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Theorem isibl2 25816
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 1912 . . . . . . 7 𝑥 𝑦𝐴
3 nfcv 2903 . . . . . . . 8 𝑥0
4 nfcv 2903 . . . . . . . 8 𝑥
5 nfcv 2903 . . . . . . . . 9 𝑥
6 nffvmpt1 6918 . . . . . . . . . 10 𝑥((𝑥𝐴𝐵)‘𝑦)
7 nfcv 2903 . . . . . . . . . 10 𝑥 /
8 nfcv 2903 . . . . . . . . . 10 𝑥(i↑𝑘)
96, 7, 8nfov 7461 . . . . . . . . 9 𝑥(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))
105, 9nffv 6917 . . . . . . . 8 𝑥(ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
113, 4, 10nfbr 5195 . . . . . . 7 𝑥0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
122, 11nfan 1897 . . . . . 6 𝑥(𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
1312, 10, 3nfif 4561 . . . . 5 𝑥if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)
14 nfcv 2903 . . . . 5 𝑦if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)
15 eleq1w 2822 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
16 fveq2 6907 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
1716fvoveq1d 7453 . . . . . . . 8 (𝑦 = 𝑥 → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))
1817breq2d 5160 . . . . . . 7 (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))))
1915, 18anbi12d 632 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))))
2019, 17ifbieq1d 4555 . . . . 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 5259 . . . 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 484 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
23 isibl2.3 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
24 eqid 2735 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2524fvmpt2 7027 . . . . . . . . 9 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2622, 23, 25syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2726fvoveq1d 7453 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
28 isibl.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
2927, 28eqtr4d 2778 . . . . . 6 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
3029ibllem 25814 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3130mpteq2dv 5250 . . . 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, 31eqtrid 2787 . . 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 2778 . 2 (𝜑𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)))
34 eqidd 2736 . 2 ((𝜑𝑦𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
3524, 23dmmptd 6714 . 2 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
36 eqidd 2736 . 2 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑦))
3733, 34, 35, 36isibl 25815 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ifcif 4531   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  ici 11155  cle 11294   / cdiv 11918  3c3 12320  ...cfz 13544  cexp 14099  cre 15133  MblFncmbf 25663  2citg2 25665  𝐿1cibl 25666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-ibl 25671
This theorem is referenced by:  iblitg  25818  iblcnlem1  25838  iblss  25855  iblss2  25856  itgeqa  25864  iblconst  25868  iblabsr  25880  iblmulc2  25881  iblmulc2nc  37672  iblsplit  45922
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