iblmulc2nc - Mathbox for Brendan Leahy

	Mathbox for Brendan Leahy		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iblmulc2nc		Structured version Visualization version GIF version

Theorem iblmulc2nc 33830

Description: Choice-free analogue of iblmulc2 23887. (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 4293	. . . . . 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 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
5		itgmulc2nc.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
6		iblmbf 23824	. . . . . . . . . . . . . . . . 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 23693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	4, 9	mulcld 10313	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
11	10	adantlr 706	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
12		elfzelz 12548	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
13	12	ad2antlr 718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 𝑘 ∈ ℤ)
14		ax-icn 10247	. . . . . . . . . . . . . . 15 ⊢ i ∈ ℂ
15		ine0 10718	. . . . . . . . . . . . . . 15 ⊢ i ≠ 0
16		expclz 13091	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
17	14, 15, 16	mp3an12 1575	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
18	13, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (i↑𝑘) ∈ ℂ)
19		expne0i 13098	. . . . . . . . . . . . . . 15 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
20	14, 15, 19	mp3an12 1575	. . . . . . . . . . . . . 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 11054	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((𝐶 · 𝐵) / (i↑𝑘)) ∈ ℂ)
23	22	recld 14220	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ∈ ℝ)
24		0re 10294	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
25		ifcl 4286	. . . . . . . . . . 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 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ)
27	26	rexrd 10342	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ ℝ^*)
28		max1 12217	. . . . . . . . . 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 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0))
30		elxrge0 12484	. . . . . . . . 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 578	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))), 0) ∈ (0[,]+∞))
32		0e0iccpnf 12486	. . . . . . . . 9 ⊢ 0 ∈ (0[,]+∞)
33	32	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
34	31, 33	ifclda 4276	. . . . . . 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 472	. . . . . 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	syl5eqel 2847	. . . . 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 6574	. . . 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 14220	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
39	38	recnd 10321	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
40	39	abscld 14461	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ ℝ)
41	9	imcld 14221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
42	41	recnd 10321	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
43	42	abscld 14461	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℝ)
44	40, 43	readdcld 10322	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ ℝ)
45	39	absge0d 14469	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℜ‘𝐵)))
46	42	absge0d 14469	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘(ℑ‘𝐵)))
47	40, 43, 45, 46	addge0d 10856	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
48		elrege0 12481	. . . . . . . . . . . 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 578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) ∈ (0[,)+∞))
50		0e0icopnf 12485	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,)+∞)
51	50	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
52	49, 51	ifclda 4276	. . . . . . . . . 10 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,)+∞))
53	52	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) ∈ (0[,)+∞))
54	53	fmpttd 6574	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)):ℝ⟶(0[,)+∞))
55		reex 10279	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
56	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
57		elrege0 12481	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘(ℜ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℜ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℜ‘𝐵))))
58	40, 45, 57	sylanbrc 578	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℜ‘𝐵)) ∈ (0[,)+∞))
59	58, 51	ifclda 4276	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
60	59	adantr 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) ∈ (0[,)+∞))
61		elrege0 12481	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘(ℑ‘𝐵)) ∈ (0[,)+∞) ↔ ((abs‘(ℑ‘𝐵)) ∈ ℝ ∧ 0 ≤ (abs‘(ℑ‘𝐵))))
62	43, 46, 61	sylanbrc 578	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ (0[,)+∞))
63	62, 51	ifclda 4276	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
64	63	adantr 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) ∈ (0[,)+∞))
65		eqidd 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)))
