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Theorem isi1f 23342
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 23293); 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 5985 . . 3 (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2 rneq 5315 . . . 4 (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
32eleq1d 2688 . . 3 (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4 cnveq 5261 . . . . . 6 (𝑔 = 𝐹𝑔 = 𝐹)
54imaeq1d 5428 . . . . 5 (𝑔 = 𝐹 → (𝑔 “ (ℝ ∖ {0})) = (𝐹 “ (ℝ ∖ {0})))
65fveq2d 6154 . . . 4 (𝑔 = 𝐹 → (vol‘(𝑔 “ (ℝ ∖ {0}))) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
76eleq1d 2688 . . 3 (𝑔 = 𝐹 → ((vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
81, 3, 73anbi123d 1396 . 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 14347 . . 3 Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(𝑓 “ {𝑥}))) ∈ V
10 df-itg1 23290 . . 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 5982 . 2 dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
128, 11elrab2 3353 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 384  w3a 1036   = wceq 1480  wcel 1992  {crab 2916  cdif 3557  {csn 4153  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  Fincfn 7900  cr 9880  0cc0 9881   · cmul 9886  Σcsu 14345  volcvol 23134  MblFncmbf 23284  1citg1 23285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-sum 14346  df-itg1 23290
This theorem is referenced by:  i1fmbf  23343  i1ff  23344  i1frn  23345  i1fima2  23347  i1fd  23349
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