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

Theorem isi1f 23660
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 ∈ dom ∫1 to represent this concept because 1 is the first preparation function for our final definition (see df-itg 23611); 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 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))

Proof of Theorem isi1f
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6187 . . 3 (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2 rneq 5506 . . . 4 (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
32eleq1d 2824 . . 3 (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4 cnveq 5451 . . . . . 6 (𝑔 = 𝐹𝑔 = 𝐹)
54imaeq1d 5623 . . . . 5 (𝑔 = 𝐹 → (𝑔 “ (ℝ ∖ {0})) = (𝐹 “ (ℝ ∖ {0})))
65fveq2d 6357 . . . 4 (𝑔 = 𝐹 → (vol‘(𝑔 “ (ℝ ∖ {0}))) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
76eleq1d 2824 . . 3 (𝑔 = 𝐹 → ((vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
81, 3, 73anbi123d 1548 . 2 (𝑔 = 𝐹 → ((𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ) ↔ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
9 sumex 14637 . . 3 Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))) ∈ V
10 df-itg1 23608 . . 3 1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))))
119, 10dmmpti 6184 . 2 dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
128, 11elrab2 3507 1 (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  {csn 4321  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  wf 6045  cfv 6049  (class class class)co 6814  Fincfn 8123  cr 10147  0cc0 10148   · cmul 10153  Σcsu 14635  volcvol 23452  MblFncmbf 23602  1citg1 23603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-sum 14636  df-itg1 23608
This theorem is referenced by:  i1fmbf  23661  i1ff  23662  i1frn  23663  i1fima2  23665  i1fd  23667
  Copyright terms: Public domain W3C validator