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Theorem iblmulc2 25788

Description: Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
itgmulc2.1	⊢ (𝜑 → 𝐶 ∈ ℂ)
itgmulc2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgmulc2.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

**Assertion**
Ref	Expression
iblmulc2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)

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

Proof of Theorem iblmulc2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgmulc2.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
2		itgmulc2.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3		itgmulc2.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
4		iblmbf 25724	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
6	1, 2, 5	mbfmulc2 25620	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
7		ifan 4533	. . . . . 6 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0)
8	1	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
9	5, 2	mbfmptcl 25593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	8, 9	mulcld 11152	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
11	10	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
12		ax-icn 11085	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
13		ine0 11572	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
14		elfzelz 13440	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
15	14	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
16		expclz 14007	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
17	12, 13, 15, 16	mp3an12i 1467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
18		expne0i 14017	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
19	12, 13, 15, 18	mp3an12i 1467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0)
20	11, 17, 19	divcld 11917	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((𝐶 · 𝐵) / (i↑𝑘)) ∈ ℂ)
21	20	recld 15117	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ)
22		0re 11134	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
23		ifcl 4525	. . . . . . . . . . 11 ⊢ (((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
24	21, 22, 23	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
25	24	rexrd 11182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^*)
26		max1 13100	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
27	22, 21, 26	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
28		elxrge0 13373	. . . . . . . . 9 ⊢ (if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞) ↔ (if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)))
29	25, 27, 28	sylanbrc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
30		0e0iccpnf 13375	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
32	29, 31	ifclda 4515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
33	32	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
34	7, 33	eqeltrid 2840	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
35	34	fmpttd 7060	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
36		reex 11117	. . . . . . . . . . 11 ⊢ ℝ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
38	1	abscld 15362	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝐶) ∈ ℝ)
40	9	abscld 15362	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
41	9	absge0d 15370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
42		elrege0 13370	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐵) ∈ (0[,)+∞) ↔ ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵)))
43	40, 41, 42	sylanbrc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,)+∞))
44		0e0icopnf 13374	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,)+∞)
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
46	43, 45	ifclda 4515	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,)+∞))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘𝐵), 0) ∈ (0[,)+∞))
48		fconstmpt 5686	. . . . . . . . . . 11 ⊢ (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶))
49	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶)))
50		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))
51	37, 39, 47, 49, 50	offval2 7642	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))))
52		ovif2 7457	. . . . . . . . . . 11 ⊢ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0))
53	8, 9	absmuld 15380	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
54	53	ifeq1da 4511	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)))
55	38	recnd 11160	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ)
56	55	mul01d 11332	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs‘𝐶) · 0) = 0)
57	56	ifeq2d 4500	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
58	54, 57	eqtr3d 2773	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
59	52, 58	eqtrid 2783	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
60	59	mpteq2dv 5192	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
61	51, 60	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
62	61	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (∫₂‘((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))))
63	47	fmpttd 7060	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,)+∞))
64	2, 3	iblabs 25786	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)
65	40, 41	iblpos 25750	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)))
66	64, 65	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ))
67	66	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0))) ∈ ℝ)
68		abscl 15201	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (abs‘𝐶) ∈ ℝ)
69		absge0 15210	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → 0 ≤ (abs‘𝐶))
70		elrege0 13370	. . . . . . . . . 10 ⊢ ((abs‘𝐶) ∈ (0[,)+∞) ↔ ((abs‘𝐶) ∈ ℝ ∧ 0 ≤ (abs‘𝐶)))
71	68, 69, 70	sylanbrc 583	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (abs‘𝐶) ∈ (0[,)+∞))
72	1, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ (0[,)+∞))
73	63, 67, 72	itg2mulc 25704	. . . . . . 7 ⊢ (𝜑 → (∫₂‘((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) = ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))))
74	62, 73	eqtr3d 2773	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))) = ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))))
75	38, 67	remulcld 11162	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) ∈ ℝ)
