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Theorem iblabsr 25880

Description: A measurable function is integrable iff its absolute value is integrable. (See iblabs 25879 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
iblabsr.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
iblabsr.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
iblabsr.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)

**Assertion**
Ref	Expression
iblabsr	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 𝑥,𝑉 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem iblabsr Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblabsr.2	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
2		ifan 4584	. . . . . . 7 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0)
3		iblabsr.1	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
4	1, 3	mbfmptcl 25685	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
5	4	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6		ax-icn 11212	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
7		ine0 11696	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
8		elfzelz 13561	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
9	8	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
10		expclz 14122	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
11	6, 7, 9, 10	mp3an12i 1464	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
12		expne0i 14132	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
13	6, 7, 9, 12	mp3an12i 1464	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0)
14	5, 11, 13	divcld 12041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐵 / (i↑𝑘)) ∈ ℂ)
15	14	recld 15230	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ)
16		0re 11261	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
17		ifcl 4576	. . . . . . . . . . 11 ⊢ (((ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ)
18	15, 16, 17	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ)
19	18	rexrd 11309	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ^*)
20		max1 13224	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))
21	16, 15, 20	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))
22		elxrge0 13494	. . . . . . . . 9 ⊢ (if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))
23	19, 21, 22	sylanbrc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,]+∞))
24		0e0iccpnf 13496	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
25	24	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
26	23, 25	ifclda 4566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
27	2, 26	eqeltrid 2843	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,]+∞))
28	27	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,]+∞))
29	28	fmpttd 7135	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
30		iblabsr.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)
31	4	abscld 15472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
32	4	absge0d 15480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
33	31, 32	iblpos 25843	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)))
34	30, 33	mpbid 232	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ))
35	34	simprd 495	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
36	35	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
37	31	rexrd 11309	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
38		elxrge0 13494	. . . . . . . . . 10 ⊢ ((abs‘𝐵) ∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ^* ∧ 0 ≤ (abs‘𝐵)))
39	37, 32, 38	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
40	24	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
41	39, 40	ifclda 4566	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
42	41	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
43	42	fmpttd 7135	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞))
44	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞))
45	14	releabsd 15487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘(𝐵 / (i↑𝑘))))
46	5, 11, 13	absdivd 15491	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = ((abs‘𝐵) / (abs‘(i↑𝑘))))
47		elfznn0 13657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
48	47	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ₀)
49		absexp 15340	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
50	6, 48, 49	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
51		absi 15322	. . . . . . . . . . . . . . . . . 18 ⊢ (abs‘i) = 1
52	51	oveq1i 7441	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
53		1exp 14129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (1↑𝑘) = 1)
54	9, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (1↑𝑘) = 1)
55	52, 54	eqtrid 2787	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1)
56	50, 55	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1)
57	56	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / (abs‘(i↑𝑘))) = ((abs‘𝐵) / 1))
58	31	recnd 11287	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ)
59	58	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ)
60	59	div1d 12033	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / 1) = (abs‘𝐵))
61	46, 57, 60	3eqtrd 2779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = (abs‘𝐵))
62	45, 61	breqtrd 5174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘𝐵))
63	5	absge0d 15480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
64		breq1 5151	. . . . . . . . . . . . 13 ⊢ ((ℜ‘(𝐵 / (i↑𝑘))) = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) → ((ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘𝐵) ↔ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵)))
65		breq1 5151	. . . . . . . . . . . . 13 ⊢ (0 = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) → (0 ≤ (abs‘𝐵) ↔ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵)))
66	64, 65	ifboth 4570	. . . . . . . . . . . 12 ⊢ (((ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘𝐵) ∧ 0 ≤ (abs‘𝐵)) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵))
67	62, 63, 66	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵))
68		iftrue 4537	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))
69	68	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))
70		iftrue 4537	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
71	70	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
72	67, 69, 71	3brtr4d 5180	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
73	72	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
74		0le0 12365	. . . . . . . . . . 11 ⊢ 0 ≤ 0
75	74	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
76		iffalse 4540	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = 0)
77		iffalse 4540	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0)
78	75, 76, 77	3brtr4d 5180	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
79	73, 78	pm2.61d1 180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
80	2, 79	eqbrtrid 5183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
81	80	ralrimivw 3148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
82		reex 11244	. . . . . . . 8 ⊢ ℝ ∈ V
83	82	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈ V)
84	37	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
85	84, 63, 38	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
86	85, 25	ifclda 4566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
87	86	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
88		eqidd 2736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))
89		eqidd 2736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
90	83, 28, 87, 88, 89	ofrfval2 7718	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
91	81, 90	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
92		itg2le 25789	. . . . 5 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))))
93	29, 44, 91, 92	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))))
94		itg2lecl 25788	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
95	29, 36, 93, 94	syl3anc 1370	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
96	95	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
97		eqidd 2736	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))
98		eqidd 2736	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
99	97, 98, 3	isibl2 25816	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)))
100	1, 96, 99	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘_r cofr 7696 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 +∞cpnf 11290 ℝ^*cxr 11292 ≤ cle 11294 / cdiv 11918 3c3 12320 ℕ₀cn0 12524 ℤcz 12611 [,]cicc 13387 ...cfz 13544 ↑cexp 14099 ℜcre 15133 abscabs 15270 MblFncmbf 25663 ∫₂citg2 25665 𝐿¹cibl 25666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-0p 25719

This theorem is referenced by: bddmulibl 25889

W3C validator