MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fd Unicode version

Theorem i1fd 19052
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 706 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  F : RR --> RR )
3 ffun 5407 . . . . . . . . 9  |-  ( F : RR --> RR  ->  Fun 
F )
4 funcnvcnv 5324 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  `' `' F )
52, 3, 43syl 18 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  Fun  `' `' F
)
6 imadif 5343 . . . . . . . 8  |-  ( Fun  `' `' F  ->  ( `' F " ( RR 
\  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
75, 6syl 15 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) ) )
8 ioof 10757 . . . . . . . . . . . . 13  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
9 frn 5411 . . . . . . . . . . . . 13  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
108, 9ax-mp 8 . . . . . . . . . . . 12  |-  ran  (,)  C_ 
~P RR
1110sseli 3189 . . . . . . . . . . 11  |-  ( x  e.  ran  (,)  ->  x  e.  ~P RR )
12 elpwi 3646 . . . . . . . . . . 11  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
1311, 12syl 15 . . . . . . . . . 10  |-  ( x  e.  ran  (,)  ->  x 
C_  RR )
1413ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  x  C_  RR )
15 dfss4 3416 . . . . . . . . 9  |-  ( x 
C_  RR  <->  ( RR  \ 
( RR  \  x
) )  =  x )
1614, 15sylib 188 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( RR  \  ( RR  \  x ) )  =  x )
1716imaeq2d 5028 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  ( RR  \  x ) ) )  =  ( `' F " x ) )
187, 17eqtr3d 2330 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  =  ( `' F " x ) )
19 fimacnv 5673 . . . . . . . . 9  |-  ( F : RR --> RR  ->  ( `' F " RR )  =  RR )
202, 19syl 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  =  RR )
21 rembl 18914 . . . . . . . 8  |-  RR  e.  dom  vol
2220, 21syl6eqel 2384 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F " RR )  e.  dom  vol )
231adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  F : RR --> RR )
24 inpreima 5668 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( `' F " ( y  i^i 
ran  F ) )  =  ( ( `' F " y )  i^i  ( `' F " ran  F ) ) )
25 iunid 3973 . . . . . . . . . . . . . . . 16  |-  U_ x  e.  ( y  i^i  ran  F ) { x }  =  ( y  i^i 
ran  F )
2625imaeq2i 5026 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  ( `' F " ( y  i^i  ran  F )
)
27 imaiun 5787 . . . . . . . . . . . . . . 15  |-  ( `' F " U_ x  e.  ( y  i^i  ran  F ) { x }
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
2826, 27eqtr3i 2318 . . . . . . . . . . . . . 14  |-  ( `' F " ( y  i^i  ran  F )
)  =  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )
29 cnvimass 5049 . . . . . . . . . . . . . . . 16  |-  ( `' F " y ) 
C_  dom  F
30 cnvimarndm 5050 . . . . . . . . . . . . . . . 16  |-  ( `' F " ran  F
)  =  dom  F
3129, 30sseqtr4i 3224 . . . . . . . . . . . . . . 15  |-  ( `' F " y ) 
C_  ( `' F " ran  F )
32 df-ss 3179 . . . . . . . . . . . . . . 15  |-  ( ( `' F " y ) 
C_  ( `' F " ran  F )  <->  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y ) )
3331, 32mpbi 199 . . . . . . . . . . . . . 14  |-  ( ( `' F " y )  i^i  ( `' F " ran  F ) )  =  ( `' F " y )
3424, 28, 333eqtr3g 2351 . . . . . . . . . . . . 13  |-  ( Fun 
F  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
3523, 3, 343syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  =  ( `' F " y ) )
36 i1fd.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  F  e.  Fin )
3736adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  0  e.  y )  ->  ran  F  e.  Fin )
38 inss2 3403 . . . . . . . . . . . . . 14  |-  ( y  i^i  ran  F )  C_ 
ran  F
39 ssfi 7099 . . . . . . . . . . . . . 14  |-  ( ( ran  F  e.  Fin  /\  ( y  i^i  ran  F )  C_  ran  F )  ->  ( y  i^i 
ran  F )  e. 
