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Theorem i1fd 19526
Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fd.1  |-  ( ph  ->  F : RR --> RR )
i1fd.2  |-  ( ph  ->  ran  F  e.  Fin )
i1fd.3  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
i1fd.4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
Assertion
Ref Expression
i1fd  |-  ( ph  ->  F  e.  dom  S.1 )
Distinct variable groups:    x, F    ph, x

Proof of Theorem i1fd
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 i1fd.1 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> RR )
21ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  F : RR --> RR )
3 ffun 5552 . . . . . . . . 9  |-  ( F : RR --> RR  ->  Fun 
F )
4 funcnvcnv 5468 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
52, 3, 43syl 19 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  Fun  `' `' F
)
6 imadif 5487 . . . . . . . 8  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
75, 6syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
8 ioof 10958 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
9 frn 5556 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
108, 9ax-mp 8 . . . . . . . . . . . 12  |-  ran  (,)  C_ 
~P RR
1110sseli 3304 . . . . . . . . . . 11  |-  ( x  e.  ran  (,)  ->  x  e.  ~P RR )
1211elpwid 3768 . . . . . . . . . 10  |-  ( x  e.  ran  (,)  ->  x 
C_  RR )
1312ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  x  C_  RR )
14 dfss4 3535 . . . . . . . . 9  |-  ( x 
C_  RR  <->  ( RR  \ 
( RR  \  x
) )  =  x )
1513, 14sylib 189 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( RR  \  ( RR  \  x ) )  =  x )
1615imaeq2d 5162 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( `' F " x ) )
177, 16eqtr3d 2438 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  =  ( `' F " x ) )
18 fimacnv 5821 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ( `' F " RR )  =  RR )
192, 18syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  =  RR )
20 rembl 19388 . . . . . . . 8  |-  RR  e.  dom  vol
2119, 20syl6eqel 2492 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  e.  dom  vol )
221adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  F : RR --> RR )
23 inpreima 5816 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( `' F " ( y  i^i 
ran  F ) )  =  ( ( `' F " y )  i^i  ( `' F " ran  F ) ) )
24 iunid 4106 . . . . . . . . . . . . . . . 16  |-  U_ x  e.  ( y  i^i  ran  F ) { x }  =  ( y  i^i 
ran  F )
2524imaeq2i 5160 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  ( `' F " ( y  i^i  ran  F )
)
26 imaiun 5951 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
2725, 26eqtr3i 2426 . . . . . . . . . . . . . 14  |-  ( `' F " ( y  i^i  ran  F )
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
28 cnvimass 5183 . . . . . . . . . . . . . . . 16  |-  ( `' F " y ) 
C_  dom  F
29 cnvimarndm 5184 . . . . . . . . . . . . . . . 16  |-  ( `' F " ran  F
)  =  dom  F
3028, 29sseqtr4i 3341 . . . . . . . . . . . . . . 15  |-  ( `' F " y ) 
C_  ( `' F " ran  F )
31 df-ss 3294 . . . . . . . . . . . . . . 15  |-  ( ( `' F " y ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y ) )
3230, 31mpbi 200 . . . . . . . . . . . . . 14  |-  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y )
3323, 27, 323eqtr3g 2459 . . . . . . . . . . . . 13  |-  ( Fun 
F  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
3422, 3, 333syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
35 i1fd.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  e.  Fin )
3635adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  0  e.  y )  ->  ran  F  e.  Fin )
37 inss2 3522 . . . . . . . . . . . . . 14  |-  ( y  i^i  ran  F )  C_ 
ran  F
38 ssfi 7288 . . . . . . . . . . . . . 14  |-  ( ( ran  F  e.  Fin  /\  ( y  i^i  ran  F )  C_  ran  F )  ->  ( y  i^i 
ran  F )  e. 
