Theorem itgneg 23570

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 23534	. . . . . . . 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 23404	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	recld 13934	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
7	5	iblcn 23565	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)))
8	1, 7	mpbid 222	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1))
9	8	simpld 475	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1)
10	6, 9	itgcl 23550	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
11		ax-icn 9995	. . . . 5 ⊢ i ∈ ℂ
12	5	imcld 13935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
13	8	simprd 479	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)
14	12, 13	itgcl 23550	. . . . 5 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
15		mulcl 10020	. . . . 5 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
16	11, 14, 15	sylancr 695	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
17	10, 16	negdid 10405	. . 3 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)))
18		0re 10040	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4130	. . . . . . . 8 ⊢ (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
20	6, 18, 19	sylancl 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
21	6	iblre 23560	. . . . . . . . 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 222	. . . . . . . 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 475	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1)
24	20, 23	itgcl 23550	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
25	6	renegcld 10457	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) ∈ ℝ)
26		ifcl 4130	. . . . . . . 8 ⊢ ((-(ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
27	25, 18, 26	sylancl 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
28	22	simprd 479	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1)
29	27, 28	itgcl 23550	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
30	24, 29	negsubdi2d 10408	. . . . 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 23563	. . . . . 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 10275	. . . . 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 10379	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
34	33	recld 13934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) ∈ ℝ)
35	4, 1	iblneg 23569	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ 𝐿1)
36	33	iblcn 23565	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ -𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)))
37	35, 36	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1))
38	37	simpld 475	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1)
39	34, 38	itgreval 23563	. . . . . 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 13949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
41	40	breq2d 4665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℜ‘-𝐵) ↔ 0 ≤ -(ℜ‘𝐵)))
42	41, 40	ifbieq1d 4109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) = if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0))
43	42	itgeq2dv 23548	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥)
44	40	negeqd 10275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = --(ℜ‘𝐵))
45	6	recnd 10068	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
46	45	negnegd 10383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℜ‘𝐵) = (ℜ‘𝐵))
47	44, 46	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = (ℜ‘𝐵))
48	47	breq2d 4665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℜ‘-𝐵) ↔ 0 ≤ (ℜ‘𝐵)))
49	48, 47	ifbieq1d 4109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) = if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0))
50	49	itgeq2dv 23548	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥)
51	43, 50	oveq12d 6668	. . . . . 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 2656	. . . . 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 2666	. . . 4 ⊢ (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = ∫𝐴(ℜ‘-𝐵) d𝑥)
54		mulneg2 10467	. . . . . 6 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
55	11, 14, 54	sylancr 695	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
56		ifcl 4130	. . . . . . . . . . 11 ⊢ (((ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
57	12, 18, 56	sylancl 694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
58	12	iblre 23560	. . . . . . . . . . . 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 222	. . . . . . . . . . 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 475	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1)
61	57, 60	itgcl 23550	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
62	12	renegcld 10457	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) ∈ ℝ)
63		ifcl 4130	. . . . . . . . . . 11 ⊢ ((-(ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
64	62, 18, 63	sylancl 694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
65	59	simprd 479	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1)
66	64, 65	itgcl 23550	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
67	61, 66	negsubdi2d 10408	. . . . . . . 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 13950	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
69	68	breq2d 4665	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℑ‘-𝐵) ↔ 0 ≤ -(ℑ‘𝐵)))
70	69, 68	ifbieq1d 4109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) = if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0))
71	70	itgeq2dv 23548	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥)
72	68	negeqd 10275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = --(ℑ‘𝐵))
73	12	recnd 10068	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
74	73	negnegd 10383	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℑ‘𝐵) = (ℑ‘𝐵))
75	72, 74	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = (ℑ‘𝐵))
76	75	breq2d 4665	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℑ‘-𝐵) ↔ 0 ≤ (ℑ‘𝐵)))
77	76, 75	ifbieq1d 4109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) = if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0))
78	77	itgeq2dv 23548	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥)
79	71, 78	oveq12d 6668	. . . . . . . 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 2659	. . . . . . 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 23563	. . . . . . . 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 10275	. . . . . . 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 13935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) ∈ ℝ)
84	37	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)
85	83, 84	itgreval 23563	. . . . . . 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 2666	. . . . . 6 ⊢ (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = ∫𝐴(ℑ‘-𝐵) d𝑥)
87	86	oveq2d 6666	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
88	55, 87	eqtr3d 2658	. . . 4 ⊢ (𝜑 → -(i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
89	53, 88	oveq12d 6668	. . 3 ⊢ (𝜑 → (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
90	17, 89	eqtrd 2656	. 2 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
91	4, 1	itgcnval 23566	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
92	91	negeqd 10275	. 2 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
93	33, 35	itgcnval 23566	. 2 ⊢ (𝜑 → ∫𝐴-𝐵 d𝑥 = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
94	90, 92, 93	3eqtr4d 2666	1 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  ici 9938   + caddc 9939   · cmul 9941   ≤ cle 10075   − cmin 10266  -cneg 10267  ℜcre 13837  ℑcim 13838  MblFncmbf 23383  𝐿¹cibl 23386  ∫citg 23387

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  df-iun 4522  df-disj 4621  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437

This theorem is referenced by:  itgsub  23592  itgsubnc  33472  itgmulc2nc  33478  sqwvfourb  40446