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Theorem isi1f 19029
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 18979); 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 5375 . . 3  |-  ( g  =  F  ->  (
g : RR --> RR  <->  F : RR
--> RR ) )
2 rneq 4904 . . . 4  |-  ( g  =  F  ->  ran  g  =  ran  F )
32eleq1d 2349 . . 3  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
4 cnveq 4855 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
54imaeq1d 5011 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( RR  \  { 0 } ) )  =  ( `' F " ( RR 
\  { 0 } ) ) )
65fveq2d 5529 . . . 4  |-  ( g  =  F  ->  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  =  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) ) )
76eleq1d 2349 . . 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 12160 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
10 df-itg1 18976 . . 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 5373 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
128, 11elrab2 2925 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 1623    e. wcel 1684   {crab 2547    \ cdif 3149   {csn 3640   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   RRcr 8736   0cc0 8737    x. cmul 8742   sum_csu 12158   volcvol 18823  MblFncmbf 18969   S.1citg1 18970
This theorem is referenced by:  i1fmbf  19030  i1ff  19031  i1frn  19032  i1fima2  19034  i1fd  19036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-sum 12159  df-itg1 18976
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