Theorem isi1f 25833

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 25782); 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 6683	. . 3 ⊢ (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ))
2		rneq 5926	. . . 4 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹)
3	2	eleq1d 2848	. . 3 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4		cnveq 5859	. . . . . 6 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹)
5	4	imaeq1d 6061	. . . . 5 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ (ℝ ∖ {0})) = (◡𝐹 “ (ℝ ∖ {0})))
6	5	fveq2d 6885	. . . 4 ⊢ (𝑔 = 𝐹 → (vol‘(◡𝑔 “ (ℝ ∖ {0}))) = (vol‘(◡𝐹 “ (ℝ ∖ {0}))))
7	6	eleq1d 2848	. . 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 15735	. . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V
10		df-itg1 25779	. . 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 6679	. 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)}
12	8, 11	elrab2 3654	1 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3902  {csn 4589  ^◡ccnv 5660  dom cdm 5661  ran crn 5662   “ cima 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  ℝcr 11094  0cc0 11095   · cmul 11100  Σcsu 15733  volcvol 25622  MblFncmbf 25773  ∫₁citg1 25774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-sum 15734  df-itg1 25779

This theorem is referenced by:  i1fmbf  25834  i1ff  25835  i1frn  25836  i1fima2  25838  i1fd  25840