iblsplit - Mathbox for Glauco Siliprandi

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Theorem iblsplit 38858

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		eqid 2610	. . . 4 ⊢ (𝑥 ∈ 𝑈 ↦ 𝐶) = (𝑥 ∈ 𝑈 ↦ 𝐶)
3	1, 2	fmptd 6292	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶):𝑈⟶ℂ)
4		ssun1 3738	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		iblsplit.2	. . . . . 6 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
6	4, 5	syl5sseqr 3617	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
7	6	resmptd 5371	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
8		iblsplit.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
9		eqidd 2611	. . . . . . 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)))
10		eqidd 2611	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
11	6	sseld 3567	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑈))
12	11	imdistani 722	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝑈))
13	12, 1	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
14	9, 10, 13	isibl2 23339	. . . . . 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)))
15	8, 14	mpbid 221	. . . . 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))
16	15	simpld 474	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
17	7, 16	eqeltrd 2688	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) ∈ MblFn)
18		ssun2 3739	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
19	18, 5	syl5sseqr 3617	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
20	19	resmptd 5371	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
21		iblsplit.5	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)
22		eqidd 2611	. . . . . . 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)))
23		eqidd 2611	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
24	19	sseld 3567	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
25	24	imdistani 722	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜑 ∧ 𝑥 ∈ 𝑈))
26	25, 1	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
27	22, 23, 26	isibl2 23339	. . . . . 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)))
28	21, 27	mpbid 221	. . . . 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))
29	28	simpld 474	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
30	20, 29	eqeltrd 2688	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) ∈ MblFn)
31	5	eqcomd 2616	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑈)
32	3, 17, 30, 31	mbfres2cn 38850	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ MblFn)
33	16, 13	mbfdm2 23211	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol)
34	33	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ∈ dom vol)
35	29, 26	mbfdm2 23211	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom vol)
36	35	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐵 ∈ dom vol)
37		iblsplit.1	. . . . . 6 ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘(𝐴 ∩ 𝐵)) = 0)
39	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑈 = (𝐴 ∪ 𝐵))
40	1	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
41		ax-icn 9874	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → i ∈ ℂ)
43		elfznn0 12302	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
44	42, 43	expcld 12870	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ)
45	44	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ∈ ℂ)
46	41	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → i ∈ ℂ)
47		ine0 10344	. . . . . . . . . . . . 13 ⊢ i ≠ 0
48	47	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → i ≠ 0)
49		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
50	49	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝑘 ∈ ℤ)
51	46, 48, 50	expne0d 12876	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ≠ 0)
52	40, 45, 51	divcld 10680	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (𝐶 / (i↑𝑘)) ∈ ℂ)
53	52	recld 13782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
54	53	rexrd 9968	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
55	54	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
56		simpr 476	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))
57		pnfge 11840	. . . . . . . 8 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^* → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
58	55, 57	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
59		0xr 9965	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
60		pnfxr 9971	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
61		elicc1 12090	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^ ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)))
62	59, 60, 61	mp2an 704	. . . . . . 7 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞) ↔ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^* ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))) ∧ (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞))
63	55, 56, 58, 62	syl3anbrc 1239	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞))
64		0e0iccpnf 12154	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
65	64	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ ¬ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ∈ (0[,]+∞))
66	63, 65	ifclda 4070	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
67		eqid 2610	. . . . 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))
68		eqid 2610	. . . . 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))
69		ifan 4084	. . . . . 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)
70	69	mpteq2i 4669	. . . . 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))
71		ifan 4084	. . . . . . . . . 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)
72	71	eqcomi 2619	. . . . . . . . 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)
73	72	mpteq2i 4669	. . . . . . . 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))
74	73	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)))
75	74	fveq2d 6107	. . . . . 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))))
76		eqidd 2611	. . . . . . . . . 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)))
77		eqidd 2611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
78	76, 77, 13	isibl2 23339	. . . . . . . . 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)))
79	8, 78	mpbid 221	. . . . . . . 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))
80	79	simprd 478	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
81	80	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
82	75, 81	eqeltrd 2688	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
83		ifan 4084	. . . . . . . . 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)
84	83	eqcomi 2619	. . . . . . . 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)
85	84	mpteq2i 4669	. . . . . . 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))
86	85	fveq2i 6106	. . . . . 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)))
87		eqidd 2611	. . . . . . . . . 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)))
88		eqidd 2611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
89	87, 88, 26	isibl2 23339	. . . . . . . . 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)))
90	21, 89	mpbid 221	. . . . . . . 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))
91	90	simprd 478	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
92	91	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
93	86, 92	syl5eqel 2692	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))) ∈ ℝ)
94	34, 36, 38, 39, 66, 67, 68, 70, 82, 93	itg2split 23322	. . . 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)))))
95	82, 93	readdcld 9948	. . . 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)
96	94, 95	eqeltrd 2688	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
97	96	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
98		eqidd 2611	. . 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)))
99		eqidd 2611	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
100	98, 99, 1	isibl2 23339	. 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)))
101	32, 97, 100	mpbir2and 959	1 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∪ cun 3538 ∩ cin 3539 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 ici 9817 + caddc 9818 +∞cpnf 9950 ℝ^*cxr 9952 ≤ cle 9954 / cdiv 10563 3c3 10948 ℤcz 11254 [,]cicc 12049 ...cfz 12197 ↑cexp 12722 ℜcre 13685 vol*covol 23038 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 ax-addf 9894

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-disj 4554 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-fi 8200 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-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 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-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197

This theorem is referenced by: iblsplitf 38862

W3C validator