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Theorem isibl2 23514
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 1841 . . . . . . 7 𝑥 𝑦𝐴
3 nfcv 2762 . . . . . . . 8 𝑥0
4 nfcv 2762 . . . . . . . 8 𝑥
5 nfcv 2762 . . . . . . . . 9 𝑥
6 nffvmpt1 6186 . . . . . . . . . 10 𝑥((𝑥𝐴𝐵)‘𝑦)
7 nfcv 2762 . . . . . . . . . 10 𝑥 /
8 nfcv 2762 . . . . . . . . . 10 𝑥(i↑𝑘)
96, 7, 8nfov 6661 . . . . . . . . 9 𝑥(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))
105, 9nffv 6185 . . . . . . . 8 𝑥(ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
113, 4, 10nfbr 4690 . . . . . . 7 𝑥0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
122, 11nfan 1826 . . . . . 6 𝑥(𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
1312, 10, 3nfif 4106 . . . . 5 𝑥if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)
14 nfcv 2762 . . . . 5 𝑦if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)
15 eleq1 2687 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
16 fveq2 6178 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
1716oveq1d 6650 . . . . . . . . 9 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)) = (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))
1817fveq2d 6182 . . . . . . . 8 (𝑦 = 𝑥 → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))
1918breq2d 4656 . . . . . . 7 (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))))
2015, 19anbi12d 746 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))))
2120, 18ifbieq1d 4100 . . . . 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 4740 . . . 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 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
24 isibl2.3 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵𝑉)
25 eqid 2620 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2625fvmpt2 6278 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2723, 24, 26syl2anc 692 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2827oveq1d 6650 . . . . . . . 8 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
2928fveq2d 6182 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
30 isibl.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
3129, 30eqtr4d 2657 . . . . . 6 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
3231ibllem 23512 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3332mpteq2dv 4736 . . . 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 2666 . . 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 2657 . 2 (𝜑𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)))
36 eqidd 2621 . 2 ((𝜑𝑦𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
3725, 24dmmptd 6011 . 2 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
38 eqidd 2621 . 2 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑦))
3935, 36, 37, 38isibl 23513 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  ifcif 4077   class class class wbr 4644  cmpt 4720  cfv 5876  (class class class)co 6635  cr 9920  0cc0 9921  ici 9923  cle 10060   / cdiv 10669  3c3 11056  ...cfz 12311  cexp 12843  cre 13818  MblFncmbf 23364  2citg2 23366  𝐿1cibl 23367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-ibl 23372
This theorem is referenced by:  iblitg  23516  iblcnlem1  23535  iblss  23552  iblss2  23553  itgeqa  23561  iblconst  23565  iblabsr  23577  iblmulc2  23578  iblmulc2nc  33446  iblsplit  39945
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