iblabsnc - Mathbox for Brendan Leahy

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Theorem iblabsnc 32644

Description: Choice-free analogue of iblabs 23401. As with ibladdnc 32637, 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 23340	. . . . . . . . . . 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 23210	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
7	6	abscld 14023	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
8	7	rexrd 9968	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
9	6	absge0d 14031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
10		elxrge0 12152	. . . . . . 7 ⊢ ((abs‘𝐵) ∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ^* ∧ 0 ≤ (abs‘𝐵)))
11	8, 9, 10	sylanbrc 695	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
12		0e0iccpnf 12154	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
13	12	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
14	11, 13	ifclda 4070	. . . . 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		eqid 2610	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))
17	15, 16	fmptd 6292	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞))
18		reex 9906	. . . . . . . . 9 ⊢ ℝ ∈ V
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
20	6	recld 13782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
21	20	recnd 9947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
22	21	abscld 14023	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ ℝ)
23	21	absge0d 14031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℜ‘𝐵)))
24		elrege0 12149	. . . . . . . . . . 11 ⊢ ((abs‘(ℜ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘𝐵))))
25	22, 23, 24	sylanbrc 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ (0[,)+∞))
26		0e0icopnf 12153	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
28	25, 27	ifclda 4070	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
30	6	imcld 13783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
31	30	recnd 9947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
32	31	abscld 14023	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
33	31	absge0d 14031	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℑ‘𝐵)))
34		elrege0 12149	. . . . . . . . . . 11 ⊢ ((abs‘(ℑ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘𝐵))))
35	32, 33, 34	sylanbrc 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ (0[,)+∞))
36	35, 27	ifclda 4070	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
38		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)))
39		eqidd 2611	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))
40	19, 29, 37, 38, 39	offval2 6812	. . . . . . 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))))
41		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = (abs‘(ℜ‘𝐵)))
42		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = (abs‘(ℑ‘𝐵)))
43	41, 42	oveq12d 6567	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
44		iftrue 4042	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
45	43, 44	eqtr4d 2647	. . . . . . . . 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))
46		00id 10090	. . . . . . . . . 10 ⊢ (0 + 0) = 0
47		iffalse 4045	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = 0)
48		iffalse 4045	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = 0)
49	47, 48	oveq12d 6567	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (0 + 0))
50		iffalse 4045	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
51	46, 49, 50	3eqtr4a 2670	. . . . . . . . 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))
52	45, 51	pm2.61i 175	. . . . . . . 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)
53	52	mpteq2i 4669	. . . . . . 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))
54	40, 53	syl6req 2661	. . . . . 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))))
55	54	fveq2d 6107	. . . . 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)))))
56		eqid 2610	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))
57	6	iblcn 23371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
58	2, 57	mpbid 221	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
59	58	simpld 474	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
60	5, 2, 56, 59, 20	iblabsnclem 32643	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ))
61	60	simpld 474	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn)
62	29, 56	fmptd 6292	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
63	60	simprd 478	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ)
64		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))
65	37, 64	fmptd 6292	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
66	58	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
67	5, 2, 64, 66, 30	iblabsnclem 32643	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ))
68	67	simprd 478	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ)
69	61, 62, 63, 65, 68	itg2addnc 32634	. . . . 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)))))
70	55, 69	eqtrd 2644	. . . 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)))))
71	63, 68	readdcld 9948	. . . 4 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))) ∈ ℝ)
72	70, 71	eqeltrd 2688	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ)
73	22, 32	readdcld 9948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ)
74	73	rexrd 9968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^*)
75	22, 32, 23, 33	addge0d 10482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
76		elxrge0 12152	. . . . . . . 8 ⊢ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞) ↔ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^* ∧ 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
77	74, 75, 76	sylanbrc 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞))
78	77, 13	ifclda 4070	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
79	78	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
80		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
81	79, 80	fmptd 6292	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,]+∞))
82		ax-icn 9874	. . . . . . . . . . 11 ⊢ i ∈ ℂ
83		mulcl 9899	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
84	82, 31, 83	sylancr 694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (i · (ℑ‘𝐵)) ∈ ℂ)
85	21, 84	abstrid 14043	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
86		iftrue 4042	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
87	86	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
88	6	replimd 13785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
89	88	fveq2d 6107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
90	87, 89	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
91	44	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
92		absmul 13882	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
93	82, 31, 92	sylancr 694	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
94		absi 13874	. . . . . . . . . . . . . 14 ⊢ (abs‘i) = 1
95	94	oveq1i 6559	. . . . . . . . . . . . 13 ⊢ ((abs‘i) · (abs‘(ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵)))
96	32	recnd 9947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℂ)
97	96	mulid2d 9937	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
98	95, 97	syl5eq 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘i) · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
99	93, 98	eqtr2d 2645	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) = (abs‘(i · (ℑ‘𝐵))))
100	99	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
101	91, 100	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
102	85, 90, 101	3brtr4d 4615	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
103	102	ex 449	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
104		0le0 10987	. . . . . . . . 9 ⊢ 0 ≤ 0
105	104	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
106		iffalse 4045	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0)
107	105, 106, 50	3brtr4d 4615	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
108	103, 107	pm2.61d1 170	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
109	108	ralrimivw 2950	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
110		eqidd 2611	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
111		eqidd 2611	. . . . . 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)))
112	19, 15, 79, 110, 111	ofrfval2 6813	. . . . 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)))
113	109, 112	mpbird 246	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) ∘_𝑟 ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
114		itg2le 23312	. . . 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))))
115	17, 81, 113, 114	syl3anc 1318	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))))
116		itg2lecl 23311	. . 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)
117	17, 72, 115, 116	syl3anc 1318	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
118	7, 9	iblpos 23365	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)))
119	1, 117, 118	mpbir2and 959	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ∘_𝑟 cofr 6794 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 +∞cpnf 9950 ℝ^*cxr 9952 ≤ cle 9954 [,)cico 12048 [,]cicc 12049 ℜcre 13685 ℑcim 13686 abscabs 13822 MblFncmbf 23189 ∫₂citg2 23191 𝐿¹cibl 23192

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197 df-0p 23243

This theorem is referenced by: itgabsnc 32649 ftc1cnnclem 32653 ftc1anclem2 32656 ftc1anclem4 32658 ftc1anclem5 32659 ftc2nc 32664

W3C validator