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Theorem iblsplit 39486
Description: The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
iblsplit.1 (𝜑 → (vol*‘(𝐴𝐵)) = 0)
iblsplit.2 (𝜑𝑈 = (𝐴𝐵))
iblsplit.3 ((𝜑𝑥𝑈) → 𝐶 ∈ ℂ)
iblsplit.4 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
iblsplit.5 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
Assertion
Ref Expression
iblsplit (𝜑 → (𝑥𝑈𝐶) ∈ 𝐿1)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑈   𝜑,𝑥
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iblsplit
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iblsplit.3 . . . 4 ((𝜑𝑥𝑈) → 𝐶 ∈ ℂ)
2 eqid 2621 . . . 4 (𝑥𝑈𝐶) = (𝑥𝑈𝐶)
31, 2fmptd 6340 . . 3 (𝜑 → (𝑥𝑈𝐶):𝑈⟶ℂ)
4 ssun1 3754 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
5 iblsplit.2 . . . . . 6 (𝜑𝑈 = (𝐴𝐵))
64, 5syl5sseqr 3633 . . . . 5 (𝜑𝐴𝑈)
76resmptd 5411 . . . 4 (𝜑 → ((𝑥𝑈𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
8 iblsplit.4 . . . . . 6 (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
9 eqidd 2622 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)))
10 eqidd 2622 . . . . . . 7 ((𝜑𝑥𝐴) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
116sseld 3582 . . . . . . . . 9 (𝜑 → (𝑥𝐴𝑥𝑈))
1211imdistani 725 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝜑𝑥𝑈))
1312, 1syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)
149, 10, 13isibl2 23439 . . . . . 6 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ)))
158, 14mpbid 222 . . . . 5 (𝜑 → ((𝑥𝐴𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ))
1615simpld 475 . . . 4 (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)
177, 16eqeltrd 2698 . . 3 (𝜑 → ((𝑥𝑈𝐶) ↾ 𝐴) ∈ MblFn)
18 ssun2 3755 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1918, 5syl5sseqr 3633 . . . . 5 (𝜑𝐵𝑈)
2019resmptd 5411 . . . 4 (𝜑 → ((𝑥𝑈𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
21 iblsplit.5 . . . . . 6 (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
22 eqidd 2622 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)))
23 eqidd 2622 . . . . . . 7 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
2419sseld 3582 . . . . . . . . 9 (𝜑 → (𝑥𝐵𝑥𝑈))
2524imdistani 725 . . . . . . . 8 ((𝜑𝑥𝐵) → (𝜑𝑥𝑈))
2625, 1syl 17 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
2722, 23, 26isibl2 23439 . . . . . 6 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ ((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ)))
2821, 27mpbid 222 . . . . 5 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ))
2928simpld 475 . . . 4 (𝜑 → (𝑥𝐵𝐶) ∈ MblFn)
3020, 29eqeltrd 2698 . . 3 (𝜑 → ((𝑥𝑈𝐶) ↾ 𝐵) ∈ MblFn)
315eqcomd 2627 . . 3 (𝜑 → (𝐴𝐵) = 𝑈)
323, 17, 30, 31mbfres2cn 39478 . 2 (𝜑 → (𝑥𝑈𝐶) ∈ MblFn)
3316, 13mbfdm2 23311 . . . . . 6 (𝜑𝐴 ∈ dom vol)
3433adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → 𝐴 ∈ dom vol)
3529, 26mbfdm2 23311 . . . . . 6 (𝜑𝐵 ∈ dom vol)
3635adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → 𝐵 ∈ dom vol)
37 iblsplit.1 . . . . . 6 (𝜑 → (vol*‘(𝐴𝐵)) = 0)
3837adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (vol*‘(𝐴𝐵)) = 0)
395adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → 𝑈 = (𝐴𝐵))
401adantlr 750 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → 𝐶 ∈ ℂ)
41 ax-icn 9939 . . . . . . . . . . . . . 14 i ∈ ℂ
4241a1i 11 . . . . . . . . . . . . 13 (𝑘 ∈ (0...3) → i ∈ ℂ)
43 elfznn0 12374 . . . . . . . . . . . . 13 (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ0)
4442, 43expcld 12948 . . . . . . . . . . . 12 (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ)
4544ad2antlr 762 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → (i↑𝑘) ∈ ℂ)
4641a1i 11 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → i ∈ ℂ)
47 ine0 10409 . . . . . . . . . . . . 13 i ≠ 0
4847a1i 11 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → i ≠ 0)
49 elfzelz 12284 . . . . . . . . . . . . 13 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
5049ad2antlr 762 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → 𝑘 ∈ ℤ)
5146, 48, 50expne0d 12954 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → (i↑𝑘) ≠ 0)
5240, 45, 51divcld 10745 . . . . . . . . . 10 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → (𝐶 / (i↑𝑘)) ∈ ℂ)
5352recld 13868 . . . . . . . . 9 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
5453rexrd 10033 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ*)
5554adantr 481 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ*)
56 simpr 477 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))
57 pnfge 11908 . . . . . . . 8 ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ* → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
