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Theorem isibl2 23578
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 1883 . . . . . . 7 𝑥 𝑦𝐴
3 nfcv 2793 . . . . . . . 8 𝑥0
4 nfcv 2793 . . . . . . . 8 𝑥
5 nfcv 2793 . . . . . . . . 9 𝑥
6 nffvmpt1 6237 . . . . . . . . . 10 𝑥((𝑥𝐴𝐵)‘𝑦)
7 nfcv 2793 . . . . . . . . . 10 𝑥 /
8 nfcv 2793 . . . . . . . . . 10 𝑥(i↑𝑘)
96, 7, 8nfov 6716 . . . . . . . . 9 𝑥(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))
105, 9nffv 6236 . . . . . . . 8 𝑥(ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
113, 4, 10nfbr 4732 . . . . . . 7 𝑥0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
122, 11nfan 1868 . . . . . 6 𝑥(𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
1312, 10, 3nfif 4148 . . . . 5 𝑥if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)
14 nfcv 2793 . . . . 5 𝑦if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)
15 eleq1 2718 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
16 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
1716oveq1d 6705 . . . . . . . . 9 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)) = (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))
1817fveq2d 6233 . . . . . . . 8 (𝑦 = 𝑥 → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))
1918breq2d 4697 . . . . . . 7 (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))))
2015, 19anbi12d 747 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))))
2120, 18ifbieq1d 4142 . . . . 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))
2213, 14, 21cbvmpt 4782 . . . 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))
23 simpr 476 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
24 isibl2.3 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵𝑉)
25 eqid 2651 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2625fvmpt2 6330 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2723, 24, 26syl2anc 694 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2827oveq1d 6705 . . . . . . . 8 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
2928fveq2d 6233 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
30 isibl.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
3129, 30eqtr4d 2688 . . . . . 6 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
3231ibllem 23576 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3332mpteq2dv 4778 . . . 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)))
3422, 33syl5eq 2697 . . 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)))
351, 34eqtr4d 2688 . 2 (𝜑𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)))
36 eqidd 2652 . 2 ((𝜑𝑦𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
3725, 24dmmptd 6062 . 2 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
38 eqidd 2652 . 2 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑦))
3935, 36, 37, 38isibl 23577 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ifcif 4119   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  ici 9976  cle 10113   / cdiv 10722  3c3 11109  ...cfz 12364  cexp 12900  cre 13881  MblFncmbf 23428  2citg2 23430  𝐿1cibl 23431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-ibl 23436
This theorem is referenced by:  iblitg  23580  iblcnlem1  23599  iblss  23616  iblss2  23617  itgeqa  23625  iblconst  23629  iblabsr  23641  iblmulc2  23642  iblmulc2nc  33605  iblsplit  40500
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