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Theorem iblmulc2 25730

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 25666	. . . 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 25562	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
7		ifan 4530	. . . . . 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 25535	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	8, 9	mulcld 11135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
11	10	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
12		ax-icn 11068	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
13		ine0 11555	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
14		elfzelz 13427	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
15	14	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
16		expclz 13991	. . . . . . . . . . . . . 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 14001	. . . . . . . . . . . . . 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 11900	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((𝐶 · 𝐵) / (i↑𝑘)) ∈ ℂ)
21	20	recld 15101	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ)
22		0re 11117	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
23		ifcl 4522	. . . . . . . . . . 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 11165	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^*)
26		max1 13087	. . . . . . . . . 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 13360	. . . . . . . . 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 13362	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
32	29, 31	ifclda 4512	. . . . . . 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 2832	. . . . 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 7049	. . . 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 11100	. . . . . . . . . . 11 ⊢ ℝ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
38	1	abscld 15346	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝐶) ∈ ℝ)
40	9	abscld 15346	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
41	9	absge0d 15354	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
42		elrege0 13357	. . . . . . . . . . . . 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 13361	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,)+∞)
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
46	43, 45	ifclda 4512	. . . . . . . . . . 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 5681	. . . . . . . . . . 11 ⊢ (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶))
49	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶)))
50		eqidd 2730	. . . . . . . . . 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 7633	. . . . . . . . 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 7448	. . . . . . . . . . 11 ⊢ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0))
53	8, 9	absmuld 15364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
54	53	ifeq1da 4508	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)))
55	38	recnd 11143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ)
56	55	mul01d 11315	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs‘𝐶) · 0) = 0)
57	56	ifeq2d 4497	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
58	54, 57	eqtr3d 2766	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
59	52, 58	eqtrid 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
60	59	mpteq2dv 5186	. . . . . . . . 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 2764	. . . . . . . 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 6826	. . . . . . 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 7049	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,)+∞))
64	2, 3	iblabs 25728	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)
65	40, 41	iblpos 25692	. . . . . . . . . 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 15185	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (abs‘𝐶) ∈ ℝ)
69		absge0 15194	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → 0 ≤ (abs‘𝐶))
70		elrege0 13357	. . . . . . . . . 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 25646	. . . . . . 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 2766	. . . . . 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 11145	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) ∈ ℝ)
76	74, 75	eqeltrd 2828	. . . . 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 15346	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
79	78	rexrd 11165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ^*)
80	10	absge0d 15354	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(𝐶 · 𝐵)))
81		elxrge0 13360	. . . . . . . . . 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 4512	. . . . . . . 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 7049	. . . . . 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 15361	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘((𝐶 · 𝐵) / (i↑𝑘))))
89	11, 17, 19	absdivd 15365	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))))
90		elfznn0 13523	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
91	90	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ₀)
92		absexp 15211	. . . . . . . . . . . . . . . . 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 15193	. . . . . . . . . . . . . . . . . 18 ⊢ (abs‘i) = 1
95	94	oveq1i 7359	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
96		1exp 13998	. . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1)
99	93, 98	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1)
100	99	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
101	78	recnd 11143	. . . . . . . . . . . . . . . 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 11892	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / 1) = (abs‘(𝐶 · 𝐵)))
104	89, 100, 103	3eqtrd 2768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = (abs‘(𝐶 · 𝐵)))
105	88, 104	breqtrd 5118	. . . . . . . . . . . 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 5095	. . . . . . . . . . . . 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 5095	. . . . . . . . . . . . 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 4516	. . . . . . . . . . . 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 4482	. . . . . . . . . . . 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 4482	. . . . . . . . . . . 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 5124	. . . . . . . . . 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 12229	. . . . . . . . . . 11 ⊢ 0 ≤ 0
118	117	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
119		iffalse 4485	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = 0)
120		iffalse 4485	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) = 0)
121	118, 119, 120	3brtr4d 5124	. . . . . . . . 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 5127	. . . . . . 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 3125	. . . . . 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 2730	. . . . . . 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 2730	. . . . . . 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 7634	. . . . . 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 25638	. . . . 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 25637	. . . 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 3121	. 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 2730	. . 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 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))))
138	136, 137, 10	isibl2 25665	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3436 ifcif 4476 {csn 4577 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∘_f cof 7611 ∘_r cofr 7612 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 ici 11011 · cmul 11014 +∞cpnf 11146 ℝ^*cxr 11148 ≤ cle 11150 / cdiv 11777 3c3 12184 ℕ₀cn0 12384 ℤcz 12471 [,)cico 13250 [,]cicc 13251 ...cfz 13410 ↑cexp 13968 ℜcre 15004 abscabs 15141 MblFncmbf 25513 ∫₂citg2 25515 𝐿¹cibl 25516

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cn 23112 df-cnp 23113 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-0p 25569

This theorem is referenced by: itgmulc2lem1 25731 itgmulc2lem2 25732 itgmulc2 25733 itgabs 25734 circlemeth 34608 3factsumint1 41994 fourierdlem83 46170 fourierdlem95 46182 sqwvfoura 46209 sqwvfourb 46210

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