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

Theorem isi1f 19558
 Description: The predicate " is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom to represent this concept because is the first preparation function for our final definition (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 MblFn

Proof of Theorem isi1f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5568 . . 3
2 rneq 5087 . . . 4
32eleq1d 2501 . . 3
4 cnveq 5038 . . . . . 6
54imaeq1d 5194 . . . . 5
65fveq2d 5724 . . . 4
76eleq1d 2501 . . 3
81, 3, 73anbi123d 1254 . 2
9 sumex 12473 . . 3
10 df-itg1 19505 . . 3 MblFn
119, 10dmmpti 5566 . 2 MblFn
128, 11elrab2 3086 1 MblFn
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  crab 2701   cdif 3309  csn 3806  ccnv 4869   cdm 4870   crn 4871  cima 4873  wf 5442  cfv 5446  (class class class)co 6073  cfn 7101  cr 8981  cc0 8982   cmul 8987  csu 12471  cvol 19352  MblFncmbf 19498  citg1 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
 Copyright terms: Public domain W3C validator