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Theorem iblss 23616
Description: A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
Hypotheses
Ref Expression
iblss.1 (𝜑𝐴𝐵)
iblss.2 (𝜑𝐴 ∈ dom vol)
iblss.3 ((𝜑𝑥𝐵) → 𝐶𝑉)
iblss.4 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
Assertion
Ref Expression
iblss (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝑥,𝑉
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iblss
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 iblss.1 . . . 4 (𝜑𝐴𝐵)
21resmptd 5487 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 iblss.4 . . . . 5 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
4 iblmbf 23579 . . . . 5 ((𝑥𝐵𝐶) ∈ 𝐿1 → (𝑥𝐵𝐶) ∈ MblFn)
53, 4syl 17 . . . 4 (𝜑 → (𝑥𝐵𝐶) ∈ MblFn)
6 iblss.2 . . . 4 (𝜑𝐴 ∈ dom vol)
7 mbfres 23456 . . . 4 (((𝑥𝐵𝐶) ∈ MblFn ∧ 𝐴 ∈ dom vol) → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
85, 6, 7syl2anc 694 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
92, 8eqeltrrd 2731 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
10 ifan 4167 . . . . . 6 if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
11 simpll 805 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → 𝜑)
121sselda 3636 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐵)
1311, 12sylan 487 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐴) → 𝑥𝐵)
14 iblss.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐵) → 𝐶𝑉)
155, 14mbfmptcl 23449 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
1611, 15sylan 487 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
17 elfzelz 12380 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
1817ad3antlr 767 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
19 ax-icn 10033 . . . . . . . . . . . . . . 15 i ∈ ℂ
20 ine0 10503 . . . . . . . . . . . . . . 15 i ≠ 0
21 expclz 12925 . . . . . . . . . . . . . . 15 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
2219, 20, 21mp3an12 1454 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
2318, 22syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
24 expne0i 12932 . . . . . . . . . . . . . . 15 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
2519, 20, 24mp3an12 1454 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
2618, 25syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
2716, 23, 26divcld 10839 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
2827recld 13978 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
29 0re 10078 . . . . . . . . . . 11 0 ∈ ℝ
30 ifcl 4163 . . . . . . . . . . 11 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
3128, 29, 30sylancl 695 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
3231rexrd 10127 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
33 max1 12054 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
3429, 28, 33sylancr 696 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
35 elxrge0 12319 . . . . . . . . 9 (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ* ∧ 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
3632, 34, 35sylanbrc 699 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3713, 36syldan 486 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐴) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
38 0e0iccpnf 12321 . . . . . . . 8 0 ∈ (0[,]+∞)
3938a1i 11 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
4037, 39ifclda 4153 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4110, 40syl5eqel 2734 . . . . 5 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
42 eqid 2651 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
4341, 42fmptd 6425 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
44 eqidd 2652 . . . . . 6 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
45 eqidd 2652 . . . . . 6 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
4644, 45, 3, 14iblitg 23580 . . . . 5 ((𝜑𝑘 ∈ ℤ) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
4717, 46sylan2 490 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
48 ifan 4167 . . . . . . 7 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
4938a1i 11 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
5036, 49ifclda 4153 . . . . . . 7 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
5148, 50syl5eqel 2734 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
52 eqid 2651 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
5351, 52fmptd 6425 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
5431leidd 10632 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
55 breq1 4688 . . . . . . . . . . . 12 (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) → (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ↔ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
56 breq1 4688 . . . . . . . . . . . 12 (0 = if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) → (0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ↔ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
5755, 56ifboth 4157 . . . . . . . . . . 11 ((if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∧ 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
5854, 34, 57syl2anc 694 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
59 iftrue 4125 . . . . . . . . . . 11 (𝑥𝐵 → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
6059adantl 481 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
6158, 60breqtrrd 4713 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
62 0le0 11148 . . . . . . . . . . 11 0 ≤ 0
6362a1i 11 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → 0 ≤ 0)
6413stoic1a 1737 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → ¬ 𝑥𝐴)
65 iffalse 4128 . . . . . . . . . . 11 𝑥𝐴 → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
6664, 65syl 17 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
67 iffalse 4128 . . . . . . . . . . 11 𝑥𝐵 → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
6867adantl 481 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
6963, 66, 683brtr4d 4717 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
7061, 69pm2.61dan 849 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
7170, 10, 483brtr4g 4719 . . . . . . 7 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
7271ralrimiva 2995 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
73 reex 10065 . . . . . . . 8 ℝ ∈ V
7473a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → ℝ ∈ V)
75 eqidd 2652 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
76 eqidd 2652 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
7774, 41, 51, 75, 76ofrfval2 6957 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
7872, 77mpbird 247 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
79 itg2le 23551 . . . . 5 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
8043, 53, 78, 79syl3anc 1366 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
81 itg2lecl 23550 . . . 4 (((𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
8243, 47, 80, 81syl3anc 1366 . . 3 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
8382ralrimiva 2995 . 2 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
84 eqidd 2652 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
85 eqidd 2652 . . 3 ((𝜑𝑥𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
8612, 15syldan 486 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
8784, 85, 86isibl2 23578 . 2 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
889, 83, 87mpbir2and 977 1 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  dom cdm 5143  cres 5145  wf 5922  cfv 5926  (class class class)co 6690  𝑟 cofr 6938  cc 9972  cr 9973  0cc0 9974  ici 9976  +∞cpnf 10109  *cxr 10111  cle 10113   / cdiv 10722  3c3 11109  cz 11415  [,]cicc 12216  ...cfz 12364  cexp 12900  cre 13881  volcvol 23278  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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  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-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435  df-ibl 23436
This theorem is referenced by:  itgss3  23626  itgless  23628  bddmulibl  23650  itgcn  23654  ditgcl  23667  ditgswap  23668  ditgsplitlem  23669  ftc1lem1  23843  ftc1lem2  23844  ftc1a  23845  ftc1lem4  23847  ftc2  23852  ftc2ditglem  23853  itgsubstlem  23856  fdvposlt  30805  fdvposle  30807  circlemeth  30846  ftc1cnnclem  33613  ftc1anc  33623  ftc2nc  33624  areacirc  33635  itgpowd  38117  lhe4.4ex1a  38845  itgsin0pilem1  40483  iblioosinexp  40486  itgsinexplem1  40487  itgsinexp  40488  itgcoscmulx  40503  itgsincmulx  40508  iblcncfioo  40512  dirkeritg  40637  fourierdlem87  40728  fourierdlem95  40736  fourierdlem103  40744  fourierdlem104  40745  fourierdlem107  40748  fourierdlem111  40752  fourierdlem112  40753  sqwvfoura  40763  sqwvfourb  40764  etransclem18  40787
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