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Theorem i1fmul 21971
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  oF  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 9589 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
21adantl 466 . . 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 21951 . . . 4  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
53, 4syl 16 . . 3  |-  ( ph  ->  F : RR --> RR )
6 i1fadd.2 . . . 4  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 21951 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9595 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3712 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6549 . 2  |-  ( ph  ->  ( F  oF  x.  G ) : RR --> RR )
13 i1frn 21952 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 21952 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7803 . . . . 5  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  e.  Fin )  ->  ( ran  F  X.  ran  G )  e. 
Fin )
1814, 16, 17syl2anc 661 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  e.  Fin )
19 eqid 2467 . . . . . 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 6320 . . . . . 6  |-  ( u  x.  v )  e. 
_V
2119, 20fnmpt2i 6864 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  x.  v
) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5807 . . . . 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 208 . . . 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 7818 . . . 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 662 . . 3  |-  ( ph  ->  ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) )  e.  Fin )
26 eqid 2467 . . . . . . . . 9  |-  ( x  x.  y )  =  ( x  x.  y
)
27 rspceov 6332 . . . . . . . . 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 1313 . . . . . . . 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 6320 . . . . . . . . 9  |-  ( x  x.  y )  e. 
_V
30 eqeq1 2471 . . . . . . . . . 10  |-  ( w  =  ( x  x.  y )  ->  (
w  =  ( u  x.  v )  <->  ( x  x.  y )  =  ( u  x.  v ) ) )
31302rexbidv 2985 . . . . . . . . 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 3255 . . . . . . . 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 212 . . . . . . 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 466 . . . . . 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 5737 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5744 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 196 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5737 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5744 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 196 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6549 . . . . 5  |-  ( ph  ->  ( F  oF  x.  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) } )
44 frn 5743 . . . . 5  |-  ( ( F  oF  x.  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) }  ->  ran  ( F  oF  x.  G )  C_  { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v
) } )
4543, 44syl 16 . . . 4  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_ 
{ w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) } )
4619rnmpt2 6407 . . . 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 3556 . . 3  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) )
48 ssfi 7752 . . 3  |-  ( ( ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) )  e.  Fin  /\  ran  ( F  oF  x.  G )  C_  ran  ( u  e.  ran  F ,  v  e.  ran  G 
|->  ( u  x.  v
) ) )  ->  ran  ( F  oF  x.  G )  e. 
Fin )
4925, 47, 48syl2anc 661 . 2  |-  ( ph  ->  ran  ( F  oF  x.  G )  e.  Fin )
50 frn 5743 . . . . . . . 8  |-  ( ( F  oF  x.  G ) : RR --> RR  ->  ran  ( F  oF  x.  G
)  C_  RR )
5112, 50syl 16 . . . . . . 7  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_  RR )
52 ax-resscn 9561 . . . . . . 7  |-  RR  C_  CC
5351, 52syl6ss 3521 . . . . . 6  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_  CC )
5453ssdifd 3645 . . . . 5  |-  ( ph  ->  ( ran  ( F  oF  x.  G
)  \  { 0 } )  C_  ( CC  \  { 0 } ) )
5554sselda 3509 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  y  e.  ( CC  \  { 0 } ) )
563, 6i1fmullem 21969 . . . 4  |-  ( (
ph  /\  y  e.  ( CC  \  { 0 } ) )  -> 
( `' ( F  oF  x.  G
) " { y } )  =  U_ z  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )
5755, 56syldan 470 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( `' ( F  oF  x.  G ) " {
y } )  = 
U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )
58 difss 3636 . . . . . 6  |-  ( ran 
G  \  { 0 } )  C_  ran  G
59 ssfi 7752 . . . . . 6  |-  ( ( ran  G  e.  Fin  /\  ( ran  G  \  { 0 } ) 
C_  ran  G )  ->  ( ran  G  \  { 0 } )  e.  Fin )
6016, 58, 59sylancl 662 . . . . 5  |-  ( ph  ->  ( ran  G  \  { 0 } )  e.  Fin )
61 i1fima 21953 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  ( `' F " { ( y  /  z ) } )  e.  dom  vol )
623, 61syl 16 . . . . . . 7  |-  ( ph  ->  ( `' F " { ( y  / 
z ) } )  e.  dom  vol )
63 i1fima 21953 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
646, 63syl 16 . . . . . . 7  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
65 inmbl 21820 . . . . . . 7  |-  ( ( ( `' F " { ( y  / 
z ) } )  e.  dom  vol  /\  ( `' G " { z } )  e.  dom  vol )  ->  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  e. 