Fin )
4037, 38, 39sylancl 643 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F )  e.  Fin )
41 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ph )
42 inss1 3402 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  i^i  ran  F )  C_  y
4342sseli 3189 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( y  i^i 
ran  F )  -> 
0  e.  y )
4443con3i 127 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  0  e.  y  ->  -.  0  e.  (
y  i^i  ran  F ) )
4544adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  -.  0  e.  y )  ->  -.  0  e.  ( y  i^i  ran  F ) )
46 disjsn 3706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( y  i^i  ran  F ) )
4745, 46sylibr 203 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
( y  i^i  ran  F )  i^i  { 0 } )  =  (/) )
48 reldisj 3511 . . . . . . . . . . . . . . . . . 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 } ) ) )
4938, 48ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  i^i  ran  F )  i^i  { 0 } )  =  (/)  <->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
5047, 49sylib 188 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  0  e.  y )  ->  (
y  i^i  ran  F ) 
C_  ( ran  F  \  { 0 } ) )
5150sselda 3193 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  x  e.  ( ran  F  \  {
0 } ) )
52 i1fd.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( `' F " { x } )  e.  dom  vol )
5341, 51, 52syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  e.  dom  vol )
5453ralrimiva 2639 . . . . . . . . . . . . 13  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
55 finiunmbl 18917 . . . . . . . . . . . . 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 )
5640, 54, 55syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  0  e.  y )  ->  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } )  e.  dom  vol )
5735, 56eqeltrrd 2371 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y )  e.  dom  vol )
5857ex 423 . . . . . . . . . 10  |-  ( ph  ->  ( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol ) )
5958alrimiv 1621 . . . . . . . . 9  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
6059ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  A. y ( -.  0  e.  y  -> 
( `' F "
y )  e.  dom  vol ) )
61 elndif 3313 . . . . . . . . 9  |-  ( 0  e.  x  ->  -.  0  e.  ( RR  \  x ) )
6261adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  -.  0  e.  ( RR  \  x ) )
63 reex 8844 . . . . . . . . . 10  |-  RR  e.  _V
64 difexg 4178 . . . . . . . . . 10  |-  ( RR  e.  _V  ->  ( RR  \  x )  e. 
_V )
6563, 64ax-mp 8 . . . . . . . . 9  |-  ( RR 
\  x )  e. 
_V
66 eleq2 2357 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  (
0  e.  y  <->  0  e.  ( RR  \  x
) ) )
6766notbid 285 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR 
\  x ) ) )
68 imaeq2 5024 . . . . . . . . . . 11  |-  ( y  =  ( RR  \  x )  ->  ( `' F " y )  =  ( `' F " ( RR  \  x
) ) )
6968eleq1d 2362 . . . . . . . . . 10  |-  ( y  =  ( RR  \  x )  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
)
7067, 69imbi12d 311 . . . . . . . . 9  |-  ( y  =  ( RR  \  x )  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  ( RR  \  x )  ->  ( `' F " ( RR 
\  x ) )  e.  dom  vol )
) )
7165, 70spcv 2887 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  ( RR  \  x
)  ->  ( `' F " ( RR  \  x ) )  e. 
dom  vol ) )
7260, 62, 71sylc 56 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
( RR  \  x
) )  e.  dom  vol )
73 difmbl 18916 . . . . . . 7  |-  ( ( ( `' F " RR )  e.  dom  vol 
/\  ( `' F " ( RR  \  x
) )  e.  dom  vol )  ->  ( ( `' F " RR ) 
\  ( `' F " ( RR  \  x
) ) )  e. 
