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

Theorem iblss 25778
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 6045 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 iblss.4 . . . . 5 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
4 iblmbf 25741 . . . . 5 ((𝑥𝐵𝐶) ∈ 𝐿1 → (𝑥𝐵𝐶) ∈ MblFn)
53, 4syl 17 . . . 4 (𝜑 → (𝑥𝐵𝐶) ∈ MblFn)
6 iblss.2 . . . 4 (𝜑𝐴 ∈ dom vol)
7 mbfres 25617 . . . 4 (((𝑥𝐵𝐶) ∈ MblFn ∧ 𝐴 ∈ dom vol) → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
85, 6, 7syl2anc 582 . . 3 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ∈ MblFn)
92, 8eqeltrrd 2826 . 2 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
10 ifan 4583 . . . . . 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)
111sselda 3976 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐵)
1211ad4ant14 750 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐴) → 𝑥𝐵)
13 iblss.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐵) → 𝐶𝑉)
145, 13mbfmptcl 25609 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
1514ad4ant14 750 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
16 ax-icn 11199 . . . . . . . . . . . . . 14 i ∈ ℂ
17 ine0 11681 . . . . . . . . . . . . . 14 i ≠ 0
18 elfzelz 13536 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
1918ad3antlr 729 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
20 expclz 14085 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
2116, 17, 19, 20mp3an12i 1461 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
22 expne0i 14095 . . . . . . . . . . . . . 14 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
2316, 17, 19, 22mp3an12i 1461 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
2415, 21, 23divcld 12023 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
2524recld 15177 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
26 0re 11248 . . . . . . . . . . 11 0 ∈ ℝ
27 ifcl 4575 . . . . . . . . . . 11 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2825, 26, 27sylancl 584 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2928rexrd 11296 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
30 max1 13199 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
3126, 25, 30sylancr 585 . . . . . . . . 9 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
32 elxrge0 13469 . . . . . . . . 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)))
3329, 31, 32sylanbrc 581 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3412, 33syldan 589 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥𝐴) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
35 0e0iccpnf 13471 . . . . . . . 8 0 ∈ (0[,]+∞)
3635a1i 11 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐴) → 0 ∈ (0[,]+∞))
3734, 36ifclda 4565 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
3810, 37eqeltrid 2829 . . . . 5 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3938fmpttd 7124 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
40 eqidd 2726 . . . . . 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)))
41 eqidd 2726 . . . . . 6 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
4240, 41, 3, 13iblitg 25742 . . . . 5 ((𝜑𝑘 ∈ ℤ) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
4318, 42sylan2 591 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
44 ifan 4583 . . . . . . 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)
4535a1i 11 . . . . . . . 8 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
4633, 45ifclda 4565 . . . . . . 7 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4744, 46eqeltrid 2829 . . . . . 6 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4847fmpttd 7124 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
4928leidd 11812 . . . . . . . . . . 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))
50 breq1 5152 . . . . . . . . . . . 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)))
51 breq1 5152 . . . . . . . . . . . 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)))
5250, 51ifboth 4569 . . . . . . . . . . 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))
5349, 31, 52syl2anc 582 . . . . . . . . . 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))
54 iftrue 4536 . . . . . . . . . . 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))
5554adantl 480 . . . . . . . . . 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))
5653, 55breqtrrd 5177 . . . . . . . . 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))
57 0le0 12346 . . . . . . . . . . 11 0 ≤ 0
5857a1i 11 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → 0 ≤ 0)
5912stoic1a 1766 . . . . . . . . . . 11 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → ¬ 𝑥𝐴)
6059iffalsed 4541 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
61 iffalse 4539 . . . . . . . . . . 11 𝑥𝐵 → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
6261adantl 480 . . . . . . . . . 10 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥𝐵) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
6358, 60, 623brtr4d 5181 . . . . . . . . 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))
6456, 63pm2.61dan 811 . . . . . . . 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))
6564, 10, 443brtr4g 5183 . . . . . . 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))
6665ralrimiva 3135 . . . . . 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))
67 reex 11231 . . . . . . . 8 ℝ ∈ V
6867a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → ℝ ∈ V)
69 eqidd 2726 . . . . . . 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)))
70 eqidd 2726 . . . . . . 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)))
7168, 38, 47, 69, 70ofrfval2 7706 . . . . . 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)))
7266, 71mpbird 256 . . . . 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)))
73 itg2le 25713 . . . . 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))))
7439, 48, 72, 73syl3anc 1368 . . . 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))))
75 itg2lecl 25712 . . . 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)
7639, 43, 74, 75syl3anc 1368 . . 3 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
7776ralrimiva 3135 . 2 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
78 eqidd 2726 . . 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)))
79 eqidd 2726 . . 3 ((𝜑𝑥𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
8011, 14syldan 589 . . 3 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
8178, 79, 80isibl2 25740 . 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)))
829, 77, 81mpbir2and 711 1 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  wss 3944  ifcif 4530   class class class wbr 5149  cmpt 5232  dom cdm 5678  cres 5680  wf 6545  cfv 6549  (class class class)co 7419  r cofr 7684  cc 11138  cr 11139  0cc0 11140  ici 11142  +∞cpnf 11277  *cxr 11279  cle 11281   / cdiv 11903  3c3 12301  cz 12591  [,]cicc 13362  ...cfz 13519  cexp 14062  cre 15080  volcvol 25436  MblFncmbf 25587  2citg2 25589  𝐿1cibl 25590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-ofr 7686  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xadd 13128  df-ioo 13363  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-mod 13871  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-xmet 21289  df-met 21290  df-ovol 25437  df-vol 25438  df-mbf 25592  df-itg1 25593  df-itg2 25594  df-ibl 25595
This theorem is referenced by:  itgss3  25788  itgless  25790  bddmulibl  25812  itgcn  25818  ditgcl  25831  ditgswap  25832  ditgsplitlem  25833  ftc1lem1  26014  ftc1lem2  26015  ftc1a  26016  ftc1lem4  26018  ftc2  26023  ftc2ditglem  26024  itgsubstlem  26027  itgpowd  26029  fdvposlt  34362  fdvposle  34364  circlemeth  34403  ftc1cnnclem  37295  ftc1anc  37305  ftc2nc  37306  areacirc  37317  lcmineqlem10  41641  lcmineqlem12  41643  lhe4.4ex1a  43908  itgsin0pilem1  45476  iblioosinexp  45479  itgsinexplem1  45480  itgsinexp  45481  itgcoscmulx  45495  itgsincmulx  45500  iblcncfioo  45504  dirkeritg  45628  fourierdlem87  45719  fourierdlem95  45727  fourierdlem103  45735  fourierdlem104  45736  fourierdlem107  45739  fourierdlem111  45743  fourierdlem112  45744  sqwvfoura  45754  sqwvfourb  45755  etransclem18  45778
  Copyright terms: Public domain W3C validator