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Theorem iblmulc2 25800

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 25736	. . . 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 25632	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
7		ifan 4535	. . . . . 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 25605	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	8, 9	mulcld 11164	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
11	10	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
12		ax-icn 11097	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
13		ine0 11584	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
14		elfzelz 13452	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
15	14	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
16		expclz 14019	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
17	12, 13, 15, 16	mp3an12i 1468	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
18		expne0i 14029	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
19	12, 13, 15, 18	mp3an12i 1468	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0)
20	11, 17, 19	divcld 11929	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((𝐶 · 𝐵) / (i↑𝑘)) ∈ ℂ)
21	20	recld 15129	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ)
22		0re 11146	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
23		ifcl 4527	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
25	24	rexrd 11194	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^*)
26		max1 13112	. . . . . . . . . 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 588	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
28		elxrge0 13385	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
30		0e0iccpnf 13387	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
32	29, 31	ifclda 4517	. . . . . . 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 2841	. . . . 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 7069	. . . 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 11129	. . . . . . . . . . 11 ⊢ ℝ ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ V)
38	1	abscld 15374	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝐶) ∈ ℝ)
40	9	abscld 15374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
41	9	absge0d 15382	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐵))
42		elrege0 13382	. . . . . . . . . . . . 13 ⊢ ((abs‘𝐵) ∈ (0[,)+∞) ↔ ((abs‘𝐵) ∈ ℝ ∧ 0 ≤ (abs‘𝐵)))
43	40, 41, 42	sylanbrc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ (0[,)+∞))
44		0e0icopnf 13386	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,)+∞)
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
46	43, 45	ifclda 4517	. . . . . . . . . . 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 5694	. . . . . . . . . . 11 ⊢ (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶))
49	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶)))
50		eqidd 2738	. . . . . . . . . 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 7652	. . . . . . . . 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 7467	. . . . . . . . . . 11 ⊢ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0))
53	8, 9	absmuld 15392	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
54	53	ifeq1da 4513	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)))
55	38	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ)
56	55	mul01d 11344	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs‘𝐶) · 0) = 0)
57	56	ifeq2d 4502	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
58	54, 57	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · (abs‘𝐵)), ((abs‘𝐶) · 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
59	52, 58	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)) = if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0))
60	59	mpteq2dv 5194	. . . . . . . . 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 2772	. . . . . . . 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 6846	. . . . . . 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 7069	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)):ℝ⟶(0[,)+∞))
64	2, 3	iblabs 25798	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿¹)
65	40, 41	iblpos 25762	. . . . . . . . . 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 15213	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → (abs‘𝐶) ∈ ℝ)
69		absge0 15222	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℂ → 0 ≤ (abs‘𝐶))
70		elrege0 13382	. . . . . . . . . 10 ⊢ ((abs‘𝐶) ∈ (0[,)+∞) ↔ ((abs‘𝐶) ∈ ℝ ∧ 0 ≤ (abs‘𝐶)))
71	68, 69, 70	sylanbrc 584	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (abs‘𝐶) ∈ (0[,)+∞))
72	1, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ (0[,)+∞))
73	63, 67, 72	itg2mulc 25716	. . . . . . 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 2774	. . . . . 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 11174	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘𝐵), 0)))) ∈ ℝ)
76	74, 75	eqeltrd 2837	. . . . 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 15374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
79	78	rexrd 11194	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ^*)
80	10	absge0d 15382	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(𝐶 · 𝐵)))
81		elxrge0 13385	. . . . . . . . . 10 ⊢ ((abs‘(𝐶 · 𝐵)) ∈ (0[,]+∞) ↔ ((abs‘(𝐶 · 𝐵)) ∈ ℝ^* ∧ 0 ≤ (abs‘(𝐶 · 𝐵))))
82	79, 80, 81	sylanbrc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ (0[,]+∞))
83	30	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
84	82, 83	ifclda 4517	. . . . . . . 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 7069	. . . . . 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 15389	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘((𝐶 · 𝐵) / (i↑𝑘))))
89	11, 17, 19	absdivd 15393	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))))
90		elfznn0 13548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
91	90	ad2antlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℕ₀)
92		absexp 15239	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
93	12, 91, 92	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
94		absi 15221	. . . . . . . . . . . . . . . . . 18 ⊢ (abs‘i) = 1
95	94	oveq1i 7378	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
96		1exp 14026	. . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘i)↑𝑘) = 1)
99	93, 98	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(i↑𝑘)) = 1)
100	99	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
101	78	recnd 11172	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
102	101	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
103	102	div1d 11921	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / 1) = (abs‘(𝐶 · 𝐵)))
104	89, 100, 103	3eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = (abs‘(𝐶 · 𝐵)))
105	88, 104	breqtrd 5126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘(𝐶 · 𝐵)))
106	80	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(𝐶 · 𝐵)))
107		breq1 5103	. . . . . . . . . . . . 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 5103	. . . . . . . . . . . . 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 4521	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 4487	. . . . . . . . . . . 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 4487	. . . . . . . . . . . 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 5132	. . . . . . . . . 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 12258	. . . . . . . . . . 11 ⊢ 0 ≤ 0
118	117	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
119		iffalse 4490	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = 0)
120		iffalse 4490	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) = 0)
121	118, 119, 120	3brtr4d 5132	. . . . . . . . 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 5135	. . . . . . 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 3134	. . . . . 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 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(𝐶 · 𝐵)), 0) ∈ (0[,]+∞))
127		eqidd 2738	. . . . . . 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 2738	. . . . . . 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 7653	. . . . . 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 25708	. . . . 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 1374	. . . 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 25707	. . . 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 1374	. . 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 3130	. 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 2738	. . 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 2738	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))))
138	136, 137, 10	isibl2 25735	. 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 714	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3442 ifcif 4481 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 × cxp 5630 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∘_f cof 7630 ∘_r cofr 7631 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 · cmul 11043 +∞cpnf 11175 ℝ^*cxr 11177 ≤ cle 11179 / cdiv 11806 3c3 12213 ℕ₀cn0 12413 ℤcz 12500 [,)cico 13275 [,]cicc 13276 ...cfz 13435 ↑cexp 13996 ℜcre 15032 abscabs 15169 MblFncmbf 25583 ∫₂citg2 25585 𝐿¹cibl 25586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cn 23183 df-cnp 23184 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-ovol 25433 df-vol 25434 df-mbf 25588 df-itg1 25589 df-itg2 25590 df-ibl 25591 df-0p 25639

This theorem is referenced by: itgmulc2lem1 25801 itgmulc2lem2 25802 itgmulc2 25803 itgabs 25804 circlemeth 34818 3factsumint1 42391 fourierdlem83 46547 fourierdlem95 46559 sqwvfoura 46586 sqwvfourb 46587

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