dom  vol )
7422, 72, 73syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( ( `' F " RR )  \  ( `' F " ( RR 
\  x ) ) )  e.  dom  vol )
7518, 74eqeltrrd 2371 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  0  e.  x )  ->  ( `' F "
x )  e.  dom  vol )
76 eleq2 2357 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
0  e.  y  <->  0  e.  x ) )
7776notbid 285 . . . . . . . . . 10  |-  ( y  =  x  ->  ( -.  0  e.  y  <->  -.  0  e.  x ) )
78 imaeq2 5024 . . . . . . . . . . 11  |-  ( y  =  x  ->  ( `' F " y )  =  ( `' F " x ) )
7978eleq1d 2362 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( `' F "
y )  e.  dom  vol  <->  ( `' F " x )  e.  dom  vol )
)
8077, 79imbi12d 311 . . . . . . . . 9  |-  ( y  =  x  ->  (
( -.  0  e.  y  ->  ( `' F " y )  e. 
dom  vol )  <->  ( -.  0  e.  x  ->  ( `' F " x )  e.  dom  vol )
) )
8180spv 1951 . . . . . . . 8  |-  ( A. y ( -.  0  e.  y  ->  ( `' F " y )  e.  dom  vol )  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8259, 81syl 15 . . . . . . 7  |-  ( ph  ->  ( -.  0  e.  x  ->  ( `' F " x )  e. 
dom  vol ) )
8382imp 418 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  x )  ->  ( `' F " x )  e.  dom  vol )
8483adantlr 695 . . . . 5  |-  ( ( ( ph  /\  x  e.  ran  (,) )  /\  -.  0  e.  x
)  ->  ( `' F " x )  e. 
dom  vol )
8575, 84pm2.61dan 766 . . . 4  |-  ( (
ph  /\  x  e.  ran  (,) )  ->  ( `' F " x )  e.  dom  vol )
8685ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol )
87 ismbf 19001 . . . 4  |-  ( F : RR --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
881, 87syl 15 . . 3  |-  ( ph  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
8986, 88mpbird 223 . 2  |-  ( ph  ->  F  e. MblFn )
90 mblvol 18905 . . . . . . . 8  |-  ( ( `' F " y )  e.  dom  vol  ->  ( vol `  ( `' F " y ) )  =  ( vol
* `  ( `' F " y ) ) )
9157, 90syl 15 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  =  ( vol * `  ( `' F " y ) ) )
92 mblss 18906 . . . . . . . . 9  |-  ( ( `' F " y )  e.  dom  vol  ->  ( `' F " y ) 
C_  RR )
9357, 92syl 15 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( `' F " y ) 
C_  RR )
94 mblvol 18905 . . . . . . . . . . 11  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( vol `  ( `' F " { x } ) )  =  ( vol * `  ( `' F " { x } ) ) )
9553, 94syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  =  ( vol
* `  ( `' F " { x }
) ) )
96 i1fd.4 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ran  F  \  {
0 } ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9741, 51, 96syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol `  ( `' F " { x } ) )  e.  RR )
9895, 97eqeltrrd 2371 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( vol * `
 ( `' F " { x } ) )  e.  RR )
9940, 98fsumrecl 12223 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) )  e.  RR )
10035fveq2d 5545 . . . . . . . . 9  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  U_ x  e.  ( y  i^i  ran  F ) ( `' F " { x } ) )  =  ( vol
* `  ( `' F " y ) ) )
101 mblss 18906 . . . . . . . . . . . . 13  |-  ( ( `' F " { x } )  e.  dom  vol 
->  ( `' F " { x } ) 
C_  RR )
10253, 101syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( `' F " { x }
)  C_  RR )
103102, 98jca 518 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  0  e.  y )  /\  x  e.  (
y  i^i  ran  F ) )  ->  ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
