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Theorem i1fd 19575
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 . . . . . . . . 9  |-  ( ph  ->  F : RR --> RR )
21ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  F : RR --> RR )
3 ffun 5595 . . . . . . . 8  |-  ( F : RR --> RR  ->  Fun 
F )
4 funcnvcnv 5511 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  `' `' F )
5 imadif 5530 . . . . . . . 8  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
62, 3, 4, 54syl 20 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
7 ioof 11004 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 frn 5599 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
97, 8ax-mp 8 . . . . . . . . . . . 12  |-  ran  (,)  C_ 
~P RR
109sseli 3346 . . . . . . . . . . 11  |-  ( x  e.  ran  (,)  ->  x  e.  ~P RR )
1110elpwid 3810 . . . . . . . . . 10  |-  ( x  e.  ran  (,)  ->  x 
C_  RR )
1211ad2antlr 709 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  x  C_  RR )
13 dfss4 3577 . . . . . . . . 9  |-  ( x 
C_  RR  <->  ( RR  \ 
( RR  \  x
) )  =  x )
1412, 13sylib 190 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( RR  \  ( RR  \  x ) )  =  x )
1514imaeq2d 5205 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( `' F " x ) )
166, 15eqtr3d 2472 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  =  ( `' F " x ) )
17 fimacnv 5864 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ( `' F " RR )  =  RR )
182, 17syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  =  RR )
19 rembl 19437 . . . . . . . 8  |-  RR  e.  dom  vol
2018, 19syl6eqel 2526 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  e.  dom  vol )
211adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  F : RR --> RR )
22 inpreima 5859 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( `' F " ( y  i^i 
ran  F ) )  =  ( ( `' F " y )  i^i  ( `' F " ran  F ) ) )
23 iunid 4148 . . . . . . . . . . . . . . . 16  |-  U_ x  e.  ( y  i^i  ran  F ) { x }  =  ( y  i^i 
ran  F )
2423imaeq2i 5203 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  ( `' F " ( y  i^i  ran  F )
)
25 imaiun 5994 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
2624, 25eqtr3i 2460 . . . . . . . . . . . . . 14  |-  ( `' F " ( y  i^i  ran  F )
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
27 cnvimass 5226 . . . . . . . . . . . . . . . 16  |-  ( `' F " y ) 
C_  dom  F
28 cnvimarndm 5227 . . . . . . . . . . . . . . . 16  |-  ( `' F " ran  F
)  =  dom  F
2927, 28sseqtr4i 3383 . . . . . . . . . . . . . . 15  |-  ( `' F " y ) 
C_  ( `' F " ran  F )
30 df-ss 3336 . . . . . . . . . . . . . . 15  |-  ( ( `' F " y ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y ) )
3129, 30mpbi 201 . . . . . . . . . . . . . 14  |-  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y )
3222, 26, 313eqtr3g 2493 . . . . . . . . . . . . 13  |-  ( Fun 
F  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
3321, 3, 323syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
34 i1fd.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  e.  Fin )
3534adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  0  e.  y )  ->  ran  F  e.  Fin )
36 inss2 3564 . . . . . . . . . . . . . 14  |-  ( y  i^i  ran  F )  C_ 
ran  F
37 ssfi 7331 . . . . . . . . . . . . . 14  |-  ( ( ran  F  e.  Fin  /\  ( y  i^i  ran  F )  C_  ran  F )  ->  ( y  i^i 
ran  F )  e. 
