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Theorem itgcnlem 25767

Description: Expand out the sum in dfitg 25746. (Contributed by Mario Carneiro, 1-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)))
itgcnlem.v	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgcnlem.i	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

**Assertion**
Ref	Expression
itgcnlem	⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))

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

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))
2	1	dfitg 25746	. . 3 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))))
3		nn0uz 12817	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4		df-3 12236	. . . . 5 ⊢ 3 = (2 + 1)
5		oveq2 7368	. . . . . . 7 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
6		i3 14156	. . . . . . 7 ⊢ (i↑3) = -i
7	5, 6	eqtrdi 2788	. . . . . 6 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
8	6	itgvallem 25762	. . . . . 6 ⊢ (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))))
9	7, 8	oveq12d 7378	. . . . 5 ⊢ (𝑘 = 3 → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))))
10		ax-icn 11088	. . . . . . . 8 ⊢ i ∈ ℂ
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → i ∈ ℂ)
12		expcl 14032	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
13	11, 12	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
14		nn0z 12539	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℤ)
15		eqidd 2738	. . . . . . . . 9 ⊢ (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)))
16		eqidd 2738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))))
17		itgcnlem.i	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
18		itgcnlem.v	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
19	15, 16, 17, 18	iblitg 25745	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
20	19	recnd 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℂ)
21	14, 20	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℂ)
22	13, 21	mulcld 11156	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) ∈ ℂ)
23		df-2 12235	. . . . . 6 ⊢ 2 = (1 + 1)
24		oveq2 7368	. . . . . . . 8 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
25		i2 14155	. . . . . . . 8 ⊢ (i↑2) = -1
26	24, 25	eqtrdi 2788	. . . . . . 7 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
27	25	itgvallem 25762	. . . . . . 7 ⊢ (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))))
28	26, 27	oveq12d 7378	. . . . . 6 ⊢ (𝑘 = 2 → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))))
29		1e0p1 12677	. . . . . . 7 ⊢ 1 = (0 + 1)
30		oveq2 7368	. . . . . . . . 9 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
31		exp1 14020	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i↑1) = i)
32	10, 31	ax-mp 5	. . . . . . . . 9 ⊢ (i↑1) = i
33	30, 32	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
34	32	itgvallem 25762	. . . . . . . 8 ⊢ (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))))
35	33, 34	oveq12d 7378	. . . . . . 7 ⊢ (𝑘 = 1 → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0)))))
36		0z 12526	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
37		itgcnlem.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
38		iblmbf 25744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
39	17, 38	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
40	39, 18	mbfmptcl 25613	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	div1d 11914	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 1) = 𝐵)
42	41	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / 1)) = (ℜ‘𝐵))
43	42	ibllem 25741	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
44	43	mpteq2dv 5180	. . . . . . . . . . . . . . 15 ⊢ (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)))
45	44	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ (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))))
46	37, 45	eqtr4id 2791	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))))
47	46	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 · 𝑅) = (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))))
48		itgcnlem.s	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
49		itgcnlem.t	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
50		itgcnlem.u	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
51	37, 48, 49, 50, 18	iblcnlem 25766	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
52	17, 51	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
53	52	simp2d 1144	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ))
54	53	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ)
55	54	recnd 11164	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℂ)
56	55	mullidd 11154	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 · 𝑅) = 𝑅)