dom  vol )
6662, 64, 65syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
6766ralrimivw 2882 . . . . 5  |-  ( ph  ->  A. z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
68 finiunmbl 21822 . . . . 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 )
6960, 67, 68syl2anc 661 . . . 4  |-  ( ph  ->  U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
7069adantr 465 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  U_ z  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
7157, 70eqeltrd 2555 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( `' ( F  oF  x.  G ) " {
y } )  e. 
dom  vol )
72 mblvol 21809 . . . 4  |-  ( ( `' ( F  oF  x.  G ) " { y } )  e.  dom  vol  ->  ( vol `  ( `' ( F  oF  x.  G ) " { y } ) )  =  ( vol* `  ( `' ( F  oF  x.  G ) " {
y } ) ) )
7371, 72syl 16 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  x.  G
) " { y } ) )  =  ( vol* `  ( `' ( F  oF  x.  G ) " { y } ) ) )
74 mblss 21810 . . . . 5  |-  ( ( `' ( F  oF  x.  G ) " { y } )  e.  dom  vol  ->  ( `' ( F  oF  x.  G ) " { y } ) 
C_  RR )
7571, 74syl 16 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( `' ( F  oF  x.  G ) " {
y } )  C_  RR )
7660adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( ran  G 
\  { 0 } )  e.  Fin )
77 inss2 3724 . . . . . . 7  |-  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } )
7877a1i 11 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  ( `' G " { z } ) )
7964ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  e.  dom  vol )
80 mblss 21810 . . . . . . 7  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( `' G " { z } ) 
C_  RR )
8179, 80syl 16 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( `' G " { z } )  C_  RR )
82 mblvol 21809 . . . . . . . 8  |-  ( ( `' G " { z } )  e.  dom  vol 
->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
8379, 82syl 16 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  =  ( vol* `  ( `' G " { z } ) ) )
846adantr 465 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  G  e.  dom  S.1 )
85 i1fima2sn 21955 . . . . . . . 8  |-  ( ( G  e.  dom  S.1  /\  z  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' G " { z } ) )  e.  RR )
8684, 85sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol `  ( `' G " { z } ) )  e.  RR )
8783, 86eqeltrrd 2556 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
88 ovolsscl 21765 . . . . . 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 )
8978, 81, 87, 88syl3anc 1228 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( ( `' F " { ( y  /  z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
9076, 89fsumrecl 13536 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol* `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR )
9157fveq2d 5876 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  x.  G ) " {
y } ) )  =  ( vol* `  U_ z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
92 mblss 21810 . . . . . . . . . 10  |-  ( ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9366, 92syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9493ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  C_  RR )
9594, 89jca 532 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  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 ) )
9695ralrimiva 2881 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  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 ) )
97 ovolfiniun 21780 . . . . . 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 } ) ) ) )
9876, 96, 97syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  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 } ) ) ) )
9991, 98eqbrtrd 4473 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  x.  G ) " {
y } ) )  <_  sum_ z  e.  ( ran  G  \  {
0 } ) ( vol* `  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )
100 ovollecl 21762 . . . 4  |-  ( ( ( `' ( F  oF  x.  G
) " { y } )  C_  RR  /\ 
sum_ z  e.  ( ran  G  \  {
0 } ) ( vol* `  (
( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) )  e.  RR  /\  ( vol* `  ( `' ( F  oF  x.  G ) " { y } ) )  <_  sum_ z  e.  ( ran  G  \  { 0 } ) ( vol* `  ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) ) ) )  -> 
( vol* `  ( `' ( F  oF  x.  G ) " { y } ) )  e.  RR )
10175, 90, 99, 100syl3anc 1228 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  x.  G ) " {
y } ) )  e.  RR )
10273, 101eqeltrd 2555 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  x.  G
) " { y } ) )  e.  RR )
10312, 49, 71, 102i1fd 21956 1  |-  ( ph  ->  ( F  oF  x.  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818   _Vcvv 3118    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4033   U_ciun 4331   class class class wbr 4453    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    oFcof 6533   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    <_ cle 9641    / cdiv 10218   sum_csu 13488   vol*covol 21742   volcvol 21743   S.1citg1 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xadd 11331  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-xmet 18282  df-met 18283  df-ovol 21744  df-vol 21745  df-mbf 21896  df-itg1 21897
This theorem is referenced by:  mbfmullem2  21999  ftc1anclem3  30019
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