iblsplit - Mathbox for Glauco Siliprandi

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Theorem iblsplit 47181

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 7139	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶):𝑈⟶ℂ)
3		ssun1 4179	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
4		iblsplit.2	. . . . . 6 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
5	3, 4	sseqtrrid 3956	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
6	5	resmptd 6044	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
7		iblsplit.4	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
8		eqidd 2764	. . . . . . 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 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
10	5	sseld 3912	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑈))
11	10	imdistani 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝑈))
12	11, 1	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
13	8, 9, 12	isibl2 26117	. . . . . 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 235	. . . . 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 500	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
16	6, 15	eqeltrd 2864	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐴) ∈ MblFn)
17		ssun2 4180	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
18	17, 4	sseqtrrid 3956	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
19	18	resmptd 6044	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))
20		iblsplit.5	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)
21		eqidd 2764	. . . . . . 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 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑦))) = (ℜ‘(𝐶 / (i↑𝑦))))
23	18	sseld 3912	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
24	23	imdistani 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜑 ∧ 𝑥 ∈ 𝑈))
25	24, 1	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
26	21, 22, 25	isibl2 26117	. . . . . 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 235	. . . . 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 500	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
29	19, 28	eqeltrd 2864	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ 𝐶) ↾ 𝐵) ∈ MblFn)
30	4	eqcomd 2769	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑈)
31	2, 16, 29, 30	mbfres2cn 47173	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ MblFn)
32	15, 12	mbfdm2 25988	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol)
33	32	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ∈ dom vol)
34	28, 25	mbfdm2 25988	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom vol)
35	34	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐵 ∈ dom vol)
36		iblsplit.1	. . . . . 6 ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
37	36	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘(𝐴 ∩ 𝐵)) = 0)
38	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑈 = (𝐴 ∪ 𝐵))
39	1	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
40		ax-icn 11286	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
41	40	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → i ∈ ℂ)
42		elfznn0 13781	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
43	41, 42	expcld 14316	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...3) → (i↑𝑘) ∈ ℂ)
44	43	ad2antlr 740	. . . . . . . . . . 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 11778	. . . . . . . . . . . . 13 ⊢ i ≠ 0
47	46	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → i ≠ 0)
48		elfzelz 13683	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
49	48	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝑘 ∈ ℤ)
50	45, 47, 49	expne0d 14322	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ≠ 0)
51	39, 44, 50	divcld 12120	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (𝐶 / (i↑𝑘)) ∈ ℂ)
52	51	recld 15390	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
53	52	rexrd 11386	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
54	53	adantr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^*)
55		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → 0 ≤ (ℜ‘(𝐶 / (i↑𝑘))))
56		pnfge 13286	. . . . . . . 8 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ^* → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
57	54, 56	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ≤ +∞)
58		0xr 11383	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
59		pnfxr 11390	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
60		elicc1 13547	. . . . . . . 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 705	. . . . . . 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 1362	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ (0[,]+∞))
63		0e0iccpnf 13617	. . . . . . 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 4521	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
66		eqid 2763	. . . . 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 2763	. . . . 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 4539	. . . . . 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 5204	. . . . 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 4539	. . . . . . . . . 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 2772	. . . . . . . . 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 5204	. . . . . . . 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 6912	. . . . . 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 2764	. . . . . . . . . 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 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
77	75, 76, 12	isibl2 26117	. . . . . . . . 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 235	. . . . . . . 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 501	. . . . . . 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 3258	. . . . . 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 2864	. . . . 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 4539	. . . . . . . . 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 2772	. . . . . . . 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 5204	. . . . . . 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 6911	. . . . . 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 2764	. . . . . . . . . 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 2764	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
88	86, 87, 25	isibl2 26117	. . . . . . . . 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 235	. . . . . . . 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 501	. . . . . . 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 3258	. . . . . 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 2868	. . . . 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 26100	. . . 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 11365	. . . 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 2864	. . 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 3158	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
97		eqidd 2764	. . 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 2764	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
99	97, 98, 1	isibl2 26117	. 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 726	1 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∩ cin 4092 ∪ cun 4152 ifcif 4485 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ↾ cres 5657 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 ici 11229 + caddc 11230 +∞cpnf 11367 ℝ^*cxr 11369 ≤ cle 11371 / cdiv 12000 3c3 12425 ℤcz 12720 [,]cicc 13506 ...cfz 13666 ↑cexp 14231 ℜcre 15293 vol*covol 25813 volcvol 25814 MblFncmbf 25965 ∫₂citg2 25967 𝐿¹cibl 25968

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5074 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-ofr 7718 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-sum 15883 df-rest 17622 df-topgen 17643 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-top 23242 df-topon 23259 df-bases 23294 df-cmp 23735 df-ovol 25815 df-vol 25816 df-mbf 25970 df-itg1 25971 df-itg2 25972 df-ibl 25973

This theorem is used by: iblsplitf 47185

W3C validator