57	47, 56	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) = 𝑅)
58	57, 55	eqeltrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) ∈ ℂ)
59		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (i↑𝑘) = (i↑0))
60		exp0 14018	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑0) = 1)
61	10, 60	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑0) = 1
62	59, 61	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (i↑𝑘) = 1)
63	61	itgvallem 25762	. . . . . . . . . . . 12 ⊢ (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))))
64	62, 63	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))))
65	64	fsum1 15700	. . . . . . . . . 10 ⊢ ((0 ∈ ℤ ∧ (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) ∈ ℂ) → Σ𝑘 ∈ (0...0)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))))
66	36, 58, 65	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))))
67	66, 57	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (0...0)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = 𝑅)
68		0nn0 12443	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
69	67, 68	jctil 519	. . . . . . 7 ⊢ (𝜑 → (0 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...0)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = 𝑅))
70		imval 15060	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) = (ℜ‘(𝐵 / i)))
71	40, 70	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) = (ℜ‘(𝐵 / i)))
72	71	ibllem 25741	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))
73	72	mpteq2dv 5180	. . . . . . . . . . 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)))
74	73	fveq2d 6838	. . . . . . . . . 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))))
75	49, 74	eqtr2id 2785	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) = 𝑇)
76	75	oveq2d 7376	. . . . . . . 8 ⊢ (𝜑 → (i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0)))) = (i · 𝑇))
77	76	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → (𝑅 + (i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))))) = (𝑅 + (i · 𝑇)))
78	3, 29, 35, 22, 69, 77	fsump1i 15722	. . . . . 6 ⊢ (𝜑 → (1 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...1)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (𝑅 + (i · 𝑇))))
79	40	renegd 15162	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
80		ax-1cn 11087	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
81	80	negnegi 11455	. . . . . . . . . . . . . . . . . . 19 ⊢ --1 = 1
82	81	oveq2i 7371	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝐵 / --1) = (-𝐵 / 1)
83	40	negcld 11483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
84	83	div1d 11914	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / 1) = -𝐵)
85	82, 84	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = -𝐵)
86	80	negcli 11453	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℂ
87		neg1ne0 12137	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
88		div2neg 11869	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℂ ∧ -1 ∈ ℂ ∧ -1 ≠ 0) → (-𝐵 / --1) = (𝐵 / -1))
89	86, 87, 88	mp3an23 1456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --1) = (𝐵 / -1))
90	40, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = (𝐵 / -1))
91	85, 90	eqtr3d 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 = (𝐵 / -1))
92	91	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = (ℜ‘(𝐵 / -1)))
93	79, 92	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) = (ℜ‘(𝐵 / -1)))
94	93	ibllem 25741	. . . . . . . . . . . . 13 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))
95	94	mpteq2dv 5180	. . . . . . . . . . . 12 ⊢ (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)))
96	95	fveq2d 6838	. . . . . . . . . . 11 ⊢ (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))))
97	48, 96	eqtrid 2784	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))
98	97	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → (-1 · 𝑆) = (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))))
99	53	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℝ)
100	99	recnd 11164	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
101	100	mulm1d 11593	. . . . . . . . 9 ⊢ (𝜑 → (-1 · 𝑆) = -𝑆)
102	98, 101	eqtr3d 2774	. . . . . . . 8 ⊢ (𝜑 → (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))) = -𝑆)
103	102	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) + (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))) = ((𝑅 + (i · 𝑇)) + -𝑆))
104	52	simp3d 1145	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))
105	104	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℝ)
106	105	recnd 11164	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
107		mulcl 11113	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝑇 ∈ ℂ) → (i · 𝑇) ∈ ℂ)
