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

Theorem isi1f 19045
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 18995); 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 5391 . . 3  |-  ( g  =  F  ->  (
g : RR --> RR  <->  F : RR
--> RR ) )
2 rneq 4920 . . . 4  |-  ( g  =  F  ->  ran  g  =  ran  F )
32eleq1d 2362 . . 3  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
4 cnveq 4871 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
54imaeq1d 5027 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( RR  \  { 0 } ) )  =  ( `' F " ( RR 
\  { 0 } ) ) )
65fveq2d 5545 . . . 4  |-  ( g  =  F  ->  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  =  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) ) )
76eleq1d 2362 . . 3  |-  ( g  =  F  ->  (
( vol `  ( `' g " ( RR  \  { 0 } ) ) )  e.  RR  <->  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) )  e.  RR ) )
81, 3, 73anbi123d 1252 . 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 12176 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
10 df-itg1 18992 . . 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 5389 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
128, 11elrab2 2938 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    \ cdif 3162   {csn 3653   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   RRcr 8752   0cc0 8753    x. cmul 8758   sum_csu 12174   volcvol 18839  MblFncmbf 18985   S.1citg1 18986
This theorem is referenced by:  i1fmbf  19046  i1ff  19047  i1frn  19048  i1fima2  19050  i1fd  19052
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-fv 5279  df-sum 12175  df-itg1 18992
  Copyright terms: Public domain W3C validator