iblabsnc - Mathbox for Brendan Leahy

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Theorem iblabsnc 37887

Description: Choice-free analogue of iblabs 25788. As with ibladdnc 37880, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)

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

**Assertion**
Ref	Expression
iblabsnc	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)

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

**Proof of Theorem iblabsnc**
Step	Hyp	Ref	Expression
1		iblabsnc.m	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)
2		iblabsnc.2	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
3		iblmbf 25726	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
5		iblabsnc.1	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
6	4, 5	mbfmptcl 25595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
7	6	abscld 15364	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
8	7	rexrd 11184	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
9	6	absge0d 15372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
10		elxrge0 13375	. . . . . . 7 ⊢ ((abs‘𝐵) ∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ^* ∧ 0 ≤ (abs‘𝐵)))
11	8, 9, 10	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
12		0e0iccpnf 13377	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
13	12	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
14	11, 13	ifclda 4515	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
16	15	fmpttd 7060	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞))
17		reex 11119	. . . . . . . . 9 ⊢ ℝ ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
19	6	recld 15119	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
20	19	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
21	20	abscld 15364	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ ℝ)
22	20	absge0d 15372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℜ‘𝐵)))
23		elrege0 13372	. . . . . . . . . . 11 ⊢ ((abs‘(ℜ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘𝐵))))
24	21, 22, 23	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ (0[,)+∞))
25		0e0icopnf 13376	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
26	25	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
27	24, 26	ifclda 4515	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
29	6	imcld 15120	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
30	29	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
31	30	abscld 15364	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
32	30	absge0d 15372	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℑ‘𝐵)))
33		elrege0 13372	. . . . . . . . . . 11 ⊢ ((abs‘(ℑ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘𝐵))))
34	31, 32, 33	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ (0[,)+∞))
35	34, 26	ifclda 4515	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
37		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)))
38		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))
39	18, 28, 36, 37, 38	offval2 7642	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) = (𝑥 ∈ ℝ ↦ (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))))
40		iftrue 4485	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = (abs‘(ℜ‘𝐵)))
41		iftrue 4485	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = (abs‘(ℑ‘𝐵)))
42	40, 41	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
43		iftrue 4485	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
44	42, 43	eqtr4d 2774	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
45		00id 11310	. . . . . . . . . 10 ⊢ (0 + 0) = 0
46		iffalse 4488	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = 0)
47		iffalse 4488	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = 0)
48	46, 47	oveq12d 7376	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (0 + 0))
49		iffalse 4488	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
50	45, 48, 49	3eqtr4a 2797	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
51	44, 50	pm2.61i 182	. . . . . . . 8 ⊢ (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)
52	51	mpteq2i 5194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
53	39, 52	eqtr2di 2788	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))))
54	53	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) = (∫₂‘((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))))
55		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))
56	6	iblcn 25758	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
57	2, 56	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
58	57	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
59	5, 2, 55, 58, 19	iblabsnclem 37886	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ))
60	59	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn)
61	28	fmpttd 7060	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
62	59	simprd 495	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ)
63	36	fmpttd 7060	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
64		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))
65	57	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
66	5, 2, 64, 65, 29	iblabsnclem 37886	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ))
67	66	simprd 495	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ)
68	60, 61, 62, 63, 67	itg2addnc 37877	. . . . 5 ⊢ (𝜑 → (∫₂‘((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∘_f + (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))))
69	54, 68	eqtrd 2771	. . . 4 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))))
70	62, 67	readdcld 11163	. . . 4 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))) ∈ ℝ)
71	69, 70	eqeltrd 2836	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ)
72	21, 31	readdcld 11163	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ)
73	72	rexrd 11184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^*)
74	21, 31, 22, 32	addge0d 11715	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
75		elxrge0 13375	. . . . . . . 8 ⊢ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞) ↔ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^* ∧ 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
76	73, 74, 75	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞))
77	76, 13	ifclda 4515	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
78	77	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
79	78	fmpttd 7060	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,]+∞))
80		ax-icn 11087	. . . . . . . . . . 11 ⊢ i ∈ ℂ
81		mulcl 11112	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
82	80, 30, 81	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (i · (ℑ‘𝐵)) ∈ ℂ)
83	20, 82	abstrid 15384	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
84		iftrue 4485	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
85	84	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
86	6	replimd 15122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
87	86	fveq2d 6838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
88	85, 87	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
89	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
90		absmul 15219	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
91	80, 30, 90	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
92		absi 15211	. . . . . . . . . . . . . 14 ⊢ (abs‘i) = 1
93	92	oveq1i 7368	. . . . . . . . . . . . 13 ⊢ ((abs‘i) · (abs‘(ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵)))
94	31	recnd 11162	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℂ)
95	94	mullidd 11152	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
96	93, 95	eqtrid 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘i) · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
97	91, 96	eqtr2d 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) = (abs‘(i · (ℑ‘𝐵))))
98	97	oveq2d 7374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
99	89, 98	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
100	83, 88, 99	3brtr4d 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
101	100	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
102		0le0 12248	. . . . . . . . 9 ⊢ 0 ≤ 0
103	102	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
104		iffalse 4488	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0)
105	103, 104, 49	3brtr4d 5130	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
106	101, 105	pm2.61d1 180	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
107	106	ralrimivw 3132	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
108		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
109		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
110	18, 15, 78, 108, 109	ofrfval2 7643	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
111	107, 110	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
112		itg2le 25698	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))))
113	16, 79, 111, 112	syl3anc 1373	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))))
114		itg2lecl 25697	. . 3 ⊢ (((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
115	16, 71, 113, 114	syl3anc 1373	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
116	7, 9	iblpos 25752	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)))
117	1, 115, 116	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 Vcvv 3440 ifcif 4479 class class class wbr 5098 ↦ cmpt 5179 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘_f cof 7620 ∘_r cofr 7621 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 ici 11030 + caddc 11031 · cmul 11033 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 [,)cico 13265 [,]cicc 13266 ℜcre 15022 ℑcim 15023 abscabs 15159 MblFncmbf 25573 ∫₂citg2 25575 𝐿¹cibl 25576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-rest 17344 df-topgen 17365 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-top 22840 df-topon 22857 df-bases 22892 df-cmp 23333 df-ovol 25423 df-vol 25424 df-mbf 25578 df-itg1 25579 df-itg2 25580 df-ibl 25581 df-0p 25629

This theorem is referenced by: itgabsnc 37892 ftc1cnnclem 37894 ftc1anclem2 37897 ftc1anclem4 37899 ftc1anclem5 37900 ftc2nc 37905

W3C validator