66		eqidd 2765	. . . . . . . . . . . . 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 7111	. . . . . . . . . . . 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 4248	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = (abs‘(ℜ‘𝐵)))
69		iftrue 4248	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = (abs‘(ℑ‘𝐵)))
70	68, 69	oveq12d 6859	. . . . . . . . . . . . . . 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 4248	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
72	70, 71	eqtr4d 2801	. . . . . . . . . . . . . 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 10464	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
74		iffalse 4251	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) = 0)
75		iffalse 4251	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0) = 0)
76	74, 75	oveq12d 6859	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ 𝐴 → (if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0) + if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (0 + 0))
77		iffalse 4251	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
78	73, 76, 77	3eqtr4a 2824	. . . . . . . . . . . . . 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 176	. . . . . . . . . . . . 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 4899	. . . . . . . . . . . 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	syl6req 2815	. . . . . . . . . . 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 6378	. . . . . . . . . 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 2764	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))
84	9	iblcn 23855	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
85	5, 84	mpbid 223	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
86	85	simpld 488	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
87	8, 5, 83, 86, 38	iblabsnclem 33828	. . . . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)) ∈ MblFn)
89	60	fmpttd 6574	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
90	87	simprd 489	. . . . . . . . . . 11 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℜ‘𝐵)), 0))) ∈ ℝ)
91	64	fmpttd 6574	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)):ℝ⟶(0[,)+∞))
92		eqid 2764	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))
93	85	simprd 489	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
94	8, 5, 92, 93, 41	iblabsnclem 33828	. . . . . . . . . . . 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 489	. . . . . . . . . . 11 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(ℑ‘𝐵)), 0))) ∈ ℝ)
96	88, 89, 90, 91, 95	itg2addnc 33819	. . . . . . . . . 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 2798	. . . . . . . . 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 10322	. . . . . . . . 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 2843	. . . . . . . 8 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0))) ∈ ℝ)
100	3	abscld 14461	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
101	3	absge0d 14469	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (abs‘𝐶))
102		elrege0 12481	. . . . . . . . 9 ⊢ ((abs‘𝐶) ∈ (0[,)+∞) ↔ ((abs‘𝐶) ∈ ℝ ∧ 0 ≤ (abs‘𝐶)))
103	100, 101, 102	sylanbrc 578	. . . . . . . 8 ⊢ (𝜑 → (abs‘𝐶) ∈ (0[,)+∞))
104	54, 99, 103	itg2mulc 23804	. . . . . . 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 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (abs‘𝐶) ∈ ℝ)
106		fconstmpt 5332	. . . . . . . . . . 11 ⊢ (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶))
107	106	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ × {(abs‘𝐶)}) = (𝑥 ∈ ℝ ↦ (abs‘𝐶)))
108		eqidd 2765	. . . . . . . . . 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 7111	. . . . . . . . 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 6857	. . . . . . . . . . . . 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 4248	. . . . . . . . . . . . 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 2801	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 10321	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (abs‘𝐶) ∈ ℂ)
115	114	mul01d 10488	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((abs‘𝐶) · 0) = 0)
116	115	adantr 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · 0) = 0)
117	77	adantl 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0) = 0)
118	117	oveq2d 6857	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · if(𝑥 ∈ 𝐴, ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))), 0)) = ((abs‘𝐶) · 0))
119		iffalse 4251	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = 0)
120	119	adantl 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) = 0)
121	116, 118, 120	3eqtr4d 2808	. . . . . . . . . . 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 847	. . . . . . . . . 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 4903	. . . . . . . . 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 2798	. . . . . . . 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 6378	. . . . . . 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 6857	. . . . . . 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 2806	. . . . . 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 10323	. . . . . 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 2843	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))) ∈ ℝ)
