Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isibl Structured version   Visualization version   GIF version

Theorem isibl 23338
 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 (𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
isibl.2 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
isibl.3 (𝜑 → dom 𝐹 = 𝐴)
isibl.4 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Assertion
Ref Expression
isibl (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ 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.
StepHypRef Expression
1 fvex 6113 . . . . . . . . 9 (ℜ‘((𝑓𝑥) / (i↑𝑘))) ∈ V
2 breq2 4587 . . . . . . . . . . 11 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))))
32anbi2d 736 . . . . . . . . . 10 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘))))))
4 id 22 . . . . . . . . . 10 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → 𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))))
53, 4ifbieq1d 4059 . . . . . . . . 9 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))), (ℜ‘((𝑓𝑥) / (i↑𝑘))), 0))
61, 5csbie 3525 . . . . . . . 8 (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))), (ℜ‘((𝑓𝑥) / (i↑𝑘))), 0)
7 dmeq 5246 . . . . . . . . . . 11 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
87eleq2d 2673 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓𝑥 ∈ dom 𝐹))
9 fveq1 6102 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
109oveq1d 6564 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓𝑥) / (i↑𝑘)) = ((𝐹𝑥) / (i↑𝑘)))
1110fveq2d 6107 . . . . . . . . . . 11 (𝑓 = 𝐹 → (ℜ‘((𝑓𝑥) / (i↑𝑘))) = (ℜ‘((𝐹𝑥) / (i↑𝑘))))
1211breq2d 4595 . . . . . . . . . 10 (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))))
138, 12anbi12d 743 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
1413, 11ifbieq1d 4059 . . . . . . . 8 (𝑓 = 𝐹 → if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))), (ℜ‘((𝑓𝑥) / (i↑𝑘))), 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))
156, 14syl5eq 2656 . . . . . . 7 (𝑓 = 𝐹(ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))
1615mpteq2dv 4673 . . . . . 6 (𝑓 = 𝐹 → (𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)))
1716fveq2d 6107 . . . . 5 (𝑓 = 𝐹 → (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))))
1817eleq1d 2672 . . . 4 (𝑓 = 𝐹 → ((∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
1918ralbidv 2969 . . 3 (𝑓 = 𝐹 → (∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ ↔ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
20 df-ibl 23197 . . 3 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
2119, 20elrab2 3333 . 2 (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ))
22 isibl.3 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
2322eleq2d 2673 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2423anbi1d 737 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
2524ifbid 4058 . . . . . . . . 9 (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))
26 isibl.4 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2726oveq1d 6564 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐹𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
2827fveq2d 6107 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
29 isibl.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
3028, 29eqtr4d 2647 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = 𝑇)
3130ibllem 23337 . . . . . . . . 9 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3225, 31eqtrd 2644 . . . . . . . 8 (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3332mpteq2dv 4673 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
34 isibl.1 . . . . . . 7 (𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
3533, 34eqtr4d 2647 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)) = 𝐺)
3635fveq2d 6107 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) = (∫2𝐺))
3736eleq1d 2672 . . . 4 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2𝐺) ∈ ℝ))
3837ralbidv 2969 . . 3 (𝜑 → (∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ))
3938anbi2d 736 . 2 (𝜑 → ((𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ) ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
4021, 39syl5bb 271 1 (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  ici 9817   ≤ cle 9954   / cdiv 10563  3c3 10948  ...cfz 12197  ↑cexp 12722  ℜcre 13685  MblFncmbf 23189  ∫2citg2 23191  𝐿1cibl 23192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-ibl 23197 This theorem is referenced by:  isibl2  23339  ibl0  23359  iblempty  38857
 Copyright terms: Public domain W3C validator