76	74, 75	eqeltrd 2836	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))) ∈ ℝ)
77	76	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))) ∈ ℝ)
78	10	abscld 15362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
79	78	rexrd 11182	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ^*)
80	10	absge0d 15370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(𝐶 · 𝐵)))
81		elxrge0 13373	. . . . . . . . . 10 ⊢ ((abs‘(𝐶 · 𝐵)) ∈ (0[,]+∞) ↔ ((abs‘(𝐶 · 𝐵)) ∈ ℝ^* ∧ 0 ≤ (abs‘(𝐶 · 𝐵))))
82	79, 80, 81	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ (0[,]+∞))
83	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
84	82, 83	ifclda 4515	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) ∈ (0[,]+∞))
85	84	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) ∈ (0[,]+∞))
86	85	fmpttd 7060	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)):ℝ⟶(0[,]+∞))
87	86	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)):ℝ⟶(0[,]+∞))
88	20	releabsd 15377	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘((𝐶 · 𝐵) / (i↑𝑘))))
89	11, 17, 19	absdivd 15381	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))))
90		elfznn0 13536	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
91	90	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ₀)
92		absexp 15227	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
93	12, 91, 92	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
94		absi 15209	. . . . . . . . . . . . . . . . . 18 ⊢ (abs‘i) = 1
95	94	oveq1i 7368	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
96		1exp 14014	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℤ → (1↑𝑘) = 1)
97	15, 96	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (1↑𝑘) = 1)
98	95, 97	eqtrid 2783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1)
99	93, 98	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1)
100	99	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
101	78	recnd 11160	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
102	101	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
103	102	div1d 11909	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / 1) = (abs‘(𝐶 · 𝐵)))
104	89, 100, 103	3eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = (abs‘(𝐶 · 𝐵)))
105	88, 104	breqtrd 5124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘(𝐶 · 𝐵)))
106	80	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(𝐶 · 𝐵)))
107		breq1 5101	. . . . . . . . . . . . 13 ⊢ ((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) → ((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘(𝐶 · 𝐵)) ↔ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ (abs‘(𝐶 · 𝐵))))
108		breq1 5101	. . . . . . . . . . . . 13 ⊢ (0 = if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) → (0 ≤ (abs‘(𝐶 · 𝐵)) ↔ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ (abs‘(𝐶 · 𝐵))))
109	107, 108	ifboth 4519	. . . . . . . . . . . 12 ⊢ (((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘(𝐶 · 𝐵)) ∧ 0 ≤ (abs‘(𝐶 · 𝐵))) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ (abs‘(𝐶 · 𝐵)))
110	105, 106, 109	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ (abs‘(𝐶 · 𝐵)))
111		iftrue 4485	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
112	111	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
113		iftrue 4485	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) = (abs‘(𝐶 · 𝐵)))
114	113	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) = (abs‘(𝐶 · 𝐵)))
115	110, 112, 114	3brtr4d 5130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
116	115	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
117		0le0 12246	. . . . . . . . . . 11 ⊢ 0 ≤ 0
118	117	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
119		iffalse 4488	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = 0)
120		iffalse 4488	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) = 0)
121	118, 119, 120	3brtr4d 5130	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
122	116, 121	pm2.61d1 180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
123	7, 122	eqbrtrid 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
124	123	ralrimivw 3132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
125	36	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈ V)
126	85	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) ∈ (0[,]+∞))
127		eqidd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)))
128		eqidd 2737	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
129	125, 34, 126, 127, 128	ofrfval2 7643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
130	124, 129	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))
131		itg2le 25696	. . . . 5 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))))
132	35, 87, 130, 131	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))))
133		itg2lecl 25695	. . . 4 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)
134	35, 77, 132, 133	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)
135	134	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)
136		eqidd 2737	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)))
137		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))))
138	136, 137, 10	isibl2 25723	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)))
139	6, 135, 138	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 Vcvv 3440 ifcif 4479 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘_f cof 7620 ∘_r cofr 7621 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 ici 11028 · cmul 11031 +∞cpnf 11163 ℝ^*cxr 11165 ≤ cle 11167 / cdiv 11794 3c3 12201 ℕ₀cn0 12401 ℤcz 12488 [,)cico 13263 [,]cicc 13264 ...cfz 13423 ↑cexp 13984 ℜcre 15020 abscabs 15157 MblFncmbf 25571 ∫₂citg2 25573 𝐿¹cibl 25574

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 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

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-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 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-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cn 23171 df-cnp 23172 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-0p 25627

This theorem is referenced by: itgmulc2lem1 25789 itgmulc2lem2 25790 itgmulc2 25791 itgabs 25792 circlemeth 34797 3factsumint1 42275 fourierdlem83 46433 fourierdlem95 46445 sqwvfoura 46472 sqwvfourb 46473

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