Fin )
3936, 37, 38sylancl 644 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F )  e.  Fin )
40 simpll 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ph )
41 inss1 3521 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  i^i  ran  F )  C_  y
4241sseli 3304 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( y  i^i 
ran  F )  -> 
0  e.  y )
4342con3i 129 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  0  e.  y  ->  -.  0  e.  (
y  i^i  ran  F ) )
4443adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  0  e.  y )  ->  -.  0  e.  ( y  i^i  ran  F ) )
45 disjsn 3828 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( y  i^i  ran  F ) )
4644, 45sylibr 204 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
( y  i^i  ran  F )  i^i  { 0 } )  =  (/) )
47 reldisj 3631 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  i^i  ran  F
)  C_  ran  F  -> 
( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  ( y  i^i 
ran  F )  C_  ( ran  F  \  {
0 } ) ) )
4837, 47ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4946, 48sylib 189 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
5049sselda 3308 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  x  e.  ( ran  F  \  {
0 } ) )
51 i1fd.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
5240, 50, 51syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  e.  dom  vol )
5352ralrimiva 2749 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
54 finiunmbl 19391 . . . . . . . . . . . . 13  |-  ( ( ( y  i^i  ran  F )  e.  Fin  /\  A. x  e.  ( y  i^i  ran  F )
( `' F " { x } )  e.  dom  vol )  ->  U_ x  e.  ( y  i^i  ran  F
) ( `' F " { x } )  e.  dom  vol )
5539, 53, 54syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
5634, 55eqeltrrd 2479 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y )  e.  dom  vol )
5756ex 424 . . . . . . . . . 10  |-  ( ph  ->  ( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol ) )
5857alrimiv 1638 . . . . . . . . 9  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
5958ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
60 elndif 3431 . . . . . . . . 9  |-  ( 0  e.  x  ->  -.  0  e.  ( RR  \  x ) )
6160adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  -.  0  e.  ( RR  \  x ) )
62 reex 9037 . . . . . . . . . 10  |-  RR  e.  _V
63 difexg 4311 . . . . . . . . . 10  |-  ( RR  e.  _V  ->  ( RR  \  x )  e. 
_V )
6462, 63ax-mp 8 . . . . . . . . 9  |-  ( RR 
\  x )  e. 
_V
65 eleq2 2465 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  (
0  e.  y  <->  0  e.  ( RR  \  x
) ) )
6665notbid 286 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR 
\  x ) ) )
67 imaeq2 5158 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  ( `' F " y )  =  ( `' F " ( RR  \  x
) ) )
6867eleq1d 2470 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
)
6966, 68imbi12d 312 . . . . . . . . 9  |-  ( y  =  ( RR  \  x )  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  ( RR  \  x )  ->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
) )
7064, 69spcv 3002 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  ( RR  \  x
)  ->  ( `' F " ( RR  \  x ) )  e. 
dom  vol ) )
7159, 61, 70sylc 58 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  x
) )  e.  dom  vol )
72 difmbl 19390 . . . . . . 7  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( RR  \  x
) )  e.  dom  vol )  ->  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) )  e. 
dom  vol )
7321, 71, 72syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  e.  dom  vol )
7417, 73eqeltrrd 2479 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
x )  e.  dom  vol )
75 eleq2 2465 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
0  e.  y  <->  0  e.  x ) )
7675notbid 286 . . . . . . . . . 10  |-  ( y  =  x  ->  ( -.  0  e.  y  <->  -.  0  e.  x ) )
77 imaeq2 5158 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( `' F " y )  =  ( `' F " x ) )
7877eleq1d 2470 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " x )  e.  dom  vol )
)
7976, 78imbi12d 312 . . . . . . . . 9  |-  ( y  =  x  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  x  ->  ( `' F " x )  e.  dom  vol )
) )
8079spv 1963 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8158, 80syl 16 . . . . . . 7  |-  ( ph  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8281imp 419 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  x )  ->  ( `' F " x )  e.  dom  vol )
8382adantlr 696 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  -.  0  e.  x
)  ->  ( `' F " x )  e. 