130	129	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0))) ∈ ℝ)
131	100	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐶) ∈ ℝ)
132	131, 44	remulcld 10323	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ)
133	132	rexrd 10342	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ^*)
134	101	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (abs‘𝐶))
135	131, 44, 134, 47	mulge0d 10857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
136		elxrge0 12484	. . . . . . . . 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 578	. . . . . . . 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 4276	. . . . . . 7 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) ∈ (0[,]+∞))
140	139	ad2antrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0) ∈ (0[,]+∞))
141	140	fmpttd 6574	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))), 0)):ℝ⟶(0[,]+∞))
142	9	abscld 14461	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ ℝ)
143	131, 142	remulcld 10323	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ∈ ℝ)
144	143	adantlr 706	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ∈ ℝ)
145	132	adantlr 706	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))) ∈ ℝ)
146	22	releabsd 14476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ (abs‘((𝐶 · 𝐵) / (i↑𝑘))))
147	11, 18, 21	absdivd 14480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))))
148		elfznn0 12639	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
149		absexp 14330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
150	14, 148, 149	sylancr 581	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...3) → (abs‘(i↑𝑘)) = ((abs‘i)↑𝑘))
151		absi 14312	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs‘i) = 1
152	151	oveq1i 6851	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((abs‘i)↑𝑘) = (1↑𝑘)
153		1exp 13095	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℤ → (1↑𝑘) = 1)
154	12, 153	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...3) → (1↑𝑘) = 1)
155	152, 154	syl5eq 2810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...3) → ((abs‘i)↑𝑘) = 1)
156	150, 155	eqtrd 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → (abs‘(i↑𝑘)) = 1)
157	156	oveq2d 6857	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...3) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
158	157	ad2antlr 718	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / (abs‘(i↑𝑘))) = ((abs‘(𝐶 · 𝐵)) / 1))
159	10	abscld 14461	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
160	159	recnd 10321	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
161	160	adantlr 706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) ∈ ℂ)
162	161	div1d 11046	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → ((abs‘(𝐶 · 𝐵)) / 1) = (abs‘(𝐶 · 𝐵)))
163	147, 158, 162	3eqtrd 2802	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = (abs‘(𝐶 · 𝐵)))
164	4, 9	absmuld 14479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
165	164	adantlr 706	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
166	163, 165	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (abs‘((𝐶 · 𝐵) / (i↑𝑘))) = ((abs‘𝐶) · (abs‘𝐵)))
167	146, 166	breqtrd 4834	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ ((abs‘𝐶) · (abs‘𝐵)))
168		mulcl 10272	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
169	14, 42, 168	sylancr 581	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (i · (ℑ‘𝐵)) ∈ ℂ)
170	39, 169	abstrid 14481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
171	9	replimd 14223	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
172	171	fveq2d 6378	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) = (abs‘((ℜ‘𝐵) + (i · (ℑ‘𝐵)))))
173		absmul 14320	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
174	14, 42, 173	sylancr 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = ((abs‘i) · (abs‘(ℑ‘𝐵))))
175	151	oveq1i 6851	. . . . . . . . . . . . . . . . . . 19 ⊢ ((abs‘i) · (abs‘(ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵)))
176	174, 175	syl6eq 2814	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(i · (ℑ‘𝐵))) = (1 · (abs‘(ℑ‘𝐵))))
177	43	recnd 10321	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) ∈ ℂ)
178	177	mulid2d 10311	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1 · (abs‘(ℑ‘𝐵))) = (abs‘(ℑ‘𝐵)))
179	176, 178	eqtr2d 2799	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘(ℑ‘𝐵)) = (abs‘(i · (ℑ‘𝐵))))
180	179	oveq2d 6857	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))) = ((abs‘(ℜ‘𝐵)) + (abs‘(i · (ℑ‘𝐵)))))
181	170, 172, 180	3brtr4d 4840	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ≤ ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵))))
182	142, 44, 131, 134, 181	lemul2ad 11217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((abs‘𝐶) · (abs‘𝐵)) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
183	182	adantlr 706	. . . . . . . . . . . . 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 10447	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
