Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iblss Structured version   Visualization version   GIF version

Theorem iblss 23377
 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 5371 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 iblss.4 . . . . 5 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
4 iblmbf 23340 . . . . 5 ((𝑥𝐵𝐶) ∈ 𝐿1 → (𝑥𝐵𝐶) ∈ MblFn)
53, 4syl 17 . . . 4 (𝜑 → (𝑥𝐵𝐶) ∈ MblFn)
6 iblss.2 . . . 4 (𝜑𝐴 ∈ dom vol)
7 mbfres 23217 . . . 4 (((𝑥𝐵𝐶) ∈ MblFn ∧ 𝐴 ∈ dom vol) → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
85, 6, 7syl2anc 691 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
92, 8eqeltrrd 2689 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
10 ifan 4084 . . . . . 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 786 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → 𝜑)
121sselda 3568 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐵)
1311, 12sylan 487 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐴) → 𝑥𝐵)
14 iblss.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐵) → 𝐶𝑉)
155, 14mbfmptcl 23210 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
1611, 15sylan 487 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
17 elfzelz 12213 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
1817ad3antlr 763 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
19 ax-icn 9874 . . . . . . . . . . . . . . 15 i ∈ ℂ
20 ine0 10344 . . . . . . . . . . . . . . 15 i ≠ 0
21 expclz 12747 . . . . . . . . . . . . . . 15 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
2219, 20, 21mp3an12 1406 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
2318, 22syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
24 expne0i 12754 . . . . . . . . . . . . . . 15 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
2519, 20, 24mp3an12 1406 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
2618, 25syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
2716, 23, 26divcld 10680 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
2827recld 13782 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
29 0re 9919 . . . . . . . . . . 11 0 ∈ ℝ
30 ifcl 4080 . . . . . . . . . . 11 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
3128, 29, 30sylancl 693 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
3231rexrd 9968 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
33 max1 11890 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
3429, 28, 33sylancr 694 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
35 elxrge0 12152 . . . . . . . . 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 695 . . . . . . . 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 12154 . . . . . . . 8 0 ∈ (0[,]+∞)
3938a1i 11 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
4037, 39ifclda 4070 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4110, 40syl5eqel 2692 . . . . 5 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
42 eqid 2610 . . . . 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 6292 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
44 eqidd 2611 . . . . . 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 2611 . . . . . 6 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
4644, 45, 3, 14iblitg 23341 . . . . 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 4084 . . . . . . 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 4070 . . . . . . 7 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
5148, 50syl5eqel 2692 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
52 eqid 2610 . . . . . 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 6292 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
5431leidd 10473 . . . . . . . . . . 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 4586 . . . . . . . . . . . 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 4586 . . . . . . . . . . . 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 4074 . . . . . . . . . . 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 691 . . . . . . . . . 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 4042 . . . . . . . . . . 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 4611 . . . . . . . . 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 10987 . . . . . . . . . . 11 0 ≤ 0
6362a1i 11 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → 0 ≤ 0)
6413stoic1a 1688 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → ¬ 𝑥𝐴)
65 iffalse 4045 . . . . . . . . . . 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 4045 . . . . . . . . . . 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 4615 . . . . . . . . 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 828 . . . . . . . 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 4617 . . . . . . 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 2949 . . . . . 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 9906 . . . . . . . 8 ℝ ∈ V
7473a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → ℝ ∈ V)
75 eqidd 2611 . . . . . . 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 2611 . . . . . . 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 6813 . . . . . 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 246 . . . . 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 23312 . . . . 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 1318 . . . 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 23311 . . . 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 1318 . . 3 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
8382ralrimiva 2949 . 2 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
84 eqidd 2611 . . 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 2611 . . 3 ((𝜑𝑥𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
8612, 15syldan 486 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
8784, 85, 86isibl2 23339 . 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 959 1 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑟 cofr 6794  ℂcc 9813  ℝcr 9814  0cc0 9815  ici 9817  +∞cpnf 9950  ℝ*cxr 9952   ≤ cle 9954   / cdiv 10563  3c3 10948  ℤcz 11254  [,]cicc 12049  ...cfz 12197  ↑cexp 12722  ℜcre 13685  volcvol 23039  MblFncmbf 23189  ∫2citg2 23191  𝐿1cibl 23192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-xmet 19560  df-met 19561  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197 This theorem is referenced by:  itgss3  23387  itgless  23389  bddmulibl  23411  itgcn  23415  ditgcl  23428  ditgswap  23429  ditgsplitlem  23430  ftc1lem1  23602  ftc1lem2  23603  ftc1a  23604  ftc1lem4  23606  ftc2  23611  ftc2ditglem  23612  itgsubstlem  23615  ftc1cnnclem  32653  ftc1anc  32663  ftc2nc  32664  areacirc  32675  itgpowd  36819  lhe4.4ex1a  37550  itgsin0pilem1  38841  iblioosinexp  38844  itgsinexplem1  38845  itgsinexp  38846  itgcoscmulx  38861  itgsincmulx  38866  iblcncfioo  38870  dirkeritg  38995  fourierdlem87  39086  fourierdlem95  39094  fourierdlem103  39102  fourierdlem104  39103  fourierdlem107  39106  fourierdlem111  39110  fourierdlem112  39111  sqwvfoura  39121  sqwvfourb  39122  etransclem18  39145
 Copyright terms: Public domain W3C validator