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Theorem i1fd 21285
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 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  F : RR --> RR )
3 ffun 5662 . . . . . . . 8  |-  ( F : RR --> RR  ->  Fun 
F )
4 funcnvcnv 5577 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  `' `' F )
5 imadif 5594 . . . . . . . 8  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
62, 3, 4, 54syl 21 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
7 ioof 11497 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
8 frn 5666 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  ran  (,)  C_ 
~P RR
109sseli 3453 . . . . . . . . . . 11  |-  ( x  e.  ran  (,)  ->  x  e.  ~P RR )
1110elpwid 3971 . . . . . . . . . 10  |-  ( x  e.  ran  (,)  ->  x 
C_  RR )
1211ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  x  C_  RR )
13 dfss4 3685 . . . . . . . . 9  |-  ( x 
C_  RR  <->  ( RR  \ 
( RR  \  x
) )  =  x )
1412, 13sylib 196 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( RR  \  ( RR  \  x ) )  =  x )
1514imaeq2d 5270 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( `' F " x ) )
166, 15eqtr3d 2494 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  =  ( `' F " x ) )
17 fimacnv 5937 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ( `' F " RR )  =  RR )
182, 17syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  =  RR )
19 rembl 21148 . . . . . . . 8  |-  RR  e.  dom  vol
2018, 19syl6eqel 2547 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  e.  dom  vol )
211adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  F : RR --> RR )
22 inpreima 5932 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( `' F " ( y  i^i 
ran  F ) )  =  ( ( `' F " y )  i^i  ( `' F " ran  F ) ) )
23 iunid 4326 . . . . . . . . . . . . . . . 16  |-  U_ x  e.  ( y  i^i  ran  F ) { x }  =  ( y  i^i 
ran  F )
2423imaeq2i 5268 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  ( `' F " ( y  i^i  ran  F )
)
25 imaiun 6064 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
2624, 25eqtr3i 2482 . . . . . . . . . . . . . 14  |-  ( `' F " ( y  i^i  ran  F )
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
27 cnvimass 5290 . . . . . . . . . . . . . . . 16  |-  ( `' F " y ) 
C_  dom  F
28 cnvimarndm 5291 . . . . . . . . . . . . . . . 16  |-  ( `' F " ran  F
)  =  dom  F
2927, 28sseqtr4i 3490 . . . . . . . . . . . . . . 15  |-  ( `' F " y ) 
C_  ( `' F " ran  F )
30 df-ss 3443 . . . . . . . . . . . . . . 15  |-  ( ( `' F " y ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y ) )
3129, 30mpbi 208 . . . . . . . . . . . . . 14  |-  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y )
3222, 26, 313eqtr3g 2515 . . . . . . . . . . . . 13  |-  ( Fun 
F  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
3321, 3, 323syl 20 . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  0  e.  y )  ->  ran  F  e.  Fin )
36 inss2 3672 . . . . . . . . . . . . . 14  |-  ( y  i^i  ran  F )  C_ 
ran  F
37 ssfi 7637 . . . . . . . . . . . . . 14  |-  ( ( ran  F  e.  Fin  /\  ( y  i^i  ran  F )  C_  ran  F )  ->  ( y  i^i 
ran  F )  e. 
Fin )
3835, 36, 37sylancl 662 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F )  e.  Fin )
39 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ph )
40 inss1 3671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  i^i  ran  F )  C_  y
4140sseli 3453 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( y  i^i 
ran  F )  -> 
0  e.  y )
4241con3i 135 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  0  e.  y  ->  -.  0  e.  (
y  i^i  ran  F ) )
4342adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  0  e.  y )  ->  -.  0  e.  ( y  i^i  ran  F ) )
44 disjsn 4037 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( y  i^i  ran  F ) )
4543, 44sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
( y  i^i  ran  F )  i^i  { 0 } )  =  (/) )
46 reldisj 3823 . . . . . . . . . . . . . . . . . 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 5 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4845, 47sylib 196 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
4948sselda 3457 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  e.  dom  vol )
5251ralrimiva 2825 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
53 finiunmbl 21151 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
5533, 54eqeltrrd 2540 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y )  e.  dom  vol )
5655ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol ) )
5756alrimiv 1686 . . . . . . . . 9  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
5857ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
59 elndif 3581 . . . . . . . . 9  |-  ( 0  e.  x  ->  -.  0  e.  ( RR  \  x ) )
6059adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  -.  0  e.  ( RR  \  x ) )
61 reex 9477 . . . . . . . . . 10  |-  RR  e.  _V
62 difexg 4541 . . . . . . . . . 10  |-  ( RR  e.  _V  ->  ( RR  \  x )  e. 
_V )
6361, 62ax-mp 5 . . . . . . . . 9  |-  ( RR 
\  x )  e. 
