iblmulc2nc - Mathbox for Brendan Leahy

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Theorem iblmulc2nc 35579

Description: Choice-free analogue of iblmulc2 24728. (Contributed by Brendan Leahy, 17-Nov-2017.)

**Hypotheses**
Ref	Expression
itgmulc2nc.1	⊢ (𝜑 → 𝐶 ∈ ℂ)
itgmulc2nc.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgmulc2nc.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
itgmulc2nc.m	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)

**Assertion**
Ref	Expression
iblmulc2nc	⊢ (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 iblmulc2nc Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgmulc2nc.m	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
2		ifan 4492	. . . . . 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)
3		itgmulc2nc.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4	3	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
5		itgmulc2nc.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
6		iblmbf 24665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
7	5, 6	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
8		itgmulc2nc.2	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
9	7, 8	mbfmptcl 24533	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	4, 9	mulcld 10853	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
11	10	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
12		elfzelz 13112	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
13	12	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
14		ax-icn 10788	. . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ
15		ine0 11267	. . . . . . . . . . . . . . 15 ⊢ i ≠ 0
16		expclz 13660	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
17	14, 15, 16	mp3an12 1453	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
18	13, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
19		expne0i 13667	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
20	14, 15, 19	mp3an12 1453	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
21	13, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ≠ 0)
22	11, 18, 21	divcld 11608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((𝐶 · 𝐵) / (i↑𝑘)) ∈ ℂ)
23	22	recld 14757	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ)
24		0re 10835	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
25		ifcl 4484	. . . . . . . . . . 11 ⊢ (((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
26	23, 24, 25	sylancl 589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
27	26	rexrd 10883	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^*)
28		max1 12775	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
29	24, 23, 28	sylancr 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
30		elxrge0 13045	. . . . . . . . 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)))
31	27, 29, 30	sylanbrc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
32		0e0iccpnf 13047	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
33	32	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
34	31, 33	ifclda 4474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
35	34	adantr 484	. . . . . 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[,]+∞))
36	2, 35	eqeltrid 2842	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
37	36	fmpttd 6932	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
38	9	recld 14757	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
39	38	recnd 10861	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
40	39	abscld 15000	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ ℝ)
41	9	imcld 14758	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
42	41	recnd 10861	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
43	42	abscld 15000	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
44	40, 43	readdcld 10862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ)
45	39	absge0d 15008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℜ‘𝐵)))
46	42	absge0d 15008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℑ‘𝐵)))
47	40, 43, 45, 46	addge0d 11408	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
48		elrege0 13042	. . . . . . . . . . . 12 ⊢ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,)+∞) ↔ (((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ ∧ 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
49	44, 47, 48	sylanbrc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,)+∞))
50		0e0icopnf 13046	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,)+∞)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
52	49, 51	ifclda 4474	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,)+∞))
53	52	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,)+∞))
54	53	fmpttd 6932	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,)+∞))
55		reex 10820	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
57		elrege0 13042	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘(ℜ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘𝐵))))
58	40, 45, 57	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ (0[,)+∞))
59	58, 51	ifclda 4474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
60	59	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
61		elrege0 13042	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘(ℑ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘𝐵))))
62	43, 46, 61	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ (0[,)+∞))
63	62, 51	ifclda 4474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
64	63	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
65		eqidd 2738	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)))
66		eqidd 2738	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))
67	56, 60, 64, 65, 66	offval2 7488	. . . . . . . . . . . 12 ⊢ (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))))
68		iftrue 4445	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = (abs‘(ℜ‘𝐵)))
69		iftrue 4445	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = (abs‘(ℑ‘𝐵)))
70	68, 69	oveq12d 7231	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
71		iftrue 4445	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
72	70, 71	eqtr4d 2780	. . . . . . . . . . . . . 14 ⊢ (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))
73		00id 11007	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
74		iffalse 4448	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = 0)
75		iffalse 4448	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = 0)
76	74, 75	oveq12d 7231	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (0 + 0))
77		iffalse 4448	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
78	73, 76, 77	3eqtr4a 2804	. . . . . . . . . . . . . 14 ⊢ (¬ 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))
