Theorem isibl 26095

Description: The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of ∫𝐴𝐵 d𝑥, which is to say that the expression ∫𝐴𝐵 d𝑥 doesn't make sense unless (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

Ref	Expression
isibl.1	⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
isibl.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
isibl.3	⊢ (𝜑 → dom 𝐹 = 𝐴)
isibl.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

Assertion

Ref	Expression
isibl	⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘   𝑘,𝐹,𝑥   𝜑,𝑘,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥, 𝑘)   𝐺(𝑥, 𝑘)

Proof of Theorem isibl

Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6903	. . . . . . . . 9 ⊢ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ∈ V
2		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))))
3	2	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))))))
4		id 23	. . . . . . . . . 10 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → 𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))))
5	3, 4	ifbieq1d 4506	. . . . . . . . 9 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0))
6	1, 5	csbie 3882	. . . . . . . 8 ⊢ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0)
7		dmeq 5886	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
8	7	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ dom 𝐹))
9		fveq1 6889	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
10	9	fvoveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) = (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))
11	10	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))))
12	8, 11	anbi12d 644	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))))
13	12, 10	ifbieq1d 4506	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))), (ℜ‘((𝑓‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))
14	6, 13	eqtrid 2807	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))
15	14	mpteq2dv 5198	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)))
16	15	fveq2d 6894	. . . . 5 ⊢ (𝑓 = 𝐹 → (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))))
17	16	eleq1d 2845	. . . 4 ⊢ (𝑓 = 𝐹 → ((∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
18	17	ralbidv 3185	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
19		df-ibl 25952	. . 3 ⊢ 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
20	18, 19	elrab2 3648	. 2 ⊢ (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
21		isibl.3	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = 𝐴)
22	21	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
23	22	anbi1d 643	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))))
24	23	ifbid 4505	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))
25		isibl.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
26	25	fvoveq1d 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐹‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
27		isibl.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
28	26, 27	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐹‘𝑥) / (i↑𝑘))) = 𝑇)
29	28	ibllem 26094	. . . . . . . . 9 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
30	24, 29	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
31	30	mpteq2dv 5198	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
32		isibl.1	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
33	31, 32	eqtr4d 2798	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) = 𝐺)
34	33	fveq2d 6894	. . . . 5 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) = (∫2‘𝐺))
35	34	eleq1d 2845	. . . 4 ⊢ (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2‘𝐺) ∈ ℝ))
36	35	ralbidv 3185	. . 3 ⊢ (𝜑 → (∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ))
37	36	anbi2d 642	. 2 ⊢ (𝜑 → ((𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0))) ∈ ℝ) ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))
38	20, 37	bitrid 286	1 ⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⦋csb 3847  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5651  ‘cfv 6543  (class class class)co 7424  ℝcr 11206  0cc0 11207  ici 11209   ≤ cle 11351   / cdiv 11980  3c3 12405  ...cfz 13646  ↑cexp 14211  ℜcre 15273  MblFncmbf 25944  ∫₂citg2 25946  𝐿¹cibl 25947

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-dm 5661  df-iota 6497  df-fv 6551  df-ov 7427  df-ibl 25952

This theorem is used by:  isibl2  26096  ibl0  26116  iblempty  47111