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Theorem itgcnlem 25782

Description: Expand out the sum in dfitg 25761. (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 2740	. . . 4 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))
2	1	dfitg 25761	. . 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 12824	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4		df-3 12243	. . . . 5 ⊢ 3 = (2 + 1)
5		oveq2 7371	. . . . . . 7 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
6		i3 14163	. . . . . . 7 ⊢ (i↑3) = -i
7	5, 6	eqtrdi 2791	. . . . . 6 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
8	6	itgvallem 25777	. . . . . 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 7381	. . . . 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 11095	. . . . . . . 8 ⊢ i ∈ ℂ
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → i ∈ ℂ)
12		expcl 14039	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
13	11, 12	sylan 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
14		nn0z 12546	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℤ)
15		eqidd 2741	. . . . . . . . 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 2741	. . . . . . . . 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 25760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
20	19	recnd 11171	. . . . . . 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 599	. . . . . 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 11163	. . . . 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 12242	. . . . . 6 ⊢ 2 = (1 + 1)
24		oveq2 7371	. . . . . . . 8 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
25		i2 14162	. . . . . . . 8 ⊢ (i↑2) = -1
26	24, 25	eqtrdi 2791	. . . . . . 7 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
27	25	itgvallem 25777	. . . . . . 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 7381	. . . . . 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 12684	. . . . . . 7 ⊢ 1 = (0 + 1)
30		oveq2 7371	. . . . . . . . 9 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
31		exp1 14027	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i↑1) = i)
32	10, 31	ax-mp 5	. . . . . . . . 9 ⊢ (i↑1) = i
33	30, 32	eqtrdi 2791	. . . . . . . 8 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
34	32	itgvallem 25777	. . . . . . . 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 7381	. . . . . . 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 12533	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
37		itgcnlem.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
38		iblmbf 25759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
39	17, 38	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
40	39, 18	mbfmptcl 25628	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	div1d 11921	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 1) = 𝐵)
42	41	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / 1)) = (ℜ‘𝐵))
43	42	ibllem 25756	. . . . . . . . . . . . . . . 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 5173	. . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))))
47	46	oveq2d 7379	. . . . . . . . . . . 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 25781	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
52	17, 51	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
53	52	simp2d 1149	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ))
54	53	simpld 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ)
55	54	recnd 11171	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℂ)
56	55	mullidd 11161	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 · 𝑅) = 𝑅)
57	47, 56	eqtr3d 2777	. . . . . . . . . . 11 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) = 𝑅)
58	57, 55	eqeltrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) ∈ ℂ)
59		oveq2 7371	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (i↑𝑘) = (i↑0))
60		exp0 14025	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑0) = 1)
61	10, 60	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑0) = 1
62	59, 61	eqtrdi 2791	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (i↑𝑘) = 1)
63	61	itgvallem 25777	. . . . . . . . . . . 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 7381	. . . . . . . . . . 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 15707	. . . . . . . . . 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 593	. . . . . . . . 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 2775	. . . . . . . 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 12450	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
69	67, 68	jctil 524	. . . . . . 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 15067	. . . . . . . . . . . . . 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 25756	. . . . . . . . . . . 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 5173	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) = 𝑇)
76	75	oveq2d 7379	. . . . . . . 8 ⊢ (𝜑 → (i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0)))) = (i · 𝑇))
77	76	oveq2d 7379	. . . . . . 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 15729	. . . . . 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 15169	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
80		ax-1cn 11094	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
81	80	negnegi 11462	. . . . . . . . . . . . . . . . . . 19 ⊢ --1 = 1
82	81	oveq2i 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝐵 / --1) = (-𝐵 / 1)
83	40	negcld 11490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
84	83	div1d 11921	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / 1) = -𝐵)
85	82, 84	eqtrid 2787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = -𝐵)
86	80	negcli 11460	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℂ
