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Theorem isibl 23255
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 (𝑥𝐴𝐵) ∈ 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 6098 . . . . . . . . 9 (ℜ‘((𝑓𝑥) / (i↑𝑘))) ∈ V
2 breq2 4581 . . . . . . . . . . 11 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))))
32anbi2d 735 . . . . . . . . . 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 4058 . . . . . . . . 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 3524 . . . . . . . 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 5233 . . . . . . . . . . 11 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
87eleq2d 2672 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓𝑥 ∈ dom 𝐹))
9 fveq1 6087 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
109oveq1d 6542 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((𝑓𝑥) / (i↑𝑘)) = ((𝐹𝑥) / (i↑𝑘)))
1110fveq2d 6092 . . . . . . . . . . 11 (𝑓 = 𝐹 → (ℜ‘((𝑓𝑥) / (i↑𝑘))) = (ℜ‘((𝐹𝑥) / (i↑𝑘))))
1211breq2d 4589 . . . . . . . . . 10 (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))))
138, 12anbi12d 742 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
1413, 11ifbieq1d 4058 . . . . . . . 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 2655 . . . . . . 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 4667 . . . . . 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 6092 . . . . 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 2671 . . . 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 2968 . . 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 23114 . . 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 3332 . 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 2672 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2423anbi1d 736 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
2524ifbid 4057 . . . . . . . . 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 6542 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐹𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
2827fveq2d 6092 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
29 isibl.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
3028, 29eqtr4d 2646 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = 𝑇)
3130ibllem 23254 . . . . . . . . 9 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3225, 31eqtrd 2643 . . . . . . . 8 (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3332mpteq2dv 4667 . . . . . . 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 2646 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)) = 𝐺)
3635fveq2d 6092 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) = (∫2𝐺))
3736eleq1d 2671 . . . 4 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2𝐺) ∈ ℝ))
3837ralbidv 2968 . . 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 735 . 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 270 1 (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  csb 3498  ifcif 4035   class class class wbr 4577  cmpt 4637  dom cdm 5028  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  ici 9794  cle 9931   / cdiv 10533  3c3 10918  ...cfz 12152  cexp 12677  cre 13631  MblFncmbf 23106  2citg2 23108  𝐿1cibl 23109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-dm 5038  df-iota 5754  df-fv 5798  df-ov 6530  df-ibl 23114
This theorem is referenced by:  isibl2  23256  ibl0  23276  iblempty  38654
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