iblabs - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iblabs		Structured version Visualization version GIF version

Theorem iblabs 25737

Description: The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
iblabs.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
iblabs.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

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

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

**Proof of Theorem iblabs**
Step	Hyp	Ref	Expression
1		absf 15311	. . . . 5 ⊢ abs:ℂ⟶ℝ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → abs:ℂ⟶ℝ)
3		iblabs.2	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
4		iblmbf 25675	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
6		iblabs.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
7	5, 6	mbfmptcl 25544	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
8	2, 7	cofmpt 7107	. . 3 ⊢ (𝜑 → (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)))
9	7	fmpttd 7090	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
10		ax-resscn 11132	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11		ssid 3972	. . . . . . 7 ⊢ ℂ ⊆ ℂ
12		cncfss 24799	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
13	10, 11, 12	mp2an 692	. . . . . 6 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
14		abscncf 24801	. . . . . 6 ⊢ abs ∈ (ℂ–cn→ℝ)
15	13, 14	sselii 3946	. . . . 5 ⊢ abs ∈ (ℂ–cn→ℂ)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℂ))
17		cncombf 25566	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ abs ∈ (ℂ–cn→ℂ)) → (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn)
18	5, 9, 16, 17	syl3anc 1373	. . 3 ⊢ (𝜑 → (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn)
19	8, 18	eqeltrrd 2830	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)
20	7	abscld 15412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
21	20	rexrd 11231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ^*)
22	7	absge0d 15420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
23		elxrge0 13425	. . . . . . 7 ⊢ ((abs‘𝐵) ∈ (0[,]+∞) ↔ ((abs‘𝐵) ∈ ℝ^* ∧ 0 ≤ (abs‘𝐵)))
24	21, 22, 23	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,]+∞))
25		0e0iccpnf 13427	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
26	25	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
27	24, 26	ifclda 4527	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
28	27	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,]+∞))
29	28	fmpttd 7090	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,]+∞))
30		reex 11166	. . . . . . . . 9 ⊢ ℝ ∈ V
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
32	7	recld 15167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
33	32	recnd 11209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
34	33	abscld 15412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ ℝ)
35	33	absge0d 15420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℜ‘𝐵)))
36		elrege0 13422	. . . . . . . . . . 11 ⊢ ((abs‘(ℜ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘𝐵))))
37	34, 35, 36	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ (0[,)+∞))
38		0e0icopnf 13426	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
40	37, 39	ifclda 4527	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
41	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
42	7	imcld 15168	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
43	42	recnd 11209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
44	43	abscld 15412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
45	43	absge0d 15420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℑ‘𝐵)))
46		elrege0 13422	. . . . . . . . . . 11 ⊢ ((abs‘(ℑ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘𝐵))))
47	44, 45, 46	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ (0[,)+∞))
48	47, 39	ifclda 4527	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
49	48	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
50		eqidd 2731	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)))
51		eqidd 2731	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))
52	31, 41, 49, 50, 51	offval2 7676	. . . . . . 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))))
53		iftrue 4497	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = (abs‘(ℜ‘𝐵)))
54		iftrue 4497	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = (abs‘(ℑ‘𝐵)))
55	53, 54	oveq12d 7408	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
56		iftrue 4497	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
57	55, 56	eqtr4d 2768	. . . . . . . . 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))
58		00id 11356	. . . . . . . . . 10 ⊢ (0 + 0) = 0
59		iffalse 4500	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = 0)
60		iffalse 4500	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = 0)
61	59, 60	oveq12d 7408	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (0 + 0))
62		iffalse 4500	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
63	58, 61, 62	3eqtr4a 2791	. . . . . . . . 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))
64	57, 63	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)
65	64	mpteq2i 5206	. . . . . . 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))
66	52, 65	eqtr2di 2782	. . . . . 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))))
67	66	fveq2d 6865	. . . . 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)))))
68		eqid 2730	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))
69	7	iblcn 25707	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
70	3, 69	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