79	72, 78	pm2.61i 185	. . . . . . . . . . . . 13 ⊢ (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)
80	79	mpteq2i 5147	. . . . . . . . . . . 12 ⊢ (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))
81	67, 80	eqtr2di 2795	. . . . . . . . . . 11 ⊢ (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))))
82	81	fveq2d 6721	. . . . . . . . . 10 ⊢ (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)))))
83		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))
84	9	iblcn 24696	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
85	5, 84	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
86	85	simpld 498	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
87	8, 5, 83, 86, 38	iblabsnclem 35577	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ))
88	87	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn)
89	60	fmpttd 6932	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
90	87	simprd 499	. . . . . . . . . . 11 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ)
91	64	fmpttd 6932	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
92		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))
93	85	simprd 499	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
94	8, 5, 92, 93, 41	iblabsnclem 35577	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) ∈ MblFn ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ))
95	94	simprd 499	. . . . . . . . . . 11 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ)
96	88, 89, 90, 91, 95	itg2addnc 35568	. . . . . . . . . 10 ⊢ (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)))))
97	82, 96	eqtrd 2777	. . . . . . . . 9 ⊢ (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)))))
98	90, 95	readdcld 10862	. . . . . . . . 9 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)))) ∈ ℝ)
99	97, 98	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ)
100	3	abscld 15000	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
101	3	absge0d 15008	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (abs‘𝐶))
102		elrege0 13042	. . . . . . . . 9 ⊢ ((abs‘𝐶) ∈ (0[,)+∞) ↔ ((abs‘𝐶) ∈ ℝ ∧ 0 ≤ (abs‘𝐶)))
103	100, 101, 102	sylanbrc 586	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ (0[,)+∞))
104	54, 99, 103	itg2mulc 24645	. . . . . . 7 ⊢ (𝜑 → (∫₂‘((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))) = ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))))
105	100	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝐶) ∈ ℝ)
106		fconstmpt 5611	. . . . . . . . . . 11 ⊢ (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶))
107	106	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶)))
108		eqidd 2738	. . . . . . . . . 10 ⊢ (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)))
109	56, 105, 53, 107, 108	offval2 7488	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) = (𝑥 ∈ ℝ ↦ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))))
110	71	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
111		iftrue 4445	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
112	110, 111	eqtr4d 2780	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
113	112	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
114	100	recnd 10861	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ)
115	114	mul01d 11031	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs‘𝐶) · 0) = 0)
116	115	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · 0) = 0)
117	77	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
118	117	oveq2d 7229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = ((abs‘𝐶) · 0))
119		iffalse 4448	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = 0)
120	119	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = 0)
121	116, 118, 120	3eqtr4d 2787	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
122	113, 121	pm2.61dan 813	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
123	122	mpteq2dv 5151	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
124	109, 123	eqtrd 2777	. . . . . . . 8 ⊢ (𝜑 → ((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
125	124	fveq2d 6721	. . . . . . 7 ⊢ (𝜑 → (∫₂‘((ℝ × {(abs‘𝐶)}) ∘_f · (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))) = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))))
126	97	oveq2d 7229	. . . . . . 7 ⊢ (𝜑 → ((abs‘𝐶) · (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)))) = ((abs‘𝐶) · ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))))))
127	104, 125, 126	3eqtr3d 2785	. . . . . 6 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))) = ((abs‘𝐶) · ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))))))
128	100, 98	remulcld 10863	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶) · ((∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))))) ∈ ℝ)
129	127, 128	eqeltrd 2838	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))) ∈ ℝ)
130	129	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))) ∈ ℝ)
131	100	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐶) ∈ ℝ)
132	131, 44	remulcld 10863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ)
133	132	rexrd 10883	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ^*)
134	101	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐶))
135	131, 44, 134, 47	mulge0d 11409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
136		elxrge0 13045	. . . . . . . . 9 ⊢ (((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ (0[,]+∞) ↔ (((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ^* ∧ 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))))
137	133, 135, 136	sylanbrc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ (0[,]+∞))
138	32	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
139	137, 138	ifclda 4474	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) ∈ (0[,]+∞))
140	139	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) ∈ (0[,]+∞))
141	140	fmpttd 6932	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)):ℝ⟶(0[,]+∞))
142	9	abscld 15000	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
143	131, 142	remulcld 10863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ∈ ℝ)
144	143	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ∈ ℝ)
145	132	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ)
146	22	releabsd 15015	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘((𝐶 · 𝐵) / (i↑𝑘))))
147	11, 18, 21	absdivd 15019	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))))
148		elfznn0 13205	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
149		absexp 14868	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
150	14, 148, 149	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...3) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
