Theorem isi1f 25647

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 25596); 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.

Step	Hyp	Ref	Expression
1		feq1 6704	. . 3 ⊢ (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2		rneq 5938	. . . 4 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
3	2	eleq1d 2810	. . 3 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4		cnveq 5876	. . . . . 6 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
5	4	imaeq1d 6063	. . . . 5 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ (ℝ ∖ {0})) = (◡𝐹 “ (ℝ ∖ {0})))
6	5	fveq2d 6900	. . . 4 ⊢ (𝑔 = 𝐹 → (vol‘(◡𝑔 “ (ℝ ∖ {0}))) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
7	6	eleq1d 2810	. . 3 ⊢ (𝑔 = 𝐹 → ((vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
8	1, 3, 7	3anbi123d 1432	. 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 15670	. . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V
10		df-itg1 25593	. . 3 ⊢ ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))))
11	9, 10	dmmpti 6700	. 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
12	8, 11	elrab2 3682	1 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3418   ∖ cdif 3941  {csn 4630  ^◡ccnv 5677  dom cdm 5678  ran crn 5679   “ cima 5681  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  Fincfn 8964  ℝcr 11139  0cc0 11140   · cmul 11145  Σcsu 15668  volcvol 25436  MblFncmbf 25587  ∫₁citg1 25588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-sum 15669  df-itg1 25593

This theorem is referenced by:  i1fmbf  25648  i1ff  25649  i1frn  25650  i1fima2  25652  i1fd  25654