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Theorem itgcnlem 25691

Description: Expand out the sum in dfitg 25670. (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 2729	. . . 4 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘)))
2	1	dfitg 25670	. . 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 12835	. . . . 5 ⊢ ℕ₀ = (ℤ_≥‘0)
4		df-3 12250	. . . . 5 ⊢ 3 = (2 + 1)
5		oveq2 7395	. . . . . . 7 ⊢ (𝑘 = 3 → (i↑𝑘) = (i↑3))
6		i3 14168	. . . . . . 7 ⊢ (i↑3) = -i
7	5, 6	eqtrdi 2780	. . . . . 6 ⊢ (𝑘 = 3 → (i↑𝑘) = -i)
8	6	itgvallem 25686	. . . . . 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 7405	. . . . 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 11127	. . . . . . . 8 ⊢ i ∈ ℂ
11	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → i ∈ ℂ)
12		expcl 14044	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
13	11, 12	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
14		nn0z 12554	. . . . . . 7 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℤ)
15		eqidd 2730	. . . . . . . . 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 2730	. . . . . . . . 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 25669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) ∈ ℝ)
20	19	recnd 11202	. . . . . . 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 593	. . . . . 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 11194	. . . . 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 12249	. . . . . 6 ⊢ 2 = (1 + 1)
24		oveq2 7395	. . . . . . . 8 ⊢ (𝑘 = 2 → (i↑𝑘) = (i↑2))
25		i2 14167	. . . . . . . 8 ⊢ (i↑2) = -1
26	24, 25	eqtrdi 2780	. . . . . . 7 ⊢ (𝑘 = 2 → (i↑𝑘) = -1)
27	25	itgvallem 25686	. . . . . . 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 7405	. . . . . 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 12691	. . . . . . 7 ⊢ 1 = (0 + 1)
30		oveq2 7395	. . . . . . . . 9 ⊢ (𝑘 = 1 → (i↑𝑘) = (i↑1))
31		exp1 14032	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i↑1) = i)
32	10, 31	ax-mp 5	. . . . . . . . 9 ⊢ (i↑1) = i
33	30, 32	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑘 = 1 → (i↑𝑘) = i)
34	32	itgvallem 25686	. . . . . . . 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 7405	. . . . . . 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 12540	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
37		itgcnlem.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
38		iblmbf 25668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
39	17, 38	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
40	39, 18	mbfmptcl 25537	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
41	40	div1d 11950	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 / 1) = 𝐵)
42	41	fveq2d 6862	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 / 1)) = (ℜ‘𝐵))
43	42	ibllem 25665	. . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0))))
47	46	oveq2d 7403	. . . . . . . . . . . 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 25690	. . . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ))
54	53	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ)
55	54	recnd 11202	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℂ)
56	55	mullidd 11192	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 · 𝑅) = 𝑅)
57	47, 56	eqtr3d 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) = 𝑅)
58	57, 55	eqeltrd 2828	. . . . . . . . . 10 ⊢ (𝜑 → (1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 1))), (ℜ‘(𝐵 / 1)), 0)))) ∈ ℂ)
59		oveq2 7395	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (i↑𝑘) = (i↑0))
60		exp0 14030	. . . . . . . . . . . . . 14 ⊢ (i ∈ ℂ → (i↑0) = 1)
61	10, 60	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (i↑0) = 1
62	59, 61	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (i↑𝑘) = 1)
63	61	itgvallem 25686	. . . . . . . . . . . 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 7405	. . . . . . . . . . 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 15713	. . . . . . . . . 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 587	. . . . . . . . 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 2764	. . . . . . . 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 12457	. . . . . . . 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 15073	. . . . . . . . . . . . . 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 25665	. . . . . . . . . . . 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 5201	. . . . . . . . . . 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 6862	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0))) = 𝑇)
76	75	oveq2d 7403	. . . . . . . 8 ⊢ (𝜑 → (i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / i))), (ℜ‘(𝐵 / i)), 0)))) = (i · 𝑇))
77	76	oveq2d 7403	. . . . . . 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 15735	. . . . . 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 15175	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
80		ax-1cn 11126	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℂ
81	80	negnegi 11492	. . . . . . . . . . . . . . . . . . 19 ⊢ --1 = 1
82	81	oveq2i 7398	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝐵 / --1) = (-𝐵 / 1)
83	40	negcld 11520	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
84	83	div1d 11950	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / 1) = -𝐵)
85	82, 84	eqtrid 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = -𝐵)
86	80	negcli 11490	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ∈ ℂ
