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Theorem isibl2 24373
Description: The predicate "𝐹 is integrable" when 𝐹 is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-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))))
isibl2.3 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
isibl2 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘   𝜑,𝑘,𝑥   𝑥,𝑉
Allowed substitution hints:   𝐵(𝑥)   𝑇(𝑥,𝑘)   𝐺(𝑥,𝑘)   𝑉(𝑘)

Proof of Theorem isibl2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isibl.1 . . 3 (𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
2 nfv 1915 . . . . . . 7 𝑥 𝑦𝐴
3 nfcv 2958 . . . . . . . 8 𝑥0
4 nfcv 2958 . . . . . . . 8 𝑥
5 nfcv 2958 . . . . . . . . 9 𝑥
6 nffvmpt1 6660 . . . . . . . . . 10 𝑥((𝑥𝐴𝐵)‘𝑦)
7 nfcv 2958 . . . . . . . . . 10 𝑥 /
8 nfcv 2958 . . . . . . . . . 10 𝑥(i↑𝑘)
96, 7, 8nfov 7169 . . . . . . . . 9 𝑥(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))
105, 9nffv 6659 . . . . . . . 8 𝑥(ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
113, 4, 10nfbr 5080 . . . . . . 7 𝑥0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
122, 11nfan 1900 . . . . . 6 𝑥(𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
1312, 10, 3nfif 4457 . . . . 5 𝑥if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)
14 nfcv 2958 . . . . 5 𝑦if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)
15 eleq1w 2875 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
16 fveq2 6649 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
1716fvoveq1d 7161 . . . . . . . 8 (𝑦 = 𝑥 → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))
1817breq2d 5045 . . . . . . 7 (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))))
1915, 18anbi12d 633 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))))
2019, 17ifbieq1d 4451 . . . . 5 (𝑦 = 𝑥 → if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0))
2113, 14, 20cbvmpt 5134 . . . 4 (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0))
22 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
23 isibl2.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
24 eqid 2801 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2524fvmpt2 6760 . . . . . . . . 9 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2622, 23, 25syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2726fvoveq1d 7161 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
28 isibl.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
2927, 28eqtr4d 2839 . . . . . 6 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
3029ibllem 24371 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3130mpteq2dv 5129 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
3221, 31syl5eq 2848 . . 3 (𝜑 → (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))
331, 32eqtr4d 2839 . 2 (𝜑𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)))
34 eqidd 2802 . 2 ((𝜑𝑦𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
3524, 23dmmptd 6469 . 2 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
36 eqidd 2802 . 2 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑦))
3733, 34, 35, 36isibl 24372 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  ifcif 4428   class class class wbr 5033  cmpt 5113  cfv 6328  (class class class)co 7139  cr 10529  0cc0 10530  ici 10532  cle 10669   / cdiv 11290  3c3 11685  ...cfz 12889  cexp 13429  cre 14451  MblFncmbf 24221  2citg2 24223  𝐿1cibl 24224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-ibl 24229
This theorem is referenced by:  iblitg  24375  iblcnlem1  24394  iblss  24411  iblss2  24412  itgeqa  24420  iblconst  24424  iblabsr  24436  iblmulc2  24437  iblmulc2nc  35115  iblsplit  42595
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