Theorem itgaddlem2 25771

Description: Lemma for itgadd 25772. (Contributed by Mario Carneiro, 17-Aug-2014.)

Hypotheses

Ref	Expression
itgadd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgadd.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)
itgadd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
itgadd.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
itgadd.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
itgadd.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)

Assertion

Ref	Expression
itgaddlem2	⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem itgaddlem2

Step	Hyp	Ref	Expression
1		itgadd.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
2		max0sub 13207	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
4		itgadd.6	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
5		max0sub 13207	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
7	3, 6	oveq12d 7434	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) + (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0))) = (𝐵 + 𝐶))
8		0re 11246	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
9		ifcl 4569	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
10	1, 8, 9	sylancl 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
11	10	recnd 11272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
12		ifcl 4569	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
13	4, 8, 12	sylancl 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
14	13	recnd 11272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
15	1	renegcld 11671	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
16		ifcl 4569	. . . . . . . . . . 11 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
17	15, 8, 16	sylancl 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
18	17	recnd 11272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
19	4	renegcld 11671	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ)
20		ifcl 4569	. . . . . . . . . . 11 ⊢ ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
21	19, 8, 20	sylancl 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
22	21	recnd 11272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
23	11, 14, 18, 22	addsub4d 11648	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) = ((if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) + (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0))))
24	1, 4	readdcld 11273	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ)
25		max0sub 13207	. . . . . . . . 9 ⊢ ((𝐵 + 𝐶) ∈ ℝ → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶))
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶))
27	7, 23, 26	3eqtr4rd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
28	24	renegcld 11671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐵 + 𝐶) ∈ ℝ)
29		ifcl 4569	. . . . . . . . . 10 ⊢ ((-(𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
30	28, 8, 29	sylancl 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
31	30	recnd 11272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℂ)
32	11, 14	addcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℂ)
33		ifcl 4569	. . . . . . . . . 10 ⊢ (((𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
34	24, 8, 33	sylancl 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
35	34	recnd 11272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℂ)
36	18, 22	addcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℂ)
37	31, 32, 35, 36	addsubeq4d 11652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ↔ (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) − (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))))
38	27, 37	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))))
39	38	itgeq2dv 25729	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥)
40		itgadd.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
41		itgadd.2	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)
42		itgadd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
43		itgadd.4	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)
44	40, 41, 42, 43	ibladd 25768	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
45	24	iblre 25741	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1)))
46	44, 45	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1))
47	46	simprd 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿1)
48	10, 13	readdcld 11273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
49	1	iblre 25741	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
50	41, 49	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))
51	50	simpld 493	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1)
52	4	iblre 25741	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)))
53	43, 52	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1))
54	53	simpld 493	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿1)
55	10, 51, 13, 54	ibladd 25768	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ 𝐿1)
56		max1 13196	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ -(𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
57	8, 28, 56	sylancr 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
58		max1 13196	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
59	8, 1, 58	sylancr 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
60		max1 13196	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
61	8, 4, 60	sylancr 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
62	10, 13, 59, 61	addge0d 11820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
63	30, 47, 48, 55, 30, 48, 57, 62	itgaddlem1 25770	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) + (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥))
64	46	simpld 493	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿1)
65	17, 21	readdcld 11273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℝ)
66	50	simprd 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)
67	53	simprd 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿1)
68	17, 66, 21, 67	ibladd 25768	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ 𝐿1)
69		max1 13196	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
70	8, 24, 69	sylancr 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
71		max1 13196	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
72	8, 15, 71	sylancr 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
73		max1 13196	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
74	8, 19, 73	sylancr 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
75	17, 21, 72, 74	addge0d 11820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)))
76	34, 64, 65, 68, 34, 65, 70, 75	itgaddlem1 25770	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) + (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
77	39, 63, 76	3eqtr3d 2773	. . . 4 ⊢ (𝜑 → (∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
78	30, 47	itgcl 25731	. . . . 5 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
79	10, 51, 13, 54, 10, 13, 59, 61	itgaddlem1 25770	. . . . . 6 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥))
80	10, 51	itgcl 25731	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
81	13, 54	itgcl 25731	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 ∈ ℂ)
82	80, 81	addcld 11263	. . . . . 6 ⊢ (𝜑 → (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) ∈ ℂ)
83	79, 82	eqeltrd 2825	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 ∈ ℂ)
84	34, 64	itgcl 25731	. . . . 5 ⊢ (𝜑 → ∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
85	17, 66, 21, 67, 17, 21, 72, 74	itgaddlem1 25770	. . . . . 6 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 = (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
86	17, 66	itgcl 25731	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
87	21, 67	itgcl 25731	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥 ∈ ℂ)
88	86, 87	addcld 11263	. . . . . 6 ⊢ (𝜑 → (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥) ∈ ℂ)
89	85, 88	eqeltrd 2825	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 ∈ ℂ)
90	78, 83, 84, 89	addsubeq4d 11652	. . . 4 ⊢ (𝜑 → ((∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥) = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 + ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) ↔ (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥)))
91	77, 90	mpbid 231	. . 3 ⊢ (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥))
92	79, 85	oveq12d 7434	. . 3 ⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
93	80, 81, 86, 87	addsub4d 11648	. . 3 ⊢ (𝜑 → ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) − (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
94	91, 92, 93	3eqtrd 2769	. 2 ⊢ (𝜑 → (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
95	24, 44	itgreval 25744	. 2 ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 − ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥))
96	1, 41	itgreval 25744	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
97	4, 43	itgreval 25744	. . 3 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
98	96, 97	oveq12d 7434	. 2 ⊢ (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) + (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥)))
99	94, 95, 98	3eqtr4d 2775	1 ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ifcif 4524   class class class wbr 5143   ↦ cmpt 5226  (class class class)co 7416  ℂcc 11136  ℝcr 11137  0cc0 11138   + caddc 11141   ≤ cle 11279   − cmin 11474  -cneg 11475  𝐿¹cibl 25564  ∫citg 25565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-cc 10458  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216  ax-addf 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5109  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-ofr 7683  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-er 8723  df-map 8845  df-pm 8846  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-fi 9434  df-sup 9465  df-inf 9466  df-oi 9533  df-dju 9924  df-card 9962  df-acn 9965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-n0 12503  df-z 12589  df-uz 12853  df-q 12963  df-rp 13007  df-xneg 13124  df-xadd 13125  df-xmul 13126  df-ioo 13360  df-ioc 13361  df-ico 13362  df-icc 13363  df-fz 13517  df-fzo 13660  df-fl 13789  df-mod 13867  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-clim 15464  df-rlim 15465  df-sum 15665  df-rest 17403  df-topgen 17424  df-psmet 21275  df-xmet 21276  df-met 21277  df-bl 21278  df-mopn 21279  df-top 22814  df-topon 22831  df-bases 22867  df-cmp 23309  df-ovol 25411  df-vol 25412  df-mbf 25566  df-itg1 25567  df-itg2 25568  df-ibl 25569  df-itg 25570  df-0p 25617

This theorem is referenced by:  itgadd  25772