5855, 57syl 17 . . . . . . 7 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
59 0xr 10030 . . . . . . . 8 0 ∈ ℝ*
60 pnfxr 10036 . . . . . . . 8 +∞ ∈ ℝ*
61 elicc1 12161 . . . . . . . 8 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ* ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)))
6259, 60, 61mp2an 707 . . . . . . 7 ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ* ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞))
6355, 56, 58, 62syl3anbrc 1244 . . . . . 6 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞))
64 0e0iccpnf 12225 . . . . . . 7 0 ∈ (0[,]+∞)
6564a1i 11 . . . . . 6 ((((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) ∧ ¬ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ∈ (0[,]+∞))
6663, 65ifclda 4092 . . . . 5 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
67 eqid 2621 . . . . 5 (𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
68 eqid 2621 . . . . 5 (𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
69 ifan 4106 . . . . . 6 if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
7069mpteq2i 4701 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
71 ifan 4106 . . . . . . . . . 10 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)
7271eqcomi 2630 . . . . . . . . 9 if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)
7372mpteq2i 4701 . . . . . . . 8 (𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
7473a1i 11 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
7574fveq2d 6152 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
76 eqidd 2622 . . . . . . . . . 10 (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)))
77 eqidd 2622 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
7876, 77, 13isibl2 23439 . . . . . . . . 9 (𝜑 → ((𝑥𝐴𝐶) ∈ 𝐿1 ↔ ((𝑥𝐴𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
798, 78mpbid 222 . . . . . . . 8 (𝜑 → ((𝑥𝐴𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))
8079simprd 479 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
8180r19.21bi 2927 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
8275, 81eqeltrd 2698 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
83 ifan 4106 . . . . . . . . 9 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)
8483eqcomi 2630 . . . . . . . 8 if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)
8584mpteq2i 4701 . . . . . . 7 (𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
8685fveq2i 6151 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
87 eqidd 2622 . . . . . . . . . 10 (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)))
88 eqidd 2622 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
8987, 88, 26isibl2 23439 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ ((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
9021, 89mpbid 222 . . . . . . . 8 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))
9190simprd 479 . . . . . . 7 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
9291r19.21bi 2927 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
9386, 92syl5eqel 2702 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
9434, 36, 38, 39, 66, 67, 68, 70, 82, 93itg2split 23422 . . . 4 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = ((∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)))))
9582, 93readdcld 10013 . . . 4 ((𝜑𝑘 ∈ (0...3)) → ((∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) + (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)))) ∈ ℝ)
9694, 95eqeltrd 2698 . . 3 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
9796ralrimiva 2960 . 2 (𝜑 → ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
98 eqidd 2622 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
99 eqidd 2622 . . 3 ((𝜑𝑥𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
10098, 99, 1isibl2 23439 . 2 (𝜑 → ((𝑥𝑈𝐶) ∈ 𝐿1 ↔ ((𝑥𝑈𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
10132, 97, 100mpbir2and 956 1 (𝜑 → (𝑥𝑈𝐶) ∈ 𝐿1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cun 3553  cin 3554  ifcif 4058   class class class wbr 4613  cmpt 4673  dom cdm 5074  cres 5076  cfv 5847  (class class class)co 6604  cc 9878  cr 9879  0cc0 9880  ici 9882   + caddc 9883  +∞cpnf 10015  *cxr 10017  cle 10019   / cdiv 10628  3c3 11015  cz 11321  [,]cicc 12120  ...cfz 12268  cexp 12800  cre 13771  vol*covol 23138  volcvol 23139  MblFncmbf 23289  2citg2 23291  𝐿1cibl 23292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-rest 16004  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-cmp 21100  df-ovol 23140  df-vol 23141  df-mbf 23294  df-itg1 23295  df-itg2 23296  df-ibl 23297
This theorem is referenced by:  iblsplitf  39490
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