dom  vol )
8474, 83pm2.61dan 767 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
8584ralrimiva 2749 . . 3  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
86 ismbf 19475 . . . 4  |-  ( F : RR --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
871, 86syl 16 . . 3  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
8885, 87mpbird 224 . 2  |-  ( ph  ->  F  e. MblFn )
89 mblvol 19379 . . . . . . . 8  |-  ( ( `' F " y )  e.  dom  vol  ->  ( vol `  ( `' F " y ) )  =  ( vol
* `  ( `' F " y ) ) )
9056, 89syl 16 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  =  ( vol * `  ( `' F " y ) ) )
91 mblss 19380 . . . . . . . . 9  |-  ( ( `' F " y )  e.  dom  vol  ->  ( `' F " y ) 
C_  RR )
9256, 91syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y ) 
C_  RR )
93 mblvol 19379 . . . . . . . . . . 11  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( vol `  ( `' F " { x } ) )  =  ( vol * `  ( `' F " { x } ) ) )
9452, 93syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol
* `  ( `' F " { x }
) ) )
95 i1fd.4 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9640, 50, 95syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9794, 96eqeltrrd 2479 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol * `
 ( `' F " { x } ) )  e.  RR )
9839, 97fsumrecl 12483 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) )  e.  RR )
9934fveq2d 5691 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  =  ( vol
* `  ( `' F " y ) ) )
100 mblss 19380 . . . . . . . . . . . . 13  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( `' F " { x } ) 
C_  RR )
10152, 100syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  C_  RR )
102101, 97jca 519 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
103102ralrimiva 2749 . . . . . . . . . 10  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
104 ovolfiniun 19350 . . . . . . . . . 10  |-  ( ( ( y  i^i  ran  F )  e.  Fin  /\  A. x  e.  ( y  i^i  ran  F )
( ( `' F " { x } ) 
C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )  -> 
( vol * `  U_ x  e.  ( y  i^i  ran  F )
( `' F " { x } ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) ) )
10539, 103, 104syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) ) )
10699, 105eqbrtrrd 4194 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) ) )
107 ovollecl 19332 . . . . . . . 8  |-  ( ( ( `' F "
y )  C_  RR  /\ 
sum_ x  e.  (
y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) )  e.  RR  /\  ( vol
* `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F
) ( vol * `  ( `' F " { x } ) ) )  ->  ( vol * `  ( `' F " y ) )  e.  RR )
10892, 98, 106, 107syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  e.  RR )
10990, 108eqeltrd 2478 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  e.  RR )
110109ex 424 . . . . 5  |-  ( ph  ->  ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR ) )
111110alrimiv 1638 . . . 4  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR ) )
112 neldifsn 3889 . . . 4  |-  -.  0  e.  ( RR  \  {
0 } )
113 difexg 4311 . . . . . 6  |-  ( RR  e.  _V  ->  ( RR  \  { 0 } )  e.  _V )
11462, 113ax-mp 8 . . . . 5  |-  ( RR 
\  { 0 } )  e.  _V
115 eleq2 2465 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( 0  e.  y  <->  0  e.  ( RR  \  { 0 } ) ) )
116115notbid 286 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR  \  { 0 } ) ) )
117 imaeq2 5158 . . . . . . . 8  |-  ( y  =  ( RR  \  { 0 } )  ->  ( `' F " y )  =  ( `' F " ( RR 
\  { 0 } ) ) )
118117fveq2d 5691 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( vol `  ( `' F " y ) )  =  ( vol `  ( `' F "
( RR  \  {
0 } ) ) ) )
119118eleq1d 2470 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( vol `  ( `' F "
y ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
120116, 119imbi12d 312 . . . . 5  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR )  <-> 
( -.  0  e.  ( RR  \  {
0 } )  -> 
( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) ) )
121114, 120spcv 3002 . . . 4  |-  ( A. y ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR )  ->  ( -.  0  e.  ( RR  \  { 0 } )  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
122111, 112, 121ee10 1382 . . 3  |-  ( ph  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
1231, 35, 1223jca 1134 . 2  |-  ( ph  ->  ( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
124 isi1f 19519 . 2  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
12588, 123, 124sylanbrc 646 1  |-  ( ph  ->  F  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U_ciun 4053   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413   Fincfn 7068   RRcr 8945   0cc0 8946   RR*cxr 9075    <_ cle 9077   (,)cioo 10872   sum_csu 12434   vol
*covol 19312   volcvol 19313  MblFncmbf 19459   S.1citg1 19460
This theorem is referenced by:  i1f0  19532  i1f1  19535  i1fadd  19540  i1fmul  19541  i1fmulc  19548  i1fres  19550  mbfi1fseqlem4  19563  itg2addnclem2  26156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xadd 10667  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-xmet 16650  df-met 16651  df-ovol 19314  df-vol 19315  df-mbf 19465  df-itg1 19466
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