iblsplit - Mathbox for Glauco Siliprandi

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Theorem iblsplit 43832

Description: The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
iblsplit.1	⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
iblsplit.2	⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
iblsplit.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
iblsplit.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
iblsplit.5	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
iblsplit	⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 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.

Step	Hyp	Ref	Expression
1		iblsplit.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
2	1	fmpttd 7039	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶):𝑈⟶ℂ)
3		ssun1 4118	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
4		iblsplit.2	. . . . . 6 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
5	3, 4	sseqtrrid 3984	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
6	5	resmptd 5974	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
7		iblsplit.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
8		eqidd 2737	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)))
9		eqidd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
10	5	sseld 3930	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑈))
11	10	imdistani 569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝑈))
12	11, 1	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
13	8, 9, 12	isibl2 25029	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ)))
14	7, 13	mpbid 231	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ))
15	14	simpld 495	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
16	6, 15	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) ∈ MblFn)
17		ssun2 4119	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17, 4	sseqtrrid 3984	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
19	18	resmptd 5974	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
20		iblsplit.5	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)
21		eqidd 2737	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0)))
22		eqidd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
23	18	sseld 3930	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
24	23	imdistani 569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜑 ∧ 𝑥 ∈ 𝑈))
25	24, 1	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
26	21, 22, 25	isibl2 25029	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ)))
27	20, 26	mpbid 231	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑦 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑦)))), (ℜ‘(𝐶 / (i↑𝑦))), 0))) ∈ ℝ))
28	27	simpld 495	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
29	19, 28	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) ∈ MblFn)
30	4	eqcomd 2742	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑈)
31	2, 16, 29, 30	mbfres2cn 43824	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ MblFn)
32	15, 12	mbfdm2 24899	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol)
33	32	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ∈ dom vol)
34	28, 25	mbfdm2 24899	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom vol)
35	34	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐵 ∈ dom vol)
36		iblsplit.1	. . . . . 6 ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
37	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘(𝐴 ∩ 𝐵)) = 0)
38	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑈 = (𝐴 ∪ 𝐵))
39	1	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
40		ax-icn 11023	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
41	40	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → i ∈ ℂ)
42		elfznn0 13442	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
43	41, 42	expcld 13957	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ)
44	43	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ∈ ℂ)
45	40	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → i ∈ ℂ)
46		ine0 11503	. . . . . . . . . . . . 13 ⊢ i ≠ 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → i ≠ 0)
48		elfzelz 13349	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
49	48	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝑘 ∈ ℤ)
50	45, 47, 49	expne0d 13963	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ≠ 0)
51	39, 44, 50	divcld 11844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (𝐶 / (i↑𝑘)) ∈ ℂ)
52	51	recld 14996	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
53	52	rexrd 11118	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
54	53	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
55		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))
56		pnfge 12959	. . . . . . . 8 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^* → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
57	54, 56	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
58		0xr 11115	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
59		pnfxr 11122	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
60		elicc1 13216	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^ ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)))
61	58, 59, 60	mp2an 689	. . . . . . 7 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^* ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞))
62	54, 55, 57, 61	syl3anbrc 1342	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞))
63		0e0iccpnf 13284	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
64	63	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ ¬ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ∈ (0[,]+∞))
65	62, 64	ifclda 4507	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
66		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
67		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
68		ifan 4525	. . . . . 6 ⊢ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
69	68	mpteq2i 5194	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
70		ifan 4525	. . . . . . . . . 10 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
71	70	eqcomi 2745	. . . . . . . . 9 ⊢ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)
72	71	mpteq2i 5194	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
73	72	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
74	73	fveq2d 6823	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
75		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
76		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
77	75, 76, 12	isibl2 25029	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
78	7, 77	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))
79	78	simprd 496	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
80	79	r19.21bi 3230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
81	74, 80	eqeltrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
82		ifan 4525	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
83	82	eqcomi 2745	. . . . . . . 8 ⊢ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)
84	83	mpteq2i 5194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
85	84	fveq2i 6822	. . . . . 6 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
86		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
87		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
88	86, 87, 25	isibl2 25029	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
89	20, 88	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))
90	89	simprd 496	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
91	90	r19.21bi 3230	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
92	85, 91	eqeltrid 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
93	33, 35, 37, 38, 65, 66, 67, 69, 81, 92	itg2split 25012	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)))))
94	81, 92	readdcld 11097	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)))) ∈ ℝ)
95	93, 94	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
96	95	ralrimiva 3139	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
97		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
98		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
99	97, 98, 1	isibl2 25029	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝑈 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
100	31, 96, 99	mpbir2and 710	1 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∀wral 3061 ∪ cun 3895 ∩ cin 3896 ifcif 4472 class class class wbr 5089 ↦ cmpt 5172 dom cdm 5614 ↾ cres 5616 ‘cfv 6473 (class class class)co 7329 ℂcc 10962 ℝcr 10963 0cc0 10964 ici 10966 + caddc 10967 +∞cpnf 11099 ℝ^*cxr 11101 ≤ cle 11103 / cdiv 11725 3c3 12122 ℤcz 12412 [,]cicc 13175 ...cfz 13332 ↑cexp 13875 ℜcre 14899 vol*covol 24724 volcvol 24725 MblFncmbf 24876 ∫₂citg2 24878 𝐿¹cibl 24879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-disj 5055 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-ofr 7588 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-2o 8360 df-er 8561 df-map 8680 df-pm 8681 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fi 9260 df-sup 9291 df-inf 9292 df-oi 9359 df-dju 9750 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-n0 12327 df-z 12413 df-uz 12676 df-q 12782 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-ioo 13176 df-ico 13178 df-icc 13179 df-fz 13333 df-fzo 13476 df-fl 13605 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-clim 15288 df-sum 15489 df-rest 17222 df-topgen 17243 df-psmet 20687 df-xmet 20688 df-met 20689 df-bl 20690 df-mopn 20691 df-top 22141 df-topon 22158 df-bases 22194 df-cmp 22636 df-ovol 24726 df-vol 24727 df-mbf 24881 df-itg1 24882 df-itg2 24883 df-ibl 24884

This theorem is referenced by: iblsplitf 43836

W3C validator