104103ralrimiva 2639 . . . . . . . . . 10  |-  ( (
ph  /\  -.  0  e.  y )  ->  A. x  e.  ( y  i^i  ran  F ) ( ( `' F " { x } )  C_  RR  /\  ( vol * `  ( `' F " { x } ) )  e.  RR ) )
105 ovolfiniun 18876 . . . . . . . . . 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 } ) ) )
10640, 104, 105syl2anc 642 . . . . . . . . 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 } ) ) )
107100, 106eqbrtrrd 4061 . . . . . . . 8  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  <_  sum_ x  e.  ( y  i^i  ran  F ) ( vol * `  ( `' F " { x } ) ) )
108 ovollecl 18858 . . . . . . . 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 )
10993, 99, 107, 108syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol * `  ( `' F " y ) )  e.  RR )
11091, 109eqeltrd 2370 . . . . . 6  |-  ( (
ph  /\  -.  0  e.  y )  ->  ( vol `  ( `' F " y ) )  e.  RR )
111110ex 423 . . . . 5  |-  ( ph  ->  ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR ) )
112111alrimiv 1621 . . . 4  |-  ( ph  ->  A. y ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR ) )
113 c0ex 8848 . . . . . 6  |-  0  e.  _V
114113snid 3680 . . . . 5  |-  0  e.  { 0 }
115 eldifn 3312 . . . . 5  |-  ( 0  e.  ( RR  \  { 0 } )  ->  -.  0  e.  { 0 } )
116114, 115mt2 170 . . . 4  |-  -.  0  e.  ( RR  \  {
0 } )
117 difexg 4178 . . . . . 6  |-  ( RR  e.  _V  ->  ( RR  \  { 0 } )  e.  _V )
11863, 117ax-mp 8 . . . . 5  |-  ( RR 
\  { 0 } )  e.  _V
119 eleq2 2357 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( 0  e.  y  <->  0  e.  ( RR  \  { 0 } ) ) )
120119notbid 285 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( -.  0  e.  y  <->  -.  0  e.  ( RR  \  { 0 } ) ) )
121 imaeq2 5024 . . . . . . . 8  |-  ( y  =  ( RR  \  { 0 } )  ->  ( `' F " y )  =  ( `' F " ( RR 
\  { 0 } ) ) )
122121fveq2d 5545 . . . . . . 7  |-  ( y  =  ( RR  \  { 0 } )  ->  ( vol `  ( `' F " y ) )  =  ( vol `  ( `' F "
( RR  \  {
0 } ) ) ) )
123122eleq1d 2362 . . . . . 6  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( vol `  ( `' F "
y ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
124120, 123imbi12d 311 . . . . 5  |-  ( y  =  ( RR  \  { 0 } )  ->  ( ( -.  0  e.  y  -> 
( vol `  ( `' F " y ) )  e.  RR )  <-> 
( -.  0  e.  ( RR  \  {
0 } )  -> 
( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) ) )
125118, 124spcv 2887 . . . 4  |-  ( A. y ( -.  0  e.  y  ->  ( vol `  ( `' F "
y ) )  e.  RR )  ->  ( -.  0  e.  ( RR  \  { 0 } )  ->  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
126112, 116, 125ee10 1366 . . 3  |-  ( ph  ->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR )
1271, 36, 1263jca 1132 . 2  |-  ( ph  ->  ( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) )
128 isi1f 19045 . 2  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
12989, 127, 128sylanbrc 645 1  |-  ( ph  ->  F  e.  dom  S.1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U_ciun 3921   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271   Fincfn 6879   RRcr 8752   0cc0 8753   RR*cxr 8882    <_ cle 8884   (,)cioo 10672   sum_csu 12174   vol
*covol 18838   volcvol 18839  MblFncmbf 18985   S.1citg1 18986
This theorem is referenced by:  i1f0  19058  i1f1  19061  i1fadd  19066  i1fmul  19067  i1fmulc  19074  i1fres  19076  mbfi1fseqlem4  19089  itg2addnclem2  25004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992
  Copyright terms: Public domain W3C validator