Theorem isibl 25820

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 6933	. . . . . . . . 9 ⊢ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ∈ V
2		breq2 5170	. . . . . . . . . . 11 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘)))))
3	2	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))))))
4		id 22	. . . . . . . . . 10 ⊢ (𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) → 𝑦 = (ℜ‘((𝑓‘𝑥) / (i↑𝑘))))
5	3, 4	ifbieq1d 4572	. . . . . . . . 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 3957	. . . . . . . 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 5928	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
8	7	eleq2d 2830	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ dom 𝐹))
9		fveq1 6919	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
10	9	fvoveq1d 7470	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) = (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))
11	10	breq2d 5178	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓‘𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))))
12	8, 11	anbi12d 631	. . . . . . . . 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 4572	. . . . . . . 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 2792	. . . . . . 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 5268	. . . . . 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 6924	. . . . 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 2829	. . . 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 3184	. . 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 25676	. . 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 3711	. 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 2830	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
23	22	anbi1d 630	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘))))))
24	23	ifbid 4571	. . . . . . . . 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 7470	. . . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐹‘𝑥) / (i↑𝑘))) = 𝑇)
29	28	ibllem 25819	. . . . . . . . 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 2780	. . . . . . . 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 5268	. . . . . . 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 2783	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹‘𝑥) / (i↑𝑘)))), (ℜ‘((𝐹‘𝑥) / (i↑𝑘))), 0)) = 𝐺)
34	33	fveq2d 6924	. . . . 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 2829	. . . 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 3184	. . 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 629	. 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 283	1 ⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ⦋csb 3921  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  ici 11186   ≤ cle 11325   / cdiv 11947  3c3 12349  ...cfz 13567  ↑cexp 14112  ℜcre 15146  MblFncmbf 25668  ∫₂citg2 25670  𝐿¹cibl 25671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451  df-ibl 25676

This theorem is referenced by:  isibl2  25821  ibl0  25842  iblempty  45886