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Theorem iblabsr 25709

Description: A measurable function is integrable iff its absolute value is integrable. (See iblabs 25708 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 4576	. . . . . . 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 25515	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
5	4	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6		ax-icn 11168	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
7		ine0 11650	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
8		elfzelz 13504	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
9	8	ad2antlr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
10		expclz 14052	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
11	6, 7, 9, 10	mp3an12i 1461	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
12		expne0i 14062	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
13	6, 7, 9, 12	mp3an12i 1461	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0)
14	5, 11, 13	divcld 11991	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐵 / (i↑𝑘)) ∈ ℂ)
15	14	recld 15144	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ∈ ℝ)
16		0re 11217	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
17		ifcl 4568	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ)
19	18	rexrd 11265	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ ℝ^*)
20		max1 13167	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0))
22		elxrge0 13437	. . . . . . . . 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 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ∈ (0[,]+∞))
24		0e0iccpnf 13439	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
25	24	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
26	23, 25	ifclda 4558	. . . . . . 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 2831	. . . . . 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 7109	. . . 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 15386	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
32	4	absge0d 15394	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
33	31, 32	iblpos 25672	. . . . . . 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 231	. . . . . 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 11265	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
38		elxrge0 13437	. . . . . . . . . 10 ⊢ ((abs‘𝐵) ∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ^* ∧ 0 ≤ (abs‘𝐵)))
39	37, 32, 38	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
40	24	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
41	39, 40	ifclda 4558	. . . . . . . 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 7109	. . . . . 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 15401	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘(𝐵 / (i↑𝑘))))
46	5, 11, 13	absdivd 15405	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = ((abs‘𝐵) / (abs‘(i↑𝑘))))
47		elfznn0 13597	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
48	47	ad2antlr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ₀)
49		absexp 15254	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
50	6, 48, 49	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
51		absi 15236	. . . . . . . . . . . . . . . . . 18 ⊢ (abs‘i) = 1
52	51	oveq1i 7414	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
53		1exp 14059	. . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1)
56	50, 55	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1)
57	56	oveq2d 7420	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / (abs‘(i↑𝑘))) = ((abs‘𝐵) / 1))
58	31	recnd 11243	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ)
59	58	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℂ)
60	59	div1d 11983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐵) / 1) = (abs‘𝐵))
61	46, 57, 60	3eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐵 / (i↑𝑘))) = (abs‘𝐵))
62	45, 61	breqtrd 5167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) ≤ (abs‘𝐵))
63	5	absge0d 15394	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
64		breq1 5144	. . . . . . . . . . . . 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 5144	. . . . . . . . . . . . 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 4562	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0) ≤ (abs‘𝐵))
68		iftrue 4529	. . . . . . . . . . . 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 4529	. . . . . . . . . . . 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 5173	. . . . . . . . . 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 12314	. . . . . . . . . . 11 ⊢ 0 ≤ 0
75	74	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
76		iffalse 4532	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐵 / (i↑𝑘))), (ℜ‘(𝐵 / (i↑𝑘))), 0), 0) = 0)
77		iffalse 4532	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0)
78	75, 76, 77	3brtr4d 5173	. . . . . . . . 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 5176	. . . . . . 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 3144	. . . . . 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 11200	. . . . . . . 8 ⊢ ℝ ∈ V
83	82	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈ V)
84	37	adantlr 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
85	84, 63, 38	sylanbrc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
86	85, 25	ifclda 4558	. . . . . . . 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 2727	. . . . . . 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 2727	. . . . . . 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 7687	. . . . . 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 25619	. . . . 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 1368	. . . 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 25618	. . . 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 1368	. . 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 3140	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
97		eqidd 2727	. . 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 2727	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
99	97, 98, 3	isibl2 25646	. 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 710	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 Vcvv 3468 ifcif 4523 class class class wbr 5141 ↦ cmpt 5224 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∘_r cofr 7665 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 ici 11111 +∞cpnf 11246 ℝ^*cxr 11248 ≤ cle 11250 / cdiv 11872 3c3 12269 ℕ₀cn0 12473 ℤcz 12559 [,]cicc 13330 ...cfz 13487 ↑cexp 14029 ℜcre 15047 abscabs 15184 MblFncmbf 25493 ∫₂citg2 25495 𝐿¹cibl 25496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-sum 15636 df-xmet 21228 df-met 21229 df-ovol 25343 df-vol 25344 df-mbf 25498 df-itg1 25499 df-itg2 25500 df-ibl 25501 df-0p 25549

This theorem is referenced by: bddmulibl 25718

W3C validator