71	70	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
72	6, 3, 68, 71, 32	iblabslem 25736	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ))
73	72	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn)
74	41	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
75	72	simprd 495	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ)
76		eqid 2730	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))
77	70	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
78	6, 3, 76, 77, 42	iblabslem 25736	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ))
79	78	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) ∈ MblFn)
80	49	fmpttd 7090	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
81	78	simprd 495	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ)
82	73, 74, 75, 79, 80, 81	itg2add 25667	. . . . 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)))))
83	67, 82	eqtrd 2765	. . . 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)))))
84	75, 81	readdcld 11210	. . . 4 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))) ∈ ℝ)
85	83, 84	eqeltrd 2829	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ)
86	34, 44	readdcld 11210	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ)
87	86	rexrd 11231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^*)
88	34, 44, 35, 45	addge0d 11761	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
89		elxrge0 13425	. . . . . . . 8 ⊢ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞) ↔ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ^* ∧ 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
90	87, 88, 89	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,]+∞))
91	90, 26	ifclda 4527	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
92	91	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,]+∞))
93	92	fmpttd 7090	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,]+∞))
94		ax-icn 11134	. . . . . . . . . . . 12 ⊢ i ∈ ℂ
95		mulcl 11159	. . . . . . . . . . . 12 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
96	94, 43, 95	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (i · (ℑ‘𝐵)) ∈ ℂ)
97	33, 96	abstrid 15432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
98	7	replimd 15170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
99	98	fveq2d 6865	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
100		absmul 15267	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
101	94, 43, 100	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
102		absi 15259	. . . . . . . . . . . . . 14 ⊢ (abs‘i) = 1
103	102	oveq1i 7400	. . . . . . . . . . . . 13 ⊢ ((abs‘i) · (abs‘(ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵)))
104	44	recnd 11209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℂ)
105	104	mullidd 11199	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
106	103, 105	eqtrid 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘i) · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
107	101, 106	eqtr2d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) = (abs‘(i · (ℑ‘𝐵))))
108	107	oveq2d 7406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
109	97, 99, 108	3brtr4d 5142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
110		iftrue 4497	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
111	110	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = (abs‘𝐵))
112	56	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
113	109, 111, 112	3brtr4d 5142	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
114	113	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))
115		0le0 12294	. . . . . . . . 9 ⊢ 0 ≤ 0
116	115	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
117		iffalse 4500	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) = 0)
118	116, 117, 62	3brtr4d 5142	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
119	114, 118	pm2.61d1 180	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
120	119	ralrimivw 3130	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))
121		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
122		eqidd 2731	. . . . . 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)))
123	31, 28, 92, 121, 122	ofrfval2 7677	. . . . 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)))
124	120, 123	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)))
125		itg2le 25647	. . . 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))))
126	29, 93, 124, 125	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))))
127		itg2lecl 25646	. . 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)
128	29, 85, 126, 127	syl3anc 1373	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
129	20, 22	iblpos 25701	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)))
130	19, 128, 129	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ⊆ wss 3917 ifcif 4491 class class class wbr 5110 ↦ cmpt 5191 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ∘_r cofr 7655 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 ici 11077 + caddc 11078 · cmul 11080 +∞cpnf 11212 ℝ^*cxr 11214 ≤ cle 11216 [,)cico 13315 [,]cicc 13316 ℜcre 15070 ℑcim 15071 abscabs 15207 –cn→ccncf 24776 MblFncmbf 25522 ∫₂citg2 25524 𝐿¹cibl 25525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-ofr 7657 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-ovol 25372 df-vol 25373 df-mbf 25527 df-itg1 25528 df-itg2 25529 df-ibl 25530 df-0p 25578

This theorem is referenced by: iblmulc2 25739 itgabs 25743 bddmulibl 25747 itgcn 25753 ftc1a 25951 ftc1lem4 25953 itgulm 26324 fourierdlem47 46158 fourierdlem87 46198 etransclem23 46262

W3C validator