87		neg1ne0 12173	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
88		div2neg 11905	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℂ ∧ -1 ∈ ℂ ∧ -1 ≠ 0) → (-𝐵 / --1) = (𝐵 / -1))
89	86, 87, 88	mp3an23 1455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --1) = (𝐵 / -1))
90	40, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --1) = (𝐵 / -1))
91	85, 90	eqtr3d 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 = (𝐵 / -1))
92	91	fveq2d 6862	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = (ℜ‘(𝐵 / -1)))
93	79, 92	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) = (ℜ‘(𝐵 / -1)))
94	93	ibllem 25665	. . . . . . . . . . . . 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 5201	. . . . . . . . . . . 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 6862	. . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0))))
98	97	oveq2d 7403	. . . . . . . . 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 11202	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ ℂ)
101	100	mulm1d 11630	. . . . . . . . 9 ⊢ (𝜑 → (-1 · 𝑆) = -𝑆)
102	98, 101	eqtr3d 2766	. . . . . . . 8 ⊢ (𝜑 → (-1 · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -1))), (ℜ‘(𝐵 / -1)), 0)))) = -𝑆)
103	102	oveq2d 7403	. . . . . . 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 1144	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))
105	104	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℝ)
106	105	recnd 11202	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ ℂ)
107		mulcl 11152	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝑇 ∈ ℂ) → (i · 𝑇) ∈ ℂ)
108	10, 106, 107	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (i · 𝑇) ∈ ℂ)
109	55, 108	addcld 11193	. . . . . . . 8 ⊢ (𝜑 → (𝑅 + (i · 𝑇)) ∈ ℂ)
110	109, 100	negsubd 11539	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) + -𝑆) = ((𝑅 + (i · 𝑇)) − 𝑆))
111	55, 108, 100	addsubd 11554	. . . . . . 7 ⊢ (𝜑 → ((𝑅 + (i · 𝑇)) − 𝑆) = ((𝑅 − 𝑆) + (i · 𝑇)))
112	103, 110, 111	3eqtrd 2768	. . . . . 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 15735	. . . . 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 15073	. . . . . . . . . . . . . 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 15176	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
117	10	negnegi 11492	. . . . . . . . . . . . . . . . 17 ⊢ --i = i
118	117	eqcomi 2738	. . . . . . . . . . . . . . . 16 ⊢ i = --i
119	118	oveq2i 7398	. . . . . . . . . . . . . . 15 ⊢ (-𝐵 / i) = (-𝐵 / --i)
120	10	negcli 11490	. . . . . . . . . . . . . . . . 17 ⊢ -i ∈ ℂ
121		ine0 11613	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
122	10, 121	negne0i 11497	. . . . . . . . . . . . . . . . 17 ⊢ -i ≠ 0
123		div2neg 11905	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ -i ∈ ℂ ∧ -i ≠ 0) → (-𝐵 / --i) = (𝐵 / -i))
124	120, 122, 123	mp3an23 1455	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℂ → (-𝐵 / --i) = (𝐵 / -i))
125	40, 124	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / --i) = (𝐵 / -i))
126	119, 125	eqtrid 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-𝐵 / i) = (𝐵 / -i))
127	126	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(-𝐵 / i)) = (ℜ‘(𝐵 / -i)))
128	115, 116, 127	3eqtr3d 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) = (ℜ‘(𝐵 / -i)))
129	128	ibllem 25665	. . . . . . . . . . 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 5201	. . . . . . . . . 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 6862	. . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0))))
133	132	oveq2d 7403	. . . . . . 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 11202	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℂ)
136		mulneg12 11616	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑈 ∈ ℂ) → (-i · 𝑈) = (i · -𝑈))
137	10, 135, 136	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (-i · 𝑈) = (i · -𝑈))
138	133, 137	eqtr3d 2766	. . . . . 6 ⊢ (𝜑 → (-i · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / -i))), (ℜ‘(𝐵 / -i)), 0)))) = (i · -𝑈))
139	138	oveq2d 7403	. . . . 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 15735	. . . 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 2776	. 2 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)))
143	55, 100	subcld 11533	. . 3 ⊢ (𝜑 → (𝑅 − 𝑆) ∈ ℂ)
144	135	negcld 11520	. . . 4 ⊢ (𝜑 → -𝑈 ∈ ℂ)
145		mulcl 11152	. . . 4 ⊢ ((i ∈ ℂ ∧ -𝑈 ∈ ℂ) → (i · -𝑈) ∈ ℂ)
146	10, 144, 145	sylancr 587	. . 3 ⊢ (𝜑 → (i · -𝑈) ∈ ℂ)
147	143, 108, 146	addassd 11196	. 2 ⊢ (𝜑 → (((𝑅 − 𝑆) + (i · 𝑇)) + (i · -𝑈)) = ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))))
148	11, 106, 144	adddid 11198	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = ((i · 𝑇) + (i · -𝑈)))
149	106, 135	negsubd 11539	. . . . 5 ⊢ (𝜑 → (𝑇 + -𝑈) = (𝑇 − 𝑈))
150	149	oveq2d 7403	. . . 4 ⊢ (𝜑 → (i · (𝑇 + -𝑈)) = (i · (𝑇 − 𝑈)))
151	148, 150	eqtr3d 2766	. . 3 ⊢ (𝜑 → ((i · 𝑇) + (i · -𝑈)) = (i · (𝑇 − 𝑈)))
152	151	oveq2d 7403	. 2 ⊢ (𝜑 → ((𝑅 − 𝑆) + ((i · 𝑇) + (i · -𝑈))) = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))
153	142, 147, 152	3eqtrd 2768	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 ici 11070 + caddc 11071 · cmul 11073 ≤ cle 11209 − cmin 11405 -cneg 11406 / cdiv 11835 2c2 12241 3c3 12242 ℕ₀cn0 12442 ℤcz 12529 ...cfz 13468 ↑cexp 14026 ℜcre 15063 ℑcim 15064 Σcsu 15652 MblFncmbf 25515 ∫₂citg2 25517 𝐿¹cibl 25518 ∫citg 25519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-mbf 25520 df-ibl 25523 df-itg 25524

This theorem is referenced by: itgrevallem1 25696 itgcnval 25701

W3C validator