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Theorem isi1f 23840
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 23789); 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 6259 . . 3 (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2 rneq 5583 . . . 4 (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
32eleq1d 2891 . . 3 (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4 cnveq 5528 . . . . . 6 (𝑔 = 𝐹𝑔 = 𝐹)
54imaeq1d 5706 . . . . 5 (𝑔 = 𝐹 → (𝑔 “ (ℝ ∖ {0})) = (𝐹 “ (ℝ ∖ {0})))
65fveq2d 6437 . . . 4 (𝑔 = 𝐹 → (vol‘(𝑔 “ (ℝ ∖ {0}))) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
76eleq1d 2891 . . 3 (𝑔 = 𝐹 → ((vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
81, 3, 73anbi123d 1566 . 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 14795 . . 3 Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))) ∈ V
10 df-itg1 23786 . . 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 6256 . 2 dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
128, 11elrab2 3589 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  {crab 3121  cdif 3795  {csn 4397  ccnv 5341  dom cdm 5342  ran crn 5343  cima 5345  wf 6119  cfv 6123  (class class class)co 6905  Fincfn 8222  cr 10251  0cc0 10252   · cmul 10257  Σcsu 14793  volcvol 23629  MblFncmbf 23780  1citg1 23781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-sum 14794  df-itg1 23786
This theorem is referenced by:  i1fmbf  23841  i1ff  23842  i1frn  23843  i1fima2  23845  i1fd  23847
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