Theorem isi1f 25903

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 25852); 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 6681	. . 3 ⊢ (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2		rneq 5920	. . . 4 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
3	2	eleq1d 2845	. . 3 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4		cnveq 5853	. . . . . 6 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
5	4	imaeq1d 6055	. . . . 5 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ (ℝ ∖ {0})) = (◡𝐹 “ (ℝ ∖ {0})))
6	5	fveq2d 6883	. . . 4 ⊢ (𝑔 = 𝐹 → (vol‘(◡𝑔 “ (ℝ ∖ {0}))) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
7	6	eleq1d 2845	. . 3 ⊢ (𝑔 = 𝐹 → ((vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
8	1, 3, 7	3anbi123d 1464	. 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 15776	. . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V
10		df-itg1 25849	. . 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 6677	. 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
12	8, 11	elrab2 3649	1 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ∖ cdif 3896  {csn 4584  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Fincfn 8953  ℝcr 11124  0cc0 11125   · cmul 11130  Σcsu 15774  volcvol 25692  MblFncmbf 25843  ∫₁citg1 25844

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-sum 15775  df-itg1 25849

This theorem is used by:  i1fmbf  25904  i1ff  25905  i1frn  25906  i1fima2  25908  i1fd  25910