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Theorem iblcnlem1 23554

Description: Lemma for iblcnlem 23555. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypotheses**
Ref	Expression
itgcnlem.r	⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
itgcnlem.s	⊢ 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
itgcnlem.t	⊢ 𝑇 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
itgcnlem.u	⊢ 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
itgcnlem1.v	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
iblcnlem1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝑅(𝑥) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥)

Proof of Theorem iblcnlem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))
2		eqidd 2623	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
3		itgcnlem1.v	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
4	1, 2, 3	isibl2 23533	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)))
5		c0ex 10034	. . . . . . . 8 ⊢ 0 ∈ V
6		1ex 10035	. . . . . . . 8 ⊢ 1 ∈ V
7		ax-icn 9995	. . . . . . . . . . 11 ⊢ i ∈ ℂ
8		exp0 12864	. . . . . . . . . . 11 ⊢ (i ∈ ℂ → (i↑0) = 1)
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ (i↑0) = 1
10	9	itgvallem 23551	. . . . . . . . 9 ⊢ (𝑘 = 0 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))))
11	10	eleq1d 2686	. . . . . . . 8 ⊢ (𝑘 = 0 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) ∈ ℝ))
12		exp1 12866	. . . . . . . . . . 11 ⊢ (i ∈ ℂ → (i↑1) = i)
13	7, 12	ax-mp 5	. . . . . . . . . 10 ⊢ (i↑1) = i
14	13	itgvallem 23551	. . . . . . . . 9 ⊢ (𝑘 = 1 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))))
15	14	eleq1d 2686	. . . . . . . 8 ⊢ (𝑘 = 1 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) ∈ ℝ))
16	5, 6, 11, 15	ralpr 4238	. . . . . . 7 ⊢ (∀𝑘 ∈ {0, 1} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) ∈ ℝ))
17	3	div1d 10793	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 1) = 𝐵)
18	17	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / 1)) = (ℜ‘𝐵))
19	18	ibllem 23531	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
20	19	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
21	20	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))))
22		itgcnlem.r	. . . . . . . . . 10 ⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
23	21, 22	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) = 𝑅)
24	23	eleq1d 2686	. . . . . . . 8 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) ∈ ℝ ↔ 𝑅 ∈ ℝ))
25		itgcnlem.t	. . . . . . . . . 10 ⊢ 𝑇 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
26		imval 13847	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) = (ℜ‘(𝐵 / i)))
27	3, 26	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) = (ℜ‘(𝐵 / i)))
28	27	ibllem 23531	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))
29	28	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0)))
30	29	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))))
31	25, 30	syl5req 2669	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) = 𝑇)
32	31	eleq1d 2686	. . . . . . . 8 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) ∈ ℝ ↔ 𝑇 ∈ ℝ))
33	24, 32	anbi12d 747	. . . . . . 7 ⊢ (𝜑 → (((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) ∈ ℝ) ↔ (𝑅 ∈ ℝ ∧ 𝑇 ∈ ℝ)))
34	16, 33	syl5bb 272	. . . . . 6 ⊢ (𝜑 → (∀𝑘 ∈ {0, 1} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (𝑅 ∈ ℝ ∧ 𝑇 ∈ ℝ)))
35		2ex 11092	. . . . . . . 8 ⊢ 2 ∈ V
36		3ex 11096	. . . . . . . 8 ⊢ 3 ∈ V
37		i2 12965	. . . . . . . . . 10 ⊢ (i↑2) = -1
38	37	itgvallem 23551	. . . . . . . . 9 ⊢ (𝑘 = 2 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))
39	38	eleq1d 2686	. . . . . . . 8 ⊢ (𝑘 = 2 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))) ∈ ℝ))
40		i3 12966	. . . . . . . . . 10 ⊢ (i↑3) = -i
41	40	itgvallem 23551	. . . . . . . . 9 ⊢ (𝑘 = 3 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))
