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Theorem itgaddlem2 25944

Description: Lemma for itgadd 25945. (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 13213	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
3	1, 2	syl 18	. . . . . . . . 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 13213	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
6	4, 5	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
7	3, 6	oveq12d 7418	. . . . . . . 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 11198	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
9		ifcl 4529	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
10	1, 8, 9	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
11	10	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
12		ifcl 4529	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
13	4, 8, 12	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
14	13	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
15	1	renegcld 11629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
16		ifcl 4529	. . . . . . . . . . 11 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
17	15, 8, 16	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
18	17	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
19	4	renegcld 11629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ)
20		ifcl 4529	. . . . . . . . . . 11 ⊢ ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
21	19, 8, 20	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
22	21	recnd 11225	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
23	11, 14, 18, 22	addsub4d 11604	. . . . . . . 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 11226	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ)
25		max0sub 13213	. . . . . . . . 9 ⊢ ((𝐵 + 𝐶) ∈ ℝ → (if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) − if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) = (𝐵 + 𝐶))
26	24, 25	syl 18	. . . . . . . 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 2811	. . . . . . 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 11629	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -(𝐵 + 𝐶) ∈ ℝ)
29		ifcl 4529	. . . . . . . . . 10 ⊢ ((-(𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
30	28, 8, 29	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℝ)
31	30	recnd 11225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) ∈ ℂ)
32	11, 14	addcld 11216	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℂ)
33		ifcl 4529	. . . . . . . . . 10 ⊢ (((𝐵 + 𝐶) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
34	24, 8, 33	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℝ)
35	34	recnd 11225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) ∈ ℂ)
36	18, 22	addcld 11216	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℂ)
37	31, 32, 35, 36	addsubeq4d 11608	. . . . . . 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 260	. . . . . 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 25902	. . . . 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 25941	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿¹)
45	24	iblre 25914	. . . . . . . 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 235	. . . . . . 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 500	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0)) ∈ 𝐿¹)
48	10, 13	readdcld 11226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
49	1	iblre 25914	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿¹))
51	50	simpld 499	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿¹)
52	4	iblre 25914	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿¹))
54	53	simpld 499	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ 𝐿¹)
55	10, 51, 13, 54	ibladd 25941	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ 𝐿¹)
56		max1 13202	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ -(𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
57	8, 28, 56	sylancr 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0))
58		max1 13202	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
59	8, 1, 58	sylancr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
60		max1 13202	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
61	8, 4, 60	sylancr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
62	10, 13, 59, 61	addge0d 11778	. . . . . 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 25943	. . . . 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 499	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0)) ∈ 𝐿¹)
65	17, 21	readdcld 11226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) ∈ ℝ)
66	50	simprd 500	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿¹)
67	53	simprd 500	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)) ∈ 𝐿¹)
68	17, 66, 21, 67	ibladd 25941	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0))) ∈ 𝐿¹)
69		max1 13202	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ (𝐵 + 𝐶) ∈ ℝ) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
70	8, 24, 69	sylancr 598	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0))
71		max1 13202	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
72	8, 15, 71	sylancr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
73		max1 13202	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
74	8, 19, 73	sylancr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
75	17, 21, 72, 74	addge0d 11778	. . . . . 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 25943	. . . . 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 2808	. . . 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 25904	. . . . 5 ⊢ (𝜑 → ∫𝐴if(0 ≤ -(𝐵 + 𝐶), -(𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
79	10, 51, 13, 54, 10, 13, 59, 61	itgaddlem1 25943	. . . . . 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 25904	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
81	13, 54	itgcl 25904	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 ∈ ℂ)
82	80, 81	addcld 11216	. . . . . 6 ⊢ (𝜑 → (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥) ∈ ℂ)
83	79, 82	eqeltrd 2865	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) d𝑥 ∈ ℂ)
84	34, 64	itgcl 25904	. . . . 5 ⊢ (𝜑 → ∫𝐴if(0 ≤ (𝐵 + 𝐶), (𝐵 + 𝐶), 0) d𝑥 ∈ ℂ)
85	17, 66, 21, 67, 17, 21, 72, 74	itgaddlem1 25943	. . . . . 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 25904	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
87	21, 67	itgcl 25904	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥 ∈ ℂ)
88	86, 87	addcld 11216	. . . . . 6 ⊢ (𝜑 → (∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 + ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥) ∈ ℂ)
89	85, 88	eqeltrd 2865	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐵, -𝐵, 0) + if(0 ≤ -𝐶, -𝐶, 0)) d𝑥 ∈ ℂ)
90	78, 83, 84, 89	addsubeq4d 11608	. . . 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 235	. . 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 7418	. . 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 11604	. . 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 2804	. 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 25917	. 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 25917	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
97	4, 43	itgreval 25917	. . 3 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴if(0 ≤ 𝐶, 𝐶, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐶, -𝐶, 0) d𝑥))
98	96, 97	oveq12d 7418	. 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 2810	1 ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 + caddc 11091 ≤ cle 11232 − cmin 11429 -cneg 11430 𝐿¹cibl 25737 ∫citg 25738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cc 10407 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-disj 5073 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-oadd 8445 df-omul 8446 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9875 df-card 9913 df-acn 9916 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-n0 12496 df-z 12583 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-clim 15529 df-rlim 15530 df-sum 15728 df-rest 17465 df-topgen 17486 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-top 23012 df-topon 23029 df-bases 23064 df-cmp 23505 df-ovol 25584 df-vol 25585 df-mbf 25739 df-itg1 25740 df-itg2 25741 df-ibl 25742 df-itg 25743 df-0p 25790

This theorem is referenced by: itgadd 25945

W3C validator