Theorem isi1f 24838

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 24787); 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 6581	. . 3 ⊢ (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2		rneq 5845	. . . 4 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
3	2	eleq1d 2823	. . 3 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4		cnveq 5782	. . . . . 6 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
5	4	imaeq1d 5968	. . . . 5 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ (ℝ ∖ {0})) = (◡𝐹 “ (ℝ ∖ {0})))
6	5	fveq2d 6778	. . . 4 ⊢ (𝑔 = 𝐹 → (vol‘(◡𝑔 “ (ℝ ∖ {0}))) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
7	6	eleq1d 2823	. . 3 ⊢ (𝑔 = 𝐹 → ((vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
8	1, 3, 7	3anbi123d 1435	. 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 15399	. . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V
10		df-itg1 24784	. . 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 6577	. 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
12	8, 11	elrab2 3627	1 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068   ∖ cdif 3884  {csn 4561  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℝcr 10870  0cc0 10871   · cmul 10876  Σcsu 15397  volcvol 24627  MblFncmbf 24778  ∫₁citg1 24779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-sum 15398  df-itg1 24784

This theorem is referenced by:  i1fmbf  24839  i1ff  24840  i1frn  24841  i1fima2  24843  i1fd  24845