Fin )
3835, 36, 37sylancl 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F )  e.  Fin )
39 simpll 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ph )
40 inss1 3563 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  i^i  ran  F )  C_  y
4140sseli 3346 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( y  i^i 
ran  F )  -> 
0  e.  y )
4241con3i 130 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  0  e.  y  ->  -.  0  e.  (
y  i^i  ran  F ) )
4342adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  0  e.  y )  ->  -.  0  e.  ( y  i^i  ran  F ) )
44 disjsn 3870 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( y  i^i  ran  F ) )
4543, 44sylibr 205 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
( y  i^i  ran  F )  i^i  { 0 } )  =  (/) )
46 reldisj 3673 . . . . . . . . . . . . . . . . . 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 } ) ) )
4736, 46ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4845, 47sylib 190 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4948sselda 3350 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  x  e.  ( ran  F  \  {
0 } ) )
50 i1fd.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
5139, 49, 50syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  e.  dom  vol )
5251ralrimiva 2791 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
53 finiunmbl 19440 . . . . . . . . . . . . 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 )
5438, 52, 53syl2anc 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
5533, 54eqeltrrd 2513 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y )  e.  dom  vol )
5655ex 425 . . . . . . . . . 10  |-  ( ph  ->  ( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol ) )
5756alrimiv 1642 . . . . . . . . 9  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
5857ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
59 elndif 3473 . . . . . . . . 9  |-  ( 0  e.  x  ->  -.  0  e.  ( RR  \  x ) )
6059adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  -.  0  e.  ( RR  \  x ) )
61 reex 9083 . . . . . . . . . 10  |-  RR  e.  _V
62 difexg 4353 . . . . . . . . . 10  |-  ( RR  e.  _V  ->  ( RR  \  x )  e. 
_V )
6361, 62ax-mp 8 . . . . . . . . 9  |-  ( RR 
\  x )  e. 
_V
64 eleq2 2499 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  (
0  e.  y  <->  0  e.  ( RR  \  x
) ) )
6564notbid 287 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR 
\  x ) ) )
66 imaeq2 5201 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  ( `' F " y )  =  ( `' F " ( RR  \  x
) ) )
6766eleq1d 2504 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
)
6865, 67imbi12d 313 . . . . . . . . 9  |-  ( y  =  ( RR  \  x )  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  ( RR  \  x )  ->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
) )
6963, 68spcv 3044 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  ( RR  \  x
)  ->  ( `' F " ( RR  \  x ) )  e. 
dom  vol ) )
7058, 60, 69sylc 59 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  x
) )  e.  dom  vol )
71 difmbl 19439 . . . . . . 7  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( RR  \  x
) )  e.  dom  vol )  ->  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) )  e. 
dom  vol )
7220, 70, 71syl2anc 644 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  e.  dom  vol )
7316, 72eqeltrrd 2513 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
x )  e.  dom  vol )
74 eleq2 2499 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
0  e.  y  <->  0  e.  x ) )
7574notbid 287 . . . . . . . . . 10  |-  ( y  =  x  ->  ( -.  0  e.  y  <->  -.  0  e.  x ) )
76 imaeq2 5201 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( `' F " y )  =  ( `' F " x ) )
7776eleq1d 2504 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " x )  e.  dom  vol )
)
7875, 77imbi12d 313 . . . . . . . . 9  |-  ( y  =  x  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  x  ->  ( `' F " x )  e.  dom  vol )
) )
7978spv 1966 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8057, 79syl 16 . . . . . . 7  |-  ( ph  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8180imp 420 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  x )  ->  ( `' F " x )  e.  dom  vol )
8281adantlr 697 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  -.  0  e.  x
)  ->  ( `' F " x )  e. 
