ) " { y } ) )  =  ( vol
* `  ( `' ( F  o F  x.  G ) " {
y } ) ) )
7472, 73syl 15 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  x.  G
) " { y } ) )  =  ( vol * `  ( `' ( F  o F  x.  G ) " { y } ) ) )
75 mblss 18890 . . . . 5  |-  ( ( `' ( F  o F  x.  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  o F  x.  G ) " { y } ) 
C_  RR )
7672, 75syl 15 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( `' ( F  o F  x.  G ) " {
y } )  C_  RR )
7761adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( ran  G 
\  { 0 } )  e.  Fin )
78 inss2 3390 . . . . . . 7  |-  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
7978a1i 10 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
8065ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
81 mblss 18890 . . . . . . 7  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
8280, 81syl 15 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
83 mblvol 18889 . . . . . . . 8  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol * `  ( `' G " { z } ) ) )
8480, 83syl 15 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol
* `  ( `' G " { z } ) ) )
856adantr 451 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
86 i1fima2sn 19035 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
8785, 86sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
8884, 87eqeltrrd 2358 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( `' G " { z } ) )  e.  RR )
89 ovolsscl 18845 . . . . . 6  |-  ( ( ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )  /\  ( `' G " { z } )  C_  RR  /\  ( vol * `  ( `' G " { z } ) )  e.  RR )  ->  ( vol * `  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
9079, 82, 88, 89syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol * `  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
9177, 90fsumrecl 12207 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
9258fveq2d 5529 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  x.  G ) " {
y } ) )  =  ( vol * `  U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
93 mblss 18890 . . . . . . . . . 10  |-  ( ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9467, 93syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9594ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9695, 90jca 518 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  o F  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol * `  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
9796ralrimiva 2626 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  A. z  e.  ( ran  G  \  { 0 } ) ( ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )
98 ovolfiniun 18860 . . . . . 6  |-  ( ( ( ran  G  \  { 0 } )  e.  Fin  /\  A. z  e.  ( ran  G 
\  { 0 } ) ( ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  C_  RR  /\  ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR ) )  ->  ( vol * `  U_ z  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  <_  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
9977, 97, 98syl2anc 642 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol * `
 U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  <_  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
10092, 99eqbrtrd 4043 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  x.  G ) " {
y } ) )  <_  sum_ z  e.  ( ran  G  \  {
0 } ) ( vol * `  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
101 ovollecl 18842 . . . 4  |-  ( ( ( `' ( F  o F  x.  G
) " { y } )  C_  RR  /\ 
sum_ z  e.  ( ran  G  \  {
0 } ) ( vol * `  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR  /\  ( vol * `  ( `' ( F  o F  x.  G ) " { y } ) )  <_  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol * `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )  -> 
( vol * `  ( `' ( F  o F  x.  G ) " { y } ) )  e.  RR )
10276, 91, 100, 101syl3anc 1182 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol * `
 ( `' ( F  o F  x.  G ) " {
y } ) )  e.  RR )
10374, 102eqeltrd 2357 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  o F  x.  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  o F  x.  G
) " { y } ) )  e.  RR )
10412, 49, 72, 103i1fd 19036 1  |-  ( ph  ->  ( F  o F  x.  G )  e. 
dom  S.1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   U_ciun 3905   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    <_ cle 8868    / cdiv 9423   sum_csu 12158   vol *covol 18822   volcvol 18823   S.1citg1 18970
This theorem is referenced by:  mbfmullem2  19079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976
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