185	135	adantlr 706	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐴) → 0 ≤ ((abs‘𝐶) · ((abs‘(ℜ‘𝐵)) + (abs‘(ℑ‘𝐵)))))
186		breq1 4811	. . . . . . . . . . . . 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 4811	. . . . . . . . . . . . 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 4280	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 4248	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 473	. . . . . . . . . . 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 4840	. . . . . . . . . 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 401	. . . . . . . . 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 11379	. . . . . . . . . . 11 ⊢ 0 ≤ 0
196	195	a1i 11	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
197		iffalse 4251	. . . . . . . . . 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 4840	. . . . . . . . 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 172	. . . . . . . 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	syl5eqbr 4843	. . . . . . 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 3113	. . . . . 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 2765	. . . . . . 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 2765	. . . . . . 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 7112	. . . . . 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 248	. . . . 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 23796	. . . . 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 1490	. . . 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 23795	. . . 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 1490	. . 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 3112	. 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 2765	. . 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 2765	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))) = (ℜ‘((𝐶 · 𝐵) / (i↑𝑘))))
214	212, 213, 10	isibl2 23823	. 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 704	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1652 ∈ wcel 2155 ≠ wne 2936 ∀wral 3054 Vcvv 3349 ifcif 4242 {csn 4333 class class class wbr 4808 ↦ cmpt 4887 × cxp 5274 ⟶wf 6063 ‘cfv 6067 (class class class)co 6841 ∘_𝑓 cof 7092 ∘_𝑟 cofr 7093 ℂcc 10186 ℝcr 10187 0cc0 10188 1c1 10189 ici 10190 + caddc 10191 · cmul 10193 +∞cpnf 10324 ℝ^*cxr 10326 ≤ cle 10328 / cdiv 10937 3c3 11327 ℕ₀cn0 11537 ℤcz 11623 [,)cico 12378 [,]cicc 12379 ...cfz 12532 ↑cexp 13066 ℜcre 14123 ℑcim 14124 abscabs 14260 MblFncmbf 23671 ∫₂citg2 23673 𝐿¹cibl 23674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2069 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2349 ax-ext 2742 ax-rep 4929 ax-sep 4940 ax-nul 4948 ax-pow 5000 ax-pr 5061 ax-un 7146 ax-inf2 8752 ax-cnex 10244 ax-resscn 10245 ax-1cn 10246 ax-icn 10247 ax-addcl 10248 ax-addrcl 10249 ax-mulcl 10250 ax-mulrcl 10251 ax-mulcom 10252 ax-addass 10253 ax-mulass 10254 ax-distr 10255 ax-i2m1 10256 ax-1ne0 10257 ax-1rid 10258 ax-rnegex 10259 ax-rrecex 10260 ax-cnre 10261 ax-pre-lttri 10262 ax-pre-lttrn 10263 ax-pre-ltadd 10264 ax-pre-mulgt0 10265 ax-pre-sup 10266 ax-addf 10267

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2062 df-mo 2564 df-eu 2581 df-clab 2751 df-cleq 2757 df-clel 2760 df-nfc 2895 df-ne 2937 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3351 df-sbc 3596 df-csb 3691 df-dif 3734 df-un 3736 df-in 3738 df-ss 3745 df-pss 3747 df-nul 4079 df-if 4243 df-pw 4316 df-sn 4334 df-pr 4336 df-tp 4338 df-op 4340 df-uni 4594 df-int 4633 df-iun 4677 df-disj 4777 df-br 4809 df-opab 4871 df-mpt 4888 df-tr 4911 df-id 5184 df-eprel 5189 df-po 5197 df-so 5198 df-fr 5235 df-se 5236 df-we 5237 df-xp 5282 df-rel 5283 df-cnv 5284 df-co 5285 df-dm 5286 df-rn 5287 df-res 5288 df-ima 5289 df-pred 5864 df-ord 5910 df-on 5911 df-lim 5912 df-suc 5913 df-iota 6030 df-fun 6069 df-fn 6070 df-f 6071 df-f1 6072 df-fo 6073 df-f1o 6074 df-fv 6075 df-isom 6076 df-riota 6802 df-ov 6844 df-oprab 6845 df-mpt2 6846 df-of 7094 df-ofr 7095 df-om 7263 df-1st 7365 df-2nd 7366 df-wrecs 7609 df-recs 7671 df-rdg 7709 df-1o 7763 df-2o 7764 df-oadd 7767 df-er 7946 df-map 8061 df-pm 8062 df-en 8160 df-dom 8161 df-sdom 8162 df-fin 8163 df-fi 8523 df-sup 8554 df-inf 8555 df-oi 8621 df-card 9015 df-cda 9242 df-pnf 10329 df-mnf 10330 df-xr 10331 df-ltxr 10332 df-le 10333 df-sub 10521 df-neg 10522 df-div 10938 df-nn 11274 df-2 11334 df-3 11335 df-n0 11538 df-z 11624 df-uz 11886 df-q 11989 df-rp 12028 df-xneg 12145 df-xadd 12146 df-xmul 12147 df-ioo 12380 df-ico 12382 df-icc 12383 df-fz 12533 df-fzo 12673 df-fl 12800 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14125 df-re 14126 df-im 14127 df-sqrt 14261 df-abs 14262 df-clim 14505 df-sum 14703 df-rest 16350 df-topgen 16371 df-psmet 20010 df-xmet 20011 df-met 20012 df-bl 20013 df-mopn 20014 df-top 20977 df-topon 20994 df-bases 21029 df-cmp 21469 df-ovol 23521 df-vol 23522 df-mbf 23676 df-itg1 23677 df-itg2 23678 df-ibl 23679 df-0p 23727

This theorem is referenced by: itgmulc2nclem1 33831 itgmulc2nclem2 33832 itgmulc2nc 33833 itgabsnc 33834 ftc1anclem6 33845

W3C validator