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Theorem i1fmul 21317
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 9482 . . . 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 21297 . . . 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 21297 . . . 4  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . 3  |-  ( ph  ->  G : RR --> RR )
9 reex 9488 . . . 4  |-  RR  e.  _V
109a1i 11 . . 3  |-  ( ph  ->  RR  e.  _V )
11 inidm 3670 . . 3  |-  ( RR 
i^i  RR )  =  RR
122, 5, 8, 10, 10, 11off 6447 . 2  |-  ( ph  ->  ( F  oF  x.  G ) : RR --> RR )
13 i1frn 21298 . . . . . 6  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
143, 13syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
15 i1frn 21298 . . . . . 6  |-  ( G  e.  dom  S.1  ->  ran 
G  e.  Fin )
166, 15syl 16 . . . . 5  |-  ( ph  ->  ran  G  e.  Fin )
17 xpfi 7697 . . . . 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 2454 . . . . . 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 6228 . . . . . 6  |-  ( u  x.  v )  e. 
_V
2119, 20fnmpt2i 6756 . . . . 5  |-  ( u  e.  ran  F , 
v  e.  ran  G  |->  ( u  x.  v
) )  Fn  ( ran  F  X.  ran  G
)
22 dffn4 5737 . . . . 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 7711 . . . 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 2454 . . . . . . . . 9  |-  ( x  x.  y )  =  ( x  x.  y
)
27 rspceov 6240 . . . . . . . . 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 1304 . . . . . . . 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 6228 . . . . . . . . 9  |-  ( x  x.  y )  e. 
_V
30 eqeq1 2458 . . . . . . . . . 10  |-  ( w  =  ( x  x.  y )  ->  (
w  =  ( u  x.  v )  <->  ( x  x.  y )  =  ( u  x.  v ) ) )
31302rexbidv 2880 . . . . . . . . 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 3213 . . . . . . . 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 5670 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
365, 35syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
37 dffn3 5677 . . . . . . 7  |-  ( F  Fn  RR  <->  F : RR
--> ran  F )
3836, 37sylib 196 . . . . . 6  |-  ( ph  ->  F : RR --> ran  F
)
39 ffn 5670 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
408, 39syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
41 dffn3 5677 . . . . . . 7  |-  ( G  Fn  RR  <->  G : RR
--> ran  G )
4240, 41sylib 196 . . . . . 6  |-  ( ph  ->  G : RR --> ran  G
)
4334, 38, 42, 10, 10, 11off 6447 . . . . 5  |-  ( ph  ->  ( F  oF  x.  G ) : RR --> { w  |  E. u  e.  ran  F E. v  e.  ran  G  w  =  ( u  x.  v ) } )
44 frn 5676 . . . . 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 6313 . . . 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 3514 . . 3  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_ 
ran  ( u  e. 
ran  F ,  v  e.  ran  G  |->  ( u  x.  v ) ) )
48 ssfi 7647 . . 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 5676 . . . . . . . 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 9454 . . . . . . 7  |-  RR  C_  CC
5351, 52syl6ss 3479 . . . . . 6  |-  ( ph  ->  ran  ( F  oF  x.  G )  C_  CC )
5453ssdifd 3603 . . . . 5  |-  ( ph  ->  ( ran  ( F  oF  x.  G
)  \  { 0 } )  C_  ( CC  \  { 0 } ) )
5554sselda 3467 . . . 4  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  y  e.  ( CC  \  { 0 } ) )
563, 6i1fmullem 21315 . . . 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 3594 . . . . . 6  |-  ( ran 
G  \  { 0 } )  C_  ran  G
59 ssfi 7647 . . . . . 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 21299 . . . . . . . 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 21299 . . . . . . . 8  |-  ( G  e.  dom  S.1  ->  ( `' G " { z } )  e.  dom  vol )
646, 63syl 16 . . . . . . 7  |-  ( ph  ->  ( `' G " { z } )  e.  dom  vol )
65 inmbl 21166 . . . . . . 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 2831 . . . . 5  |-  ( ph  ->  A. z  e.  ( ran  G  \  {
0 } ) ( ( `' F " { ( y  / 
z ) } )  i^i  ( `' G " { z } ) )  e.  dom  vol )
68 finiunmbl 21168 . . . . 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 2542 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( `' ( F  oF  x.  G ) " {
y } )  e. 
dom  vol )
72 mblvol 21155 . . . 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 21156 . . . . 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 3682 . . . . . . 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 21156 . . . . . . 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 21155 . . . . . . . 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 21301 . . . . . . . 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 2543 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( ran  ( F  oF  x.  G
)  \  { 0 } ) )  /\  z  e.  ( ran  G 
\  { 0 } ) )  ->  ( vol* `  ( `' G " { z } ) )  e.  RR )
88 ovolsscl 21111 . . . . . 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 1219 . . . . 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 13333 . . . 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 5806 . . . . 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 21156 . . . . . . . . . 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 2830 . . . . . 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 21126 . . . . . 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 4423 . . . 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 21108 . . . 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 1219 . . 3  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol* `  ( `' ( F  oF  x.  G ) " {
y } ) )  e.  RR )
10273, 101eqeltrd 2542 . 2  |-  ( (
ph  /\  y  e.  ( ran  ( F  oF  x.  G )  \  { 0 } ) )  ->  ( vol `  ( `' ( F  oF  x.  G
) " { y } ) )  e.  RR )
10312, 49, 71, 102i1fd 21302 1  |-  ( ph  ->  ( F  oF  x.  G )  e. 
dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   _Vcvv 3078    \ cdif 3436    i^i cin 3438    C_ wss 3439   {csn 3988   U_ciun 4282   class class class wbr 4403    X. cxp 4949   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954    Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205    oFcof 6431   Fincfn 7423   CCcc 9395   RRcr 9396   0cc0 9397    x. cmul 9402    <_ cle 9534    / cdiv 10108   sum_csu 13285   vol*covol 21088   volcvol 21089   S.1citg1 21238
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 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
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 7806  df-oi 7839  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-q 11069  df-rp 11107  df-xadd 11205  df-ioo 11419  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286  df-xmet 17945  df-met 17946  df-ovol 21090  df-vol 21091  df-mbf 21242  df-itg1 21243
This theorem is referenced by:  mbfmullem2  21345  ftc1anclem3  28640
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