87		neg1ne0 12144	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
88		div2neg 11876	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℂ ∧ -1 ∈ ℂ ∧ -1 ≠ 0) → (-𝐵 / --1) = (𝐵 / -1))
89	86, 87, 88	mp3an23 1461	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --1) = (𝐵 / -1))
90	40, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = (𝐵 / -1))
91	85, 90	eqtr3d 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 = (𝐵 / -1))
92	91	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = (ℜ‘(𝐵 / -1)))
93	79, 92	eqtr3d 2777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) = (ℜ‘(𝐵 / -1)))
94	93	ibllem 25756	. . . . . . . . . . . . 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 5173	. . . . . . . . . . . 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 2787	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))
98	97	oveq2d 7379	. . . . . . . . 9 ⊢ (𝜑 → (-1 · 𝑆) = (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))))
99	53	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℝ)
100	99	recnd 11171	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
101	100	mulm1d 11600	. . . . . . . . 9 ⊢ (𝜑 → (-1 · 𝑆) = -𝑆)
102	98, 101	eqtr3d 2777	. . . . . . . 8 ⊢ (𝜑 → (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))) = -𝑆)
103	102	oveq2d 7379	. . . . . . 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 1150	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))
105	104	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℝ)
106	105	recnd 11171	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
107		mulcl 11120	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝑇 ∈ ℂ) → (i · 𝑇) ∈ ℂ)
108	10, 106, 107	sylancr 593	. . . . . . . . 9 ⊢ (𝜑 → (i · 𝑇) ∈ ℂ)
109	55, 108	addcld 11162	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + (i · 𝑇)) ∈ ℂ)
110	109, 100	negsubd 11509	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) + -𝑆) = ((𝑅 + (i · 𝑇)) − 𝑆))
111	55, 108, 100	addsubd 11524	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) − 𝑆) = ((𝑅 − 𝑆) + (i · 𝑇)))
112	103, 110, 111	3eqtrd 2779	. . . . . 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 15729	. . . . 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 15067	. . . . . . . . . . . . . 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 15170	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
117	10	negnegi 11462	. . . . . . . . . . . . . . . . 17 ⊢ --i = i
118	117	eqcomi 2749	. . . . . . . . . . . . . . . 16 ⊢ i = --i
119	118	oveq2i 7374	. . . . . . . . . . . . . . 15 ⊢ (-𝐵 / i) = (-𝐵 / --i)
120	10	negcli 11460	. . . . . . . . . . . . . . . . 17 ⊢ -i ∈ ℂ
121		ine0 11583	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
122	10, 121	negne0i 11467	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
123		div2neg 11876	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ -i ∈ ℂ ∧ -i ≠ 0) → (-𝐵 / --i) = (𝐵 / -i))
124	120, 122, 123	mp3an23 1461	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --i) = (𝐵 / -i))
125	40, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --i) = (𝐵 / -i))
126	119, 125	eqtrid 2787	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) = (ℜ‘(𝐵 / -i)))
129	128	ibllem 25756	. . . . . . . . . . 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 5173	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))
133	132	oveq2d 7379	. . . . . . 7 ⊢ (𝜑 → (-i · 𝑈) = (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))))
134	104	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
135	134	recnd 11171	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℂ)
136		mulneg12 11586	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑈 ∈ ℂ) → (-i · 𝑈) = (i · -𝑈))
137	10, 135, 136	sylancr 593	. . . . . . 7 ⊢ (𝜑 → (-i · 𝑈) = (i · -𝑈))
138	133, 137	eqtr3d 2777	. . . . . 6 ⊢ (𝜑 → (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))) = (i · -𝑈))
139	138	oveq2d 7379	. . . . 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 15729	. . . 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 496	. . 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 2787	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)))
143	55, 100	subcld 11503	. . 3 ⊢ (𝜑 → (𝑅 − 𝑆) ∈ ℂ)
144	135	negcld 11490	. . . 4 ⊢ (𝜑 → -𝑈 ∈ ℂ)
145		mulcl 11120	. . . 4 ⊢ ((i ∈ ℂ ∧ -𝑈 ∈ ℂ) → (i · -𝑈) ∈ ℂ)
146	10, 144, 145	sylancr 593	. . 3 ⊢ (𝜑 → (i · -𝑈) ∈ ℂ)
147	143, 108, 146	addassd 11165	. 2 ⊢ (𝜑 → (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)) = ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))))
148	11, 106, 144	adddid 11167	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = ((i · 𝑇) + (i · -𝑈)))
149	106, 135	negsubd 11509	. . . . 5 ⊢ (𝜑 → (𝑇 + -𝑈) = (𝑇 − 𝑈))
150	149	oveq2d 7379	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = (i · (𝑇 − 𝑈)))
151	148, 150	eqtr3d 2777	. . 3 ⊢ (𝜑 → ((i · 𝑇) + (i · -𝑈)) = (i · (𝑇 − 𝑈)))
152	151	oveq2d 7379	. 2 ⊢ (𝜑 → ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))) = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))
153	142, 147, 152	3eqtrd 2779	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ifcif 4461 class class class wbr 5079 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 ici 11038 + caddc 11039 · cmul 11041 ≤ cle 11178 − cmin 11375 -cneg 11376 / cdiv 11805 2c2 12234 3c3 12235 ℕ₀cn0 12435 ℤcz 12522 ...cfz 13459 ↑cexp 14021 ℜcre 15057 ℑcim 15058 Σcsu 15646 MblFncmbf 25606 ∫₂citg2 25608 𝐿¹cibl 25609 ∫citg 25610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-n0 12436 df-z 12523 df-uz 12787 df-rp 12941 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-clim 15448 df-sum 15647 df-mbf 25611 df-ibl 25614 df-itg 25615

This theorem is referenced by: itgrevallem1 25787 itgcnval 25792

W3C validator