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Theorem iblss 24874

Description: A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Hypotheses**
Ref	Expression
iblss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
iblss.2	⊢ (𝜑 → 𝐴 ∈ dom vol)
iblss.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
iblss.4	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
iblss	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 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.

Step	Hyp	Ref	Expression
1		iblss.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	resmptd 5937	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
3		iblss.4	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)
4		iblmbf 24837	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
6		iblss.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ dom vol)
7		mbfres 24713	. . . 4 ⊢ (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ 𝐴 ∈ dom vol) → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ MblFn)
8	5, 6, 7	syl2anc 583	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ∈ MblFn)
9	2, 8	eqeltrrd 2840	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
10		ifan 4509	. . . . . 6 ⊢ 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)
11	1	sselda 3917	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
12	11	ad4ant14 748	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
13		iblss.3	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
14	5, 13	mbfmptcl 24705	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
15	14	ad4ant14 748	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
16		ax-icn 10861	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
17		ine0 11340	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
18		elfzelz 13185	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
19	18	ad3antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℤ)
20		expclz 13735	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
21	16, 17, 19, 20	mp3an12i 1463	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ∈ ℂ)
22		expne0i 13743	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
23	16, 17, 19, 22	mp3an12i 1463	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ≠ 0)
24	15, 21, 23	divcld 11681	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
25	24	recld 14833	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
26		0re 10908	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
27		ifcl 4501	. . . . . . . . . . 11 ⊢ (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
28	25, 26, 27	sylancl 585	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
29	28	rexrd 10956	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ^*)
30		max1 12848	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
31	26, 25, 30	sylancr 586	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
32		elxrge0 13118	. . . . . . . . 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)))
33	29, 31, 32	sylanbrc 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
34	12, 33	syldan 590	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
35		0e0iccpnf 13120	. . . . . . . 8 ⊢ 0 ∈ (0[,]+∞)
36	35	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
37	34, 36	ifclda 4491	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
38	10, 37	eqeltrid 2843	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
39	38	fmpttd 6971	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
40		eqidd 2739	. . . . . 6 ⊢ (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)))
41		eqidd 2739	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
42	40, 41, 3, 13	iblitg 24838	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
43	18, 42	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
44		ifan 4509	. . . . . . 7 ⊢ 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)
45	35	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ∈ (0[,]+∞))
46	33, 45	ifclda 4491	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
47	44, 46	eqeltrid 2843	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
48	47	fmpttd 6971	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
49	28	leidd 11471	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
50		breq1 5073	. . . . . . . . . . . 12 ⊢ (if(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) → (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if(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) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
51		breq1 5073	. . . . . . . . . . . 12 ⊢ (0 = if(𝑥 ∈ 𝐴, 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(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
52	50, 51	ifboth 4495	. . . . . . . . . . 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(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
53	49, 31, 52	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, 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 4462	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
55	54	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
56	53, 55	breqtrrd 5098	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
57		0le0 12004	. . . . . . . . . . 11 ⊢ 0 ≤ 0
58	57	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ≤ 0)
59	12	stoic1a 1776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴)
60	59	iffalsed 4467	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
61		iffalse 4465	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐵 → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
62	61	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) = 0)
63	58, 60, 62	3brtr4d 5102	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
64	56, 63	pm2.61dan 809	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
65	64, 10, 44	3brtr4g 5104	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
66	65	ralrimiva 3107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ≤ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
67		reex 10893	. . . . . . . 8 ⊢ ℝ ∈ V
68	67	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈ V)
69		eqidd 2739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (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)))
70		eqidd 2739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (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)))
71	68, 38, 47, 69, 70	ofrfval2 7532	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 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) ≤ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
72	66, 71	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
73		itg2le 24809	. . . . 5 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
74	39, 48, 72, 73	syl3anc 1369	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
75		itg2lecl 24808	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
76	39, 43, 74, 75	syl3anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
77	76	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
78		eqidd 2739	. . 3 ⊢ (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)))
79		eqidd 2739	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
80	11, 14	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
81	78, 79, 80	isibl2 24836	. 2 ⊢ (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)))
82	9, 77, 81	mpbir2and 709	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ⊆ wss 3883 ifcif 4456 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_r cofr 7510 ℂcc 10800 ℝcr 10801 0cc0 10802 ici 10804 +∞cpnf 10937 ℝ^*cxr 10939 ≤ cle 10941 / cdiv 11562 3c3 11959 ℤcz 12249 [,]cicc 13011 ...cfz 13168 ↑cexp 13710 ℜcre 14736 volcvol 24532 MblFncmbf 24683 ∫₂citg2 24685 𝐿¹cibl 24686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xadd 12778 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-xmet 20503 df-met 20504 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691

This theorem is referenced by: itgss3 24884 itgless 24886 bddmulibl 24908 itgcn 24914 ditgcl 24927 ditgswap 24928 ditgsplitlem 24929 ftc1lem1 25104 ftc1lem2 25105 ftc1a 25106 ftc1lem4 25108 ftc2 25113 ftc2ditglem 25114 itgsubstlem 25117 itgpowd 25119 fdvposlt 32479 fdvposle 32481 circlemeth 32520 ftc1cnnclem 35775 ftc1anc 35785 ftc2nc 35786 areacirc 35797 lcmineqlem10 39974 lcmineqlem12 39976 lhe4.4ex1a 41836 itgsin0pilem1 43381 iblioosinexp 43384 itgsinexplem1 43385 itgsinexp 43386 itgcoscmulx 43400 itgsincmulx 43405 iblcncfioo 43409 dirkeritg 43533 fourierdlem87 43624 fourierdlem95 43632 fourierdlem103 43640 fourierdlem104 43641 fourierdlem107 43644 fourierdlem111 43648 fourierdlem112 43649 sqwvfoura 43659 sqwvfourb 43660 etransclem18 43683

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