151		absi 14850	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs‘i) = 1
152	151	oveq1i 7223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
153		1exp 13664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℤ → (1↑𝑘) = 1)
154	12, 153	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...3) → (1↑𝑘) = 1)
155	152, 154	syl5eq 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...3) → ((abs‘i)↑𝑘) = 1)
156	150, 155	eqtrd 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → (abs‘(i↑𝑘)) = 1)
157	156	oveq2d 7229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...3) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
158	157	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
159	10	abscld 15000	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
160	159	recnd 10861	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
161	160	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
162	161	div1d 11600	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / 1) = (abs‘(𝐶 · 𝐵)))
163	147, 158, 162	3eqtrd 2781	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = (abs‘(𝐶 · 𝐵)))
164	4, 9	absmuld 15018	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
165	164	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
166	163, 165	eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘𝐶) · (abs‘𝐵)))
167	146, 166	breqtrd 5079	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ ((abs‘𝐶) · (abs‘𝐵)))
168		mulcl 10813	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
169	14, 42, 168	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (i · (ℑ‘𝐵)) ∈ ℂ)
170	39, 169	abstrid 15020	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
171	9	replimd 14760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
172	171	fveq2d 6721	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
173		absmul 14858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
174	14, 42, 173	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
175	151	oveq1i 7223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘i) · (abs‘(ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵)))
176	174, 175	eqtrdi 2794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵))))
177	43	recnd 10861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℂ)
178	177	mulid2d 10851	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
179	176, 178	eqtr2d 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) = (abs‘(i · (ℑ‘𝐵))))
180	179	oveq2d 7229	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
181	170, 172, 180	3brtr4d 5085	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
182	142, 44, 131, 134, 181	lemul2ad 11772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
183	182	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
184	23, 144, 145, 167, 183	letrd 10989	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
185	135	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
186		breq1 5056	. . . . . . . . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ↔ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))))
187		breq1 5056	. . . . . . . . . . . . 13 ⊢ (0 = if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) → (0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ↔ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))))
188	186, 187	ifboth 4478	. . . . . . . . . . . 12 ⊢ (((ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∧ 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
189	184, 185, 188	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
190		iftrue 4445	. . . . . . . . . . . 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))
191	190	adantl 485	. . . . . . . . . . 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))
192	111	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
193	189, 191, 192	3brtr4d 5085	. . . . . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
194	193	ex 416	. . . . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
195		0le0 11931	. . . . . . . . . . 11 ⊢ 0 ≤ 0
196	195	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
197		iffalse 4448	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0), 0) = 0)
198	196, 197, 119	3brtr4d 5085	. . . . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
199	194, 198	pm2.61d1 183	. . . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
200	2, 199	eqbrtrid 5088	. . . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
201	200	ralrimivw 3106	. . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))
202	55	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ℝ ∈ V)
203		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)))
204		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
205	202, 36, 140, 203, 204	ofrfval2 7489	. . . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ≤ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
206	201, 205	mpbird 260	. . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))
207		itg2le 24637	. . . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))))
208	37, 141, 206, 207	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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))))
209		itg2lecl 24636	. . . 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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘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) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)
210	37, 130, 208, 209	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)
211	210	ralrimiva 3105	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘)))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))) ∈ ℝ)
212		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)))
213		eqidd 2738	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))))
214	212, 213, 10	isibl2 24664	. 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)))
215	1, 211, 214	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 Vcvv 3408 ifcif 4439 {csn 4541 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ∘_f cof 7467 ∘_r cofr 7468 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 ici 10731 + caddc 10732 · cmul 10734 +∞cpnf 10864 ℝ^*cxr 10866 ≤ cle 10868 / cdiv 11489 3c3 11886 ℕ₀cn0 12090 ℤcz 12176 [,)cico 12937 [,]cicc 12938 ...cfz 13095 ↑cexp 13635 ℜcre 14660 ℑcim 14661 abscabs 14797 MblFncmbf 24511 ∫₂citg2 24513 𝐿¹cibl 24514

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-cmp 22284 df-ovol 24361 df-vol 24362 df-mbf 24516 df-itg1 24517 df-itg2 24518 df-ibl 24519 df-0p 24567

This theorem is referenced by: itgmulc2nclem1 35580 itgmulc2nclem2 35581 itgmulc2nc 35582 itgabsnc 35583 ftc1anclem6 35592

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