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Theorem isibl 23823
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 6388 . . . . . . . . 9 (ℜ‘((𝑓𝑥) / (i↑𝑘))) ∈ V
2 breq2 4813 . . . . . . . . . . 11 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))))
32anbi2d 622 . . . . . . . . . 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 4266 . . . . . . . . 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 3717 . . . . . . . 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 5492 . . . . . . . . . . 11 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
87eleq2d 2830 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓𝑥 ∈ dom 𝐹))
9 fveq1 6374 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
109fvoveq1d 6864 . . . . . . . . . . 11 (𝑓 = 𝐹 → (ℜ‘((𝑓𝑥) / (i↑𝑘))) = (ℜ‘((𝐹𝑥) / (i↑𝑘))))
1110breq2d 4821 . . . . . . . . . 10 (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))))
128, 11anbi12d 624 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
1312, 10ifbieq1d 4266 . . . . . . . 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))
146, 13syl5eq 2811 . . . . . . 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))
1514mpteq2dv 4904 . . . . . 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)))
1615fveq2d 6379 . . . . 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))))
1716eleq1d 2829 . . . 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))
1817ralbidv 3133 . . 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))
19 df-ibl 23680 . . 3 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘((𝑓𝑥) / (i↑𝑘))) / 𝑦if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ}
2018, 19elrab2 3523 . 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))
21 isibl.3 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
2221eleq2d 2830 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2322anbi1d 623 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
2423ifbid 4265 . . . . . . . . 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))
25 isibl.4 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
2625fvoveq1d 6864 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
27 isibl.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
2826, 27eqtr4d 2802 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = 𝑇)
2928ibllem 23822 . . . . . . . . 9 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3024, 29eqtrd 2799 . . . . . . . 8 (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3130mpteq2dv 4904 . . . . . . 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)))
32 isibl.1 . . . . . . 7 (𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
3331, 32eqtr4d 2802 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)) = 𝐺)
3433fveq2d 6379 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) = (∫2𝐺))
3534eleq1d 2829 . . . 4 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2𝐺) ∈ ℝ))
3635ralbidv 3133 . . 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))
3736anbi2d 622 . 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)))
3820, 37syl5bb 274 1 (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  csb 3691  ifcif 4243   class class class wbr 4809  cmpt 4888  dom cdm 5277  cfv 6068  (class class class)co 6842  cr 10188  0cc0 10189  ici 10191  cle 10329   / cdiv 10938  3c3 11328  ...cfz 12533  cexp 13067  cre 14124  MblFncmbf 23672  2citg2 23674  𝐿1cibl 23675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4949
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-dm 5287  df-iota 6031  df-fv 6076  df-ov 6845  df-ibl 23680
This theorem is referenced by:  isibl2  23824  ibl0  23844  iblempty  40750
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