42	41	eleq1d 2686	. . . . . . . 8 ⊢ (𝑘 = 3 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))) ∈ ℝ))
43	35, 36, 39, 42	ralpr 4238	. . . . . . 7 ⊢ (∀𝑘 ∈ {2, 3} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))) ∈ ℝ))
44		itgcnlem.s	. . . . . . . . . 10 ⊢ 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
45	3	renegd 13949	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
46		ax-1cn 9994	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
47	46	negnegi 10351	. . . . . . . . . . . . . . . . . 18 ⊢ --1 = 1
48	47	oveq2i 6661	. . . . . . . . . . . . . . . . 17 ⊢ (-𝐵 / --1) = (-𝐵 / 1)
49	3	negcld 10379	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
50	49	div1d 10793	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / 1) = -𝐵)
51	48, 50	syl5eq 2668	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = -𝐵)
52	46	negcli 10349	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
53		neg1ne0 11126	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ≠ 0
54		div2neg 10748	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℂ ∧ -1 ∈ ℂ ∧ -1 ≠ 0) → (-𝐵 / --1) = (𝐵 / -1))
55	52, 53, 54	mp3an23 1416	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --1) = (𝐵 / -1))
56	3, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = (𝐵 / -1))
57	51, 56	eqtr3d 2658	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 = (𝐵 / -1))
58	57	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = (ℜ‘(𝐵 / -1)))
59	45, 58	eqtr3d 2658	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) = (ℜ‘(𝐵 / -1)))
60	59	ibllem 23531	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))
61	60	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))
62	61	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))
63	44, 62	syl5req 2669	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))) = 𝑆)
64	63	eleq1d 2686	. . . . . . . 8 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))) ∈ ℝ ↔ 𝑆 ∈ ℝ))
65		itgcnlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
66		imval 13847	. . . . . . . . . . . . . . 15 ⊢ (-𝐵 ∈ ℂ → (ℑ‘-𝐵) = (ℜ‘(-𝐵 / i)))
67	49, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = (ℜ‘(-𝐵 / i)))
68	3	imnegd 13950	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
69	7	negnegi 10351	. . . . . . . . . . . . . . . . . 18 ⊢ --i = i
70	69	eqcomi 2631	. . . . . . . . . . . . . . . . 17 ⊢ i = --i
71	70	oveq2i 6661	. . . . . . . . . . . . . . . 16 ⊢ (-𝐵 / i) = (-𝐵 / --i)
72	7	negcli 10349	. . . . . . . . . . . . . . . . . 18 ⊢ -i ∈ ℂ
73		ine0 10465	. . . . . . . . . . . . . . . . . . 19 ⊢ i ≠ 0
74	7, 73	negne0i 10356	. . . . . . . . . . . . . . . . . 18 ⊢ -i ≠ 0
75		div2neg 10748	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℂ ∧ -i ∈ ℂ ∧ -i ≠ 0) → (-𝐵 / --i) = (𝐵 / -i))
76	72, 74, 75	mp3an23 1416	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --i) = (𝐵 / -i))
77	3, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --i) = (𝐵 / -i))
78	71, 77	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / i) = (𝐵 / -i))
79	78	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(-𝐵 / i)) = (ℜ‘(𝐵 / -i)))
80	67, 68, 79	3eqtr3d 2664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) = (ℜ‘(𝐵 / -i)))
81	80	ibllem 23531	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))
82	81	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))
83	82	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))
84	65, 83	syl5req 2669	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))) = 𝑈)
85	84	eleq1d 2686	. . . . . . . 8 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))) ∈ ℝ ↔ 𝑈 ∈ ℝ))
86	64, 85	anbi12d 747	. . . . . . 7 ⊢ (𝜑 → (((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))) ∈ ℝ) ↔ (𝑆 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
87	43, 86	syl5bb 272	. . . . . 6 ⊢ (𝜑 → (∀𝑘 ∈ {2, 3} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (𝑆 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
