MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isi1f Unicode version

Theorem isi1f 19433
Description: The predicate " F is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom  F  e.  dom  S.1 to represent this concept because  S.1 is the first preparation function for our final definition  S. (see df-itg 19383); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
isi1f  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )

Proof of Theorem isi1f
Dummy variables  f 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5516 . . 3  |-  ( g  =  F  ->  (
g : RR --> RR  <->  F : RR
--> RR ) )
2 rneq 5035 . . . 4  |-  ( g  =  F  ->  ran  g  =  ran  F )
32eleq1d 2453 . . 3  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
4 cnveq 4986 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
54imaeq1d 5142 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( RR  \  { 0 } ) )  =  ( `' F " ( RR 
\  { 0 } ) ) )
65fveq2d 5672 . . . 4  |-  ( g  =  F  ->  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  =  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) ) )
76eleq1d 2453 . . 3  |-  ( g  =  F  ->  (
( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
81, 3, 73anbi123d 1254 . 2  |-  ( g  =  F  ->  (
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR )  <-> 
( F : RR --> RR  /\  ran  F  e. 
Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
9 sumex 12408 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
10 df-itg1 19380 . . 3  |-  S.1  =  ( f  e.  {
g  e. MblFn  |  (
g : RR --> RR  /\  ran  g  e.  Fin  /\  ( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR ) }  |->  sum_
x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f " {
x } ) ) ) )
119, 10dmmpti 5514 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
128, 11elrab2 3037 1  |-  ( F  e.  dom  S.1  <->  ( F  e. MblFn  /\  ( F : RR
--> RR  /\  ran  F  e.  Fin  /\  ( vol `  ( `' F "
( RR  \  {
0 } ) ) )  e.  RR ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653    \ cdif 3260   {csn 3757   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821   -->wf 5390   ` cfv 5394  (class class class)co 6020   Fincfn 7045   RRcr 8922   0cc0 8923    x. cmul 8928   sum_csu 12406   volcvol 19227  MblFncmbf 19373   S.1citg1 19374
This theorem is referenced by:  i1fmbf  19434  i1ff  19435  i1frn  19436  i1fima2  19438  i1fd  19440
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-sum 12407  df-itg1 19380
  Copyright terms: Public domain W3C validator