_V
64 eleq2 2524 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  (
0  e.  y  <->  0  e.  ( RR  \  x
) ) )
6564notbid 294 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR 
\  x ) ) )
66 imaeq2 5266 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  ( `' F " y )  =  ( `' F " ( RR  \  x
) ) )
6766eleq1d 2520 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
)
6865, 67imbi12d 320 . . . . . . . . 9  |-  ( y  =  ( RR  \  x )  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  ( RR  \  x )  ->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
) )
6963, 68spcv 3162 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  ( RR  \  x
)  ->  ( `' F " ( RR  \  x ) )  e. 
dom  vol ) )
7058, 60, 69sylc 60 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  x
) )  e.  dom  vol )
71 difmbl 21150 . . . . . . 7  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( RR  \  x
) )  e.  dom  vol )  ->  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) )  e. 
dom  vol )
7220, 70, 71syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  e.  dom  vol )
7316, 72eqeltrrd 2540 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
x )  e.  dom  vol )
74 eleq2 2524 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
0  e.  y  <->  0  e.  x ) )
7574notbid 294 . . . . . . . . . 10  |-  ( y  =  x  ->  ( -.  0  e.  y  <->  -.  0  e.  x ) )
76 imaeq2 5266 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( `' F " y )  =  ( `' F " x ) )
7776eleq1d 2520 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " x )  e.  dom  vol )
)
7875, 77imbi12d 320 . . . . . . . . 9  |-  ( y  =  x  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  x  ->  ( `' F " x )  e.  dom  vol )
) )
7978spv 1964 . . . . . . . 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 429 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  x )  ->  ( `' F " x )  e.  dom  vol )
8281adantlr 714 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  -.  0  e.  x
)  ->  ( `' F " x )  e. 
dom  vol )
8373, 82pm2.61dan 789 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
8483ralrimiva 2825 . . 3  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
85 ismbf 21234 . . . 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 232 . 2  |-  ( ph  ->  F  e. MblFn )
88 mblvol 21138 . . . . . . . 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 21139 . . . . . . . . 9  |-  ( ( `' F " y )  e.  dom  vol  ->  ( `' F " y ) 
C_  RR )
9155, 90syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y ) 
C_  RR )
92 mblvol 21138 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9693, 95eqeltrrd 2540 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol* `  ( `' F " { x } ) )  e.  RR )
9738, 96fsumrecl 13322 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) )  e.  RR )
9833fveq2d 5796 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  =  ( vol* `  ( `' F " y ) ) )
99 mblss 21139 . . . . . . . . . . . . 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 532 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( ( `' F " { x } )  C_  RR  /\  ( vol* `  ( `' F " { x } ) )  e.  RR ) )
102101ralrimiva 2825 . . . . . . . . . 10  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( ( `' F " { x } )  C_  RR  /\  ( vol* `  ( `' F " { x } ) )  e.  RR ) )
103 ovolfiniun 21109 . . . . . . . . . 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 661 . . . . . . . . 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 4415 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol* `  ( `' F " { x } ) ) )
106 ovollecl 21091 . . . . . . . 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 1219 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol* `  ( `' F " y ) )  e.  RR )
10889, 107eqeltrd 2539 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  e.  RR )
109108ex 434 . . . . 5  |-  ( ph  ->  ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR ) )
110109alrimiv 1686 . . . 4  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR ) )
111 neldifsn 4103 . . . 4  |-  -.  0  e.  ( RR  \  {
0 } )
112 difexg 4541 . . . . . 6  |-  ( RR  e.  _V  ->  ( RR  \  { 0 } )  e.  _V )
11361, 112ax-mp 5 . . . . 5  |-  ( RR 
\  { 0 } )  e.  _V
114 eleq2 2524 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( 0  e.  y  <->  0  e.  ( RR  \  { 0 } ) ) )
115114notbid 294 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR  \  { 0 } ) ) )
116 imaeq2 5266 . . . . . . . 8  |-  ( y  =  ( RR  \  { 0 } )  ->  ( `' F " y )  =  ( `' F " ( RR 
\  { 0 } ) ) )
117116fveq2d 5796 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( vol `  ( `' F " y ) )  =  ( vol `  ( `' F "
( RR  \  {
0 } ) ) ) )
118117eleq1d 2520 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( vol `  ( `' F "
y ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
119115, 118imbi12d 320 . . . . 5  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR )  <-> 
( -.  0  e.  ( RR  \  {
0 } )  -> 
( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) ) )
120113, 119spcv 3162 . . . 4  |-  ( A. y ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR )  ->  ( -.  0  e.  ( RR  \  { 0 } )  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
121110, 111, 120mpisyl 18 . . 3  |-  ( ph  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
1221, 34, 1213jca 1168 . 2  |-  ( ph  ->  ( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
123 isi1f 21278 . 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 664 1  |-  ( ph  ->  F  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071    \ cdif 3426    i^i cin 3428    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   {csn 3978   U_ciun 4272   class class class wbr 4393    X. cxp 4939   `'ccnv 4940   dom cdm 4941   ran crn 4942   "cima 4944   Fun wfun 5513   -->wf 5515   ` cfv 5519   Fincfn 7413   RRcr 9385   0cc0 9386   RR*cxr 9521    <_ cle 9523   (,)cioo 11404   sum_csu 13274   vol*covol 21071   volcvol 21072  MblFncmbf 21220   S.1citg1 21221
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-xadd 11194  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-xmet 17928  df-met 17929  df-ovol 21073  df-vol 21074  df-mbf 21225  df-itg1 21226
This theorem is referenced by:  i1f0  21291  i1f1  21294  i1fadd  21299  i1fmul  21300  i1fmulc  21307  i1fres  21309  mbfi1fseqlem4  21322  itg2addnclem2  28585  ftc1anclem3  28610
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