88	34, 87	anbi12d 747	. . . . 5 ⊢ (𝜑 → ((∀𝑘 ∈ {0, 1} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ∧ ∀𝑘 ∈ {2, 3} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) ↔ ((𝑅 ∈ ℝ ∧ 𝑇 ∈ ℝ) ∧ (𝑆 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
89		1le3 11244	. . . . . . . . . 10 ⊢ 1 ≤ 3
90		1eluzge0 11732	. . . . . . . . . . 11 ⊢ 1 ∈ (ℤ_≥‘0)
91		3z 11410	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
92		elfz5 12334	. . . . . . . . . . 11 ⊢ ((1 ∈ (ℤ_≥‘0) ∧ 3 ∈ ℤ) → (1 ∈ (0...3) ↔ 1 ≤ 3))
93	90, 91, 92	mp2an 708	. . . . . . . . . 10 ⊢ (1 ∈ (0...3) ↔ 1 ≤ 3)
94	89, 93	mpbir 221	. . . . . . . . 9 ⊢ 1 ∈ (0...3)
95		fzsplit 12367	. . . . . . . . 9 ⊢ (1 ∈ (0...3) → (0...3) = ((0...1) ∪ ((1 + 1)...3)))
96	94, 95	ax-mp 5	. . . . . . . 8 ⊢ (0...3) = ((0...1) ∪ ((1 + 1)...3))
97		0z 11388	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
98		fzpr 12396	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
99	97, 98	ax-mp 5	. . . . . . . . . 10 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
100		1e0p1 11552	. . . . . . . . . . 11 ⊢ 1 = (0 + 1)
101	100	oveq2i 6661	. . . . . . . . . 10 ⊢ (0...1) = (0...(0 + 1))
102	100	preq2i 4272	. . . . . . . . . 10 ⊢ {0, 1} = {0, (0 + 1)}
103	99, 101, 102	3eqtr4i 2654	. . . . . . . . 9 ⊢ (0...1) = {0, 1}
104		2z 11409	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
105		fzpr 12396	. . . . . . . . . . 11 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
106	104, 105	ax-mp 5	. . . . . . . . . 10 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
107		1p1e2 11134	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
108		df-3 11080	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
109	107, 108	oveq12i 6662	. . . . . . . . . 10 ⊢ ((1 + 1)...3) = (2...(2 + 1))
110	108	preq2i 4272	. . . . . . . . . 10 ⊢ {2, 3} = {2, (2 + 1)}
111	106, 109, 110	3eqtr4i 2654	. . . . . . . . 9 ⊢ ((1 + 1)...3) = {2, 3}
112	103, 111	uneq12i 3765	. . . . . . . 8 ⊢ ((0...1) ∪ ((1 + 1)...3)) = ({0, 1} ∪ {2, 3})
113	96, 112	eqtri 2644	. . . . . . 7 ⊢ (0...3) = ({0, 1} ∪ {2, 3})
114	113	raleqi 3142	. . . . . 6 ⊢ (∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈ ({0, 1} ∪ {2, 3})(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
115		ralunb 3794	. . . . . 6 ⊢ (∀𝑘 ∈ ({0, 1} ∪ {2, 3})(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∀𝑘 ∈ {0, 1} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ∧ ∀𝑘 ∈ {2, 3} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ))
116	114, 115	bitri 264	. . . . 5 ⊢ (∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∀𝑘 ∈ {0, 1} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ∧ ∀𝑘 ∈ {2, 3} (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ))
117		an4 865	. . . . 5 ⊢ (((𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) ↔ ((𝑅 ∈ ℝ ∧ 𝑇 ∈ ℝ) ∧ (𝑆 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
118	88, 116, 117	3bitr4g 303	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ ↔ ((𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
119	118	anbi2d 740	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ((𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
120		3anass 1042	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ((𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
121	119, 120	syl6bbr 278	. 2 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
122	4, 121	bitrd 268	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∪ cun 3572 ifcif 4086 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 ici 9938 + caddc 9939 ≤ cle 10075 -cneg 10267 / cdiv 10684 2c2 11070 3c3 11071 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ↑cexp 12860 ℜcre 13837 ℑcim 13838 MblFncmbf 23383 ∫₂citg2 23385 𝐿¹cibl 23386

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-ibl 23391

This theorem is referenced by: iblcnlem 23555 iblcn 23565 bddiblnc 33480

W3C validator