dom  vol )
8373, 82pm2.61dan 768 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
8483ralrimiva 2791 . . 3  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
85 ismbf 19524 . . . 4  |-  ( F : RR --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
861, 85syl 16 . . 3  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
8784, 86mpbird 225 . 2  |-  ( ph  ->  F  e. MblFn )
88 mblvol 19428 . . . . . . . 8  |-  ( ( `' F " y )  e.  dom  vol  ->  ( vol `  ( `' F " y ) )  =  ( vol
* `  ( `' F " y ) ) )
8955, 88syl 16 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  =  ( vol * `  ( `' F " y ) ) )
90 mblss 19429 . . . . . . . . 9  |-  ( ( `' F " y )  e.  dom  vol  ->  ( `' F " y ) 
C_  RR )
9155, 90syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y ) 
C_  RR )
92 mblvol 19428 . . . . . . . . . . 11  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( vol `  ( `' F " { x } ) )  =  ( vol * `  ( `' F " { x } ) ) )
9351, 92syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol
* `  ( `' F " { x }
) ) )
94 i1fd.4 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9539, 49, 94syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9693, 95eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol * `
 ( `' F " { x } ) )  e.  RR )
9738, 96fsumrecl 12530 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) )  e.  RR )
9833fveq2d 5734 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  =  ( vol
* `  ( `' F " y ) ) )
99 mblss 19429 . . . . . . . . . . . . 13  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( `' F " { x } ) 
C_  RR )
10051, 99syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  C_  RR )
101100, 96jca 520 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
102101ralrimiva 2791 . . . . . . . . . 10  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
103 ovolfiniun 19399 . . . . . . . . . 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 } ) ) )
10438, 102, 103syl2anc 644 . . . . . . . . 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 } ) ) )
10598, 104eqbrtrrd 4236 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) ) )
106 ovollecl 19381 . . . . . . . 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 )
10791, 97, 105, 106syl3anc 1185 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  e.  RR )
10889, 107eqeltrd 2512 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  e.  RR )
109108ex 425 . . . . 5  |-  ( ph  ->  ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR ) )
110109alrimiv 1642 . . . 4  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR ) )
111 neldifsn 3931 . . . 4  |-  -.  0  e.  ( RR  \  {
0 } )
112 difexg 4353 . . . . . 6  |-  ( RR  e.  _V  ->  ( RR  \  { 0 } )  e.  _V )
11361, 112ax-mp 8 . . . . 5  |-  ( RR 
\  { 0 } )  e.  _V
114 eleq2 2499 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( 0  e.  y  <->  0  e.  ( RR  \  { 0 } ) ) )
115114notbid 287 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR  \  { 0 } ) ) )
116 imaeq2 5201 . . . . . . . 8  |-  ( y  =  ( RR  \  { 0 } )  ->  ( `' F " y )  =  ( `' F " ( RR 
\  { 0 } ) ) )
117116fveq2d 5734 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( vol `  ( `' F " y ) )  =  ( vol `  ( `' F "
( RR  \  {
0 } ) ) ) )
118117eleq1d 2504 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( vol `  ( `' F "
y ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
119115, 118imbi12d 313 . . . . 5  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR )  <-> 
( -.  0  e.  ( RR  \  {
0 } )  -> 
( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) ) )
120113, 119spcv 3044 . . . 4  |-  ( A. y ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR )  ->  ( -.  0  e.  ( RR  \  { 0 } )  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
121110, 111, 120ee10 1386 . . 3  |-  ( ph  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
1221, 34, 1213jca 1135 . 2  |-  ( ph  ->  ( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
123 isi1f 19568 . 2  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
12487, 122, 123sylanbrc 647 1  |-  ( ph  ->  F  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U_ciun 4095   class class class wbr 4214    X. cxp 4878   `'ccnv 4879   dom cdm 4880   ran crn 4881   "cima 4883   Fun wfun 5450   -->wf 5452   ` cfv 5456   Fincfn 7111   RRcr 8991   0cc0 8992   RR*cxr 9121    <_ cle 9123   (,)cioo 10918   sum_csu 12481   vol
*covol 19361   volcvol 19362  MblFncmbf 19508   S.1citg1 19509
This theorem is referenced by:  i1f0  19581  i1f1  19584  i1fadd  19589  i1fmul  19590  i1fmulc  19597  i1fres  19599  mbfi1fseqlem4  19612  itg2addnclem2  26259  ftc1anclem3  26284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364  df-mbf 19514  df-itg1 19515
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