108	10, 106, 107	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → (i · 𝑇) ∈ ℂ)
109	55, 108	addcld 11155	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + (i · 𝑇)) ∈ ℂ)
110	109, 100	negsubd 11502	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) + -𝑆) = ((𝑅 + (i · 𝑇)) − 𝑆))
111	55, 108, 100	addsubd 11517	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) − 𝑆) = ((𝑅 − 𝑆) + (i · 𝑇)))
112	103, 110, 111	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) + (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))) = ((𝑅 − 𝑆) + (i · 𝑇)))
113	3, 23, 28, 22, 78, 112	fsump1i 15722	. . . . 5 ⊢ (𝜑 → (2 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...2)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = ((𝑅 − 𝑆) + (i · 𝑇))))
114		imval 15060	. . . . . . . . . . . . . 14 ⊢ (-𝐵 ∈ ℂ → (ℑ‘-𝐵) = (ℜ‘(-𝐵 / i)))
115	83, 114	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = (ℜ‘(-𝐵 / i)))
116	40	imnegd 15163	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
117	10	negnegi 11455	. . . . . . . . . . . . . . . . 17 ⊢ --i = i
118	117	eqcomi 2746	. . . . . . . . . . . . . . . 16 ⊢ i = --i
119	118	oveq2i 7371	. . . . . . . . . . . . . . 15 ⊢ (-𝐵 / i) = (-𝐵 / --i)
120	10	negcli 11453	. . . . . . . . . . . . . . . . 17 ⊢ -i ∈ ℂ
121		ine0 11576	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
122	10, 121	negne0i 11460	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
123		div2neg 11869	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ -i ∈ ℂ ∧ -i ≠ 0) → (-𝐵 / --i) = (𝐵 / -i))
124	120, 122, 123	mp3an23 1456	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --i) = (𝐵 / -i))
125	40, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --i) = (𝐵 / -i))
126	119, 125	eqtrid 2784	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / i) = (𝐵 / -i))
127	126	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(-𝐵 / i)) = (ℜ‘(𝐵 / -i)))
128	115, 116, 127	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) = (ℜ‘(𝐵 / -i)))
129	128	ibllem 25741	. . . . . . . . . . 11 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))
130	129	mpteq2dv 5180	. . . . . . . . . 10 ⊢ (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)))
131	130	fveq2d 6838	. . . . . . . . 9 ⊢ (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))))
132	50, 131	eqtrid 2784	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))
133	132	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → (-i · 𝑈) = (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))))
134	104	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
135	134	recnd 11164	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℂ)
136		mulneg12 11579	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑈 ∈ ℂ) → (-i · 𝑈) = (i · -𝑈))
137	10, 135, 136	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (-i · 𝑈) = (i · -𝑈))
138	133, 137	eqtr3d 2774	. . . . . 6 ⊢ (𝜑 → (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))) = (i · -𝑈))
139	138	oveq2d 7376	. . . . 5 ⊢ (𝜑 → (((𝑅 − 𝑆) + (i · 𝑇)) + (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))) = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)))
140	3, 4, 9, 22, 113, 139	fsump1i 15722	. . . 4 ⊢ (𝜑 → (3 ∈ ℕ₀ ∧ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈))))
141	140	simprd 495	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)))
142	2, 141	eqtrid 2784	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)))
143	55, 100	subcld 11496	. . 3 ⊢ (𝜑 → (𝑅 − 𝑆) ∈ ℂ)
144	135	negcld 11483	. . . 4 ⊢ (𝜑 → -𝑈 ∈ ℂ)
145		mulcl 11113	. . . 4 ⊢ ((i ∈ ℂ ∧ -𝑈 ∈ ℂ) → (i · -𝑈) ∈ ℂ)
146	10, 144, 145	sylancr 588	. . 3 ⊢ (𝜑 → (i · -𝑈) ∈ ℂ)
147	143, 108, 146	addassd 11158	. 2 ⊢ (𝜑 → (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)) = ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))))
148	11, 106, 144	adddid 11160	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = ((i · 𝑇) + (i · -𝑈)))
149	106, 135	negsubd 11502	. . . . 5 ⊢ (𝜑 → (𝑇 + -𝑈) = (𝑇 − 𝑈))
150	149	oveq2d 7376	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = (i · (𝑇 − 𝑈)))
151	148, 150	eqtr3d 2774	. . 3 ⊢ (𝜑 → ((i · 𝑇) + (i · -𝑈)) = (i · (𝑇 − 𝑈)))
152	151	oveq2d 7376	. 2 ⊢ (𝜑 → ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))) = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))
153	142, 147, 152	3eqtrd 2776	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 ici 11031 + caddc 11032 · cmul 11034 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 3c3 12228 ℕ₀cn0 12428 ℤcz 12515 ...cfz 13452 ↑cexp 14014 ℜcre 15050 ℑcim 15051 Σcsu 15639 MblFncmbf 25591 ∫₂citg2 25593 𝐿¹cibl 25594 ∫citg 25595

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-mbf 25596 df-ibl 25599 df-itg 25600

This theorem is referenced by: itgrevallem1 25772 itgcnval 25777

W3C validator