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Theorem isi1f 19558
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 19508); 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 5568 . . 3  |-  ( g  =  F  ->  (
g : RR --> RR  <->  F : RR
--> RR ) )
2 rneq 5087 . . . 4  |-  ( g  =  F  ->  ran  g  =  ran  F )
32eleq1d 2501 . . 3  |-  ( g  =  F  ->  ( ran  g  e.  Fin  <->  ran  F  e.  Fin ) )
4 cnveq 5038 . . . . . 6  |-  ( g  =  F  ->  `' g  =  `' F
)
54imaeq1d 5194 . . . . 5  |-  ( g  =  F  ->  ( `' g " ( RR  \  { 0 } ) )  =  ( `' F " ( RR 
\  { 0 } ) ) )
65fveq2d 5724 . . . 4  |-  ( g  =  F  ->  ( vol `  ( `' g
" ( RR  \  { 0 } ) ) )  =  ( vol `  ( `' F " ( RR 
\  { 0 } ) ) ) )
76eleq1d 2501 . . 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 12473 . . 3  |-  sum_ x  e.  ( ran  f  \  { 0 } ) ( x  x.  ( vol `  ( `' f
" { x }
) ) )  e. 
_V
10 df-itg1 19505 . . 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 5566 . 2  |-  dom  S.1  =  { g  e. MblFn  | 
( g : RR --> RR  /\  ran  g  e. 
Fin  /\  ( vol `  ( `' g "
( RR  \  {
0 } ) ) )  e.  RR ) }
128, 11elrab2 3086 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 1652    e. wcel 1725   {crab 2701    \ cdif 3309   {csn 3806   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   Fincfn 7101   RRcr 8981   0cc0 8982    x. cmul 8987   sum_csu 12471   volcvol 19352  MblFncmbf 19498   S.1citg1 19499
This theorem is referenced by:  i1fmbf  19559  i1ff  19560  i1frn  19561  i1fima2  19563  i1fd  19565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-sum 12472  df-itg1 19505
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