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Theorem itgneg 25312

Description: Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
itgcnval.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgcnval.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)

**Assertion**
Ref	Expression
itgneg	⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 𝑥,𝑉 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem itgneg**
Step	Hyp	Ref	Expression
1		itgcnval.2	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
2		iblmbf 25276	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4		itgcnval.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
5	3, 4	mbfmptcl 25144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	recld 15137	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
7	5	iblcn 25307	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
8	1, 7	mpbid 231	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
9	8	simpld 495	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
10	6, 9	itgcl 25292	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
11		ax-icn 11165	. . . . 5 ⊢ i ∈ ℂ
12	5	imcld 15138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
13	8	simprd 496	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
14	12, 13	itgcl 25292	. . . . 5 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
15		mulcl 11190	. . . . 5 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
16	11, 14, 15	sylancr 587	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
17	10, 16	negdid 11580	. . 3 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)))
18		0re 11212	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4572	. . . . . . . 8 ⊢ (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
20	6, 18, 19	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
21	6	iblre 25302	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿¹)))
22	9, 21	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿¹))
23	22	simpld 495	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿¹)
24	20, 23	itgcl 25292	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
25	6	renegcld 11637	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) ∈ ℝ)
26		ifcl 4572	. . . . . . . 8 ⊢ ((-(ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
27	25, 18, 26	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
28	22	simprd 496	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿¹)
29	27, 28	itgcl 25292	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
30	24, 29	negsubdi2d 11583	. . . . 5 ⊢ (𝜑 → -(∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
31	6, 9	itgreval 25305	. . . . . 6 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
32	31	negeqd 11450	. . . . 5 ⊢ (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥))
33	5	negcld 11554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
34	33	recld 15137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) ∈ ℝ)
35	4, 1	iblneg 25311	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ 𝐿¹)
36	33	iblcn 25307	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ -𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿¹)))
37	35, 36	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿¹))
38	37	simpld 495	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿¹)
39	34, 38	itgreval 25305	. . . . . 6 ⊢ (𝜑 → ∫𝐴(ℜ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥))
40	5	renegd 15152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
41	40	breq2d 5159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℜ‘-𝐵) ↔ 0 ≤ -(ℜ‘𝐵)))
42	41, 40	ifbieq1d 4551	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) = if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0))
43	42	itgeq2dv 25290	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥)
44	40	negeqd 11450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = --(ℜ‘𝐵))
45	6	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
46	45	negnegd 11558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℜ‘𝐵) = (ℜ‘𝐵))
47	44, 46	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = (ℜ‘𝐵))
48	47	breq2d 5159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℜ‘-𝐵) ↔ 0 ≤ (ℜ‘𝐵)))
49	48, 47	ifbieq1d 4551	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) = if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0))
50	49	itgeq2dv 25290	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥)
51	43, 50	oveq12d 7423	. . . . . 6 ⊢ (𝜑 → (∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
52	39, 51	eqtrd 2772	. . . . 5 ⊢ (𝜑 → ∫𝐴(ℜ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥))
53	30, 32, 52	3eqtr4d 2782	. . . 4 ⊢ (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = ∫𝐴(ℜ‘-𝐵) d𝑥)
54		mulneg2 11647	. . . . . 6 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
55	11, 14, 54	sylancr 587	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
56		ifcl 4572	. . . . . . . . . . 11 ⊢ (((ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
57	12, 18, 56	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
58	12	iblre 25302	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿¹)))
59	13, 58	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿¹))
60	59	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿¹)
61	57, 60	itgcl 25292	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
62	12	renegcld 11637	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) ∈ ℝ)
63		ifcl 4572	. . . . . . . . . . 11 ⊢ ((-(ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
64	62, 18, 63	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
65	59	simprd 496	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿¹)
66	64, 65	itgcl 25292	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
67	61, 66	negsubdi2d 11583	. . . . . . . 8 ⊢ (𝜑 → -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥))
68	5	imnegd 15153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
69	68	breq2d 5159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℑ‘-𝐵) ↔ 0 ≤ -(ℑ‘𝐵)))
70	69, 68	ifbieq1d 4551	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) = if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0))
71	70	itgeq2dv 25290	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥)
72	68	negeqd 11450	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = --(ℑ‘𝐵))
73	12	recnd 11238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
74	73	negnegd 11558	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℑ‘𝐵) = (ℑ‘𝐵))
75	72, 74	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = (ℑ‘𝐵))
76	75	breq2d 5159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℑ‘-𝐵) ↔ 0 ≤ (ℑ‘𝐵)))
77	76, 75	ifbieq1d 4551	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) = if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0))
78	77	itgeq2dv 25290	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥)
79	71, 78	oveq12d 7423	. . . . . . . 8 ⊢ (𝜑 → (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥))
80	67, 79	eqtr4d 2775	. . . . . . 7 ⊢ (𝜑 → -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥) = (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥))
81	12, 13	itgreval 25305	. . . . . . . 8 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
82	81	negeqd 11450	. . . . . . 7 ⊢ (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = -(∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥))
83	33	imcld 15138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) ∈ ℝ)
84	37	simprd 496	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿¹)
85	83, 84	itgreval 25305	. . . . . . 7 ⊢ (𝜑 → ∫𝐴(ℑ‘-𝐵) d𝑥 = (∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 − ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥))
86	80, 82, 85	3eqtr4d 2782	. . . . . 6 ⊢ (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = ∫𝐴(ℑ‘-𝐵) d𝑥)
87	86	oveq2d 7421	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
88	55, 87	eqtr3d 2774	. . . 4 ⊢ (𝜑 → -(i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
89	53, 88	oveq12d 7423	. . 3 ⊢ (𝜑 → (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
90	17, 89	eqtrd 2772	. 2 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
91	4, 1	itgcnval 25308	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
92	91	negeqd 11450	. 2 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
93	33, 35	itgcnval 25308	. 2 ⊢ (𝜑 → ∫𝐴-𝐵 d𝑥 = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
94	90, 92, 93	3eqtr4d 2782	1 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 ici 11108 + caddc 11109 · cmul 11111 ≤ cle 11245 − cmin 11440 -cneg 11441 ℜcre 15040 ℑcim 15041 MblFncmbf 25122 𝐿¹cibl 25125 ∫citg 25126

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xadd 13089 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-xmet 20929 df-met 20930 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-itg 25131 df-0p 25178

This theorem is referenced by: itgsub 25334 itgsubnc 36538 itgmulc2nc 36544 sqwvfourb 44931

W3C validator