Theorem itgneg 24873

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 24837	. . . . . . . 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 24705	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6	5	recld 14833	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
7	5	iblcn 24868	. . . . . . 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 494	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1)
10	6, 9	itgcl 24853	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
11		ax-icn 10861	. . . . 5 ⊢ i ∈ ℂ
12	5	imcld 14834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
13	8	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)
14	12, 13	itgcl 24853	. . . . 5 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
15		mulcl 10886	. . . . 5 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
16	11, 14, 15	sylancr 586	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
17	10, 16	negdid 11275	. . 3 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)))
18		0re 10908	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4501	. . . . . . . 8 ⊢ (((ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
20	6, 18, 19	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) ∈ ℝ)
21	6	iblre 24863	. . . . . . . . 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 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0)) ∈ 𝐿1)
24	20, 23	itgcl 24853	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
25	6	renegcld 11332	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘𝐵) ∈ ℝ)
26		ifcl 4501	. . . . . . . 8 ⊢ ((-(ℜ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
27	25, 18, 26	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) ∈ ℝ)
28	22	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0)) ∈ 𝐿1)
29	27, 28	itgcl 24853	. . . . . 6 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥 ∈ ℂ)
30	24, 29	negsubdi2d 11278	. . . . 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 24866	. . . . . 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 11145	. . . . 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 11249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℂ)
34	33	recld 14833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) ∈ ℝ)
35	4, 1	iblneg 24872	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ 𝐿1)
36	33	iblcn 24868	. . . . . . . . 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 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘-𝐵)) ∈ 𝐿1)
39	34, 38	itgreval 24866	. . . . . 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 14848	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘-𝐵) = -(ℜ‘𝐵))
41	40	breq2d 5082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℜ‘-𝐵) ↔ 0 ≤ -(ℜ‘𝐵)))
42	41, 40	ifbieq1d 4480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) = if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0))
43	42	itgeq2dv 24851	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℜ‘-𝐵), (ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℜ‘𝐵), -(ℜ‘𝐵), 0) d𝑥)
44	40	negeqd 11145	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = --(ℜ‘𝐵))
45	6	recnd 10934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℂ)
46	45	negnegd 11253	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℜ‘𝐵) = (ℜ‘𝐵))
47	44, 46	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℜ‘-𝐵) = (ℜ‘𝐵))
48	47	breq2d 5082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℜ‘-𝐵) ↔ 0 ≤ (ℜ‘𝐵)))
49	48, 47	ifbieq1d 4480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) = if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0))
50	49	itgeq2dv 24851	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℜ‘-𝐵), -(ℜ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℜ‘𝐵), (ℜ‘𝐵), 0) d𝑥)
51	43, 50	oveq12d 7273	. . . . . 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 2778	. . . . 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 2788	. . . 4 ⊢ (𝜑 → -∫𝐴(ℜ‘𝐵) d𝑥 = ∫𝐴(ℜ‘-𝐵) d𝑥)
54		mulneg2 11342	. . . . . 6 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
55	11, 14, 54	sylancr 586	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = -(i · ∫𝐴(ℑ‘𝐵) d𝑥))
56		ifcl 4501	. . . . . . . . . . 11 ⊢ (((ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
57	12, 18, 56	sylancl 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) ∈ ℝ)
58	12	iblre 24863	. . . . . . . . . . . 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 494	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0)) ∈ 𝐿1)
61	57, 60	itgcl 24853	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
62	12	renegcld 11332	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘𝐵) ∈ ℝ)
63		ifcl 4501	. . . . . . . . . . 11 ⊢ ((-(ℑ‘𝐵) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
64	62, 18, 63	sylancl 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) ∈ ℝ)
65	59	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0)) ∈ 𝐿1)
66	64, 65	itgcl 24853	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥 ∈ ℂ)
67	61, 66	negsubdi2d 11278	. . . . . . . 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 14849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) = -(ℑ‘𝐵))
69	68	breq2d 5082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ (ℑ‘-𝐵) ↔ 0 ≤ -(ℑ‘𝐵)))
70	69, 68	ifbieq1d 4480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) = if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0))
71	70	itgeq2dv 24851	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ (ℑ‘-𝐵), (ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ -(ℑ‘𝐵), -(ℑ‘𝐵), 0) d𝑥)
72	68	negeqd 11145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = --(ℑ‘𝐵))
73	12	recnd 10934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℂ)
74	73	negnegd 11253	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --(ℑ‘𝐵) = (ℑ‘𝐵))
75	72, 74	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(ℑ‘-𝐵) = (ℑ‘𝐵))
76	75	breq2d 5082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -(ℑ‘-𝐵) ↔ 0 ≤ (ℑ‘𝐵)))
77	76, 75	ifbieq1d 4480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) = if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0))
78	77	itgeq2dv 24851	. . . . . . . . 9 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(ℑ‘-𝐵), -(ℑ‘-𝐵), 0) d𝑥 = ∫𝐴if(0 ≤ (ℑ‘𝐵), (ℑ‘𝐵), 0) d𝑥)
79	71, 78	oveq12d 7273	. . . . . . . 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 2781	. . . . . . 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 24866	. . . . . . . 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 11145	. . . . . . 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 14834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘-𝐵) ∈ ℝ)
84	37	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘-𝐵)) ∈ 𝐿1)
85	83, 84	itgreval 24866	. . . . . . 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 2788	. . . . . 6 ⊢ (𝜑 → -∫𝐴(ℑ‘𝐵) d𝑥 = ∫𝐴(ℑ‘-𝐵) d𝑥)
87	86	oveq2d 7271	. . . . 5 ⊢ (𝜑 → (i · -∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
88	55, 87	eqtr3d 2780	. . . 4 ⊢ (𝜑 → -(i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ∫𝐴(ℑ‘-𝐵) d𝑥))
89	53, 88	oveq12d 7273	. . 3 ⊢ (𝜑 → (-∫𝐴(ℜ‘𝐵) d𝑥 + -(i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
90	17, 89	eqtrd 2778	. 2 ⊢ (𝜑 → -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
91	4, 1	itgcnval 24869	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
92	91	negeqd 11145	. 2 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = -(∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
93	33, 35	itgcnval 24869	. 2 ⊢ (𝜑 → ∫𝐴-𝐵 d𝑥 = (∫𝐴(ℜ‘-𝐵) d𝑥 + (i · ∫𝐴(ℑ‘-𝐵) d𝑥)))
94	90, 92, 93	3eqtr4d 2788	1 ⊢ (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  0cc0 10802  ici 10804   + caddc 10805   · cmul 10807   ≤ cle 10941   − cmin 11135  -cneg 11136  ℜcre 14736  ℑcim 14737  MblFncmbf 24683  𝐿¹cibl 24686  ∫citg 24687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689  df-itg2 24690  df-ibl 24691  df-itg 24692  df-0p 24739

This theorem is referenced by:  itgsub  24895  itgsubnc  35766  itgmulc2nc  35772  sqwvfourb  43660