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Theorem isibl 24300
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 6682 . . . . . . . . 9 (ℜ‘((𝑓𝑥) / (i↑𝑘))) ∈ V
2 breq2 5067 . . . . . . . . . . 11 (𝑦 = (ℜ‘((𝑓𝑥) / (i↑𝑘))) → (0 ≤ 𝑦 ↔ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))))
32anbi2d 628 . . . . . . . . . 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 4493 . . . . . . . . 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 3922 . . . . . . . 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 5771 . . . . . . . . . . 11 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
87eleq2d 2903 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑥 ∈ dom 𝑓𝑥 ∈ dom 𝐹))
9 fveq1 6668 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
109fvoveq1d 7172 . . . . . . . . . . 11 (𝑓 = 𝐹 → (ℜ‘((𝑓𝑥) / (i↑𝑘))) = (ℜ‘((𝐹𝑥) / (i↑𝑘))))
1110breq2d 5075 . . . . . . . . . 10 (𝑓 = 𝐹 → (0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))))
128, 11anbi12d 630 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑥 ∈ dom 𝑓 ∧ 0 ≤ (ℜ‘((𝑓𝑥) / (i↑𝑘)))) ↔ (𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
1312, 10ifbieq1d 4493 . . . . . . . 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 2873 . . . . . . 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 5159 . . . . . 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 6673 . . . . 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 2902 . . . 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 3202 . . 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 24157 . . 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 3687 . 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 2903 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ dom 𝐹𝑥𝐴))
2322anbi1d 629 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘))))))
2423ifbid 4492 . . . . . . . . 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 7172 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
27 isibl.2 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
2826, 27eqtr4d 2864 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘((𝐹𝑥) / (i↑𝑘))) = 𝑇)
2928ibllem 24299 . . . . . . . . 9 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3024, 29eqtrd 2861 . . . . . . . 8 (𝜑 → if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3130mpteq2dv 5159 . . . . . . 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 2864 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0)) = 𝐺)
3433fveq2d 6673 . . . . 5 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) = (∫2𝐺))
3534eleq1d 2902 . . . 4 (𝜑 → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ dom 𝐹 ∧ 0 ≤ (ℜ‘((𝐹𝑥) / (i↑𝑘)))), (ℜ‘((𝐹𝑥) / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2𝐺) ∈ ℝ))
3635ralbidv 3202 . . 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 628 . 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 284 1 (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  csb 3887  ifcif 4470   class class class wbr 5063  cmpt 5143  dom cdm 5554  cfv 6354  (class class class)co 7150  cr 10530  0cc0 10531  ici 10533  cle 10670   / cdiv 11291  3c3 11687  ...cfz 12887  cexp 13424  cre 14451  MblFncmbf 24149  2citg2 24151  𝐿1cibl 24152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-dm 5564  df-iota 6313  df-fv 6362  df-ov 7153  df-ibl 24157
This theorem is referenced by:  isibl2  24301  ibl0  24321  iblempty  42134
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