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Theorem isibl2 23256
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 1829 . . . . . . 7 𝑥 𝑦𝐴
3 nfcv 2750 . . . . . . . 8 𝑥0
4 nfcv 2750 . . . . . . . 8 𝑥
5 nfcv 2750 . . . . . . . . 9 𝑥
6 nffvmpt1 6096 . . . . . . . . . 10 𝑥((𝑥𝐴𝐵)‘𝑦)
7 nfcv 2750 . . . . . . . . . 10 𝑥 /
8 nfcv 2750 . . . . . . . . . 10 𝑥(i↑𝑘)
96, 7, 8nfov 6553 . . . . . . . . 9 𝑥(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))
105, 9nffv 6095 . . . . . . . 8 𝑥(ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
113, 4, 10nfbr 4623 . . . . . . 7 𝑥0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))
122, 11nfan 1815 . . . . . 6 𝑥(𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
1312, 10, 3nfif 4064 . . . . 5 𝑥if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)
14 nfcv 2750 . . . . 5 𝑦if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0)
15 eleq1 2675 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
16 fveq2 6088 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑥))
1716oveq1d 6542 . . . . . . . . 9 (𝑦 = 𝑥 → (((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)) = (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))
1817fveq2d 6092 . . . . . . . 8 (𝑦 = 𝑥 → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))
1918breq2d 4589 . . . . . . 7 (𝑦 = 𝑥 → (0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))))
2015, 19anbi12d 742 . . . . . 6 (𝑦 = 𝑥 → ((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))))))
2120, 18ifbieq1d 4058 . . . . 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 4671 . . . 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 475 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
24 isibl2.3 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵𝑉)
25 eqid 2609 . . . . . . . . . . 11 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
2625fvmpt2 6185 . . . . . . . . . 10 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2723, 24, 26syl2anc 690 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
2827oveq1d 6542 . . . . . . . 8 ((𝜑𝑥𝐴) → (((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)) = (𝐵 / (i↑𝑘)))
2928fveq2d 6092 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
30 isibl.2 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))
3129, 30eqtr4d 2646 . . . . . 6 ((𝜑𝑥𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))) = 𝑇)
3231ibllem 23254 . . . . 5 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑥) / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))
3332mpteq2dv 4667 . . . 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 2655 . . 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 2646 . 2 (𝜑𝐺 = (𝑦 ∈ ℝ ↦ if((𝑦𝐴 ∧ 0 ≤ (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘)))), (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))), 0)))
36 eqidd 2610 . 2 ((𝜑𝑦𝐴) → (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))) = (ℜ‘(((𝑥𝐴𝐵)‘𝑦) / (i↑𝑘))))
3725, 24dmmptd 5923 . 2 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
38 eqidd 2610 . 2 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐵)‘𝑦) = ((𝑥𝐴𝐵)‘𝑦))
3935, 36, 37, 38isibl 23255 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  ici 9794  cle 9931   / cdiv 10533  3c3 10918  ...cfz 12152  cexp 12677  cre 13631  MblFncmbf 23106  2citg2 23108  𝐿1cibl 23109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-ibl 23114
This theorem is referenced by:  iblitg  23258  iblcnlem1  23277  iblss  23294  iblss2  23295  itgeqa  23303  iblconst  23307  iblabsr  23319  iblmulc2  23320  iblmulc2nc  32441  iblsplit  38655
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