itgsplit - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itgsplit		Structured version Visualization version GIF version

Theorem itgsplit 25793

Description: The ∫ integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)

**Hypotheses**
Ref	Expression
itgsplit.i	⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
itgsplit.u	⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
itgsplit.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ 𝑉)
itgsplit.a	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
itgsplit.b	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
itgsplit	⊢ (𝜑 → ∫𝑈𝐶 d𝑥 = (∫𝐴𝐶 d𝑥 + ∫𝐵𝐶 d𝑥))

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

Proof of Theorem itgsplit Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgsplit.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
2		iblmbf 25724	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
4		ssun1 4130	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
5		itgsplit.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))
6	4, 5	sseqtrrid 3977	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
7	6	sselda 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑈)
8		itgsplit.c	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ 𝑉)
9	7, 8	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
10	3, 9	mbfdm2 25594	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ dom vol)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ∈ dom vol)
12		itgsplit.b	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹)
13		iblmbf 25724	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn)
15		ssun2 4131	. . . . . . . . . . . 12 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
16	15, 5	sseqtrrid 3977	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
17	16	sselda 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈)
18	17, 8	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
19	14, 18	mbfdm2 25594	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ dom vol)
20	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐵 ∈ dom vol)
21		itgsplit.i	. . . . . . . 8 ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘(𝐴 ∩ 𝐵)) = 0)
23	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑈 = (𝐴 ∪ 𝐵))
24	5	eleq2d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ 𝑥 ∈ (𝐴 ∪ 𝐵)))
25		elun 4105	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
26	24, 25	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)))
27	26	biimpa 476	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
28	3, 9	mbfmptcl 25593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
29	14, 18	mbfmptcl 25593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
30	28, 29	jaodan 959	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → 𝐶 ∈ ℂ)
31	27, 30	syldan 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
32	31	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)
33		ax-icn 11085	. . . . . . . . . . . . . 14 ⊢ i ∈ ℂ
34		elfznn0 13536	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℕ₀)
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑘 ∈ ℕ₀)
36		expcl 14002	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (i↑𝑘) ∈ ℂ)
37	33, 35, 36	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (i↑𝑘) ∈ ℂ)
38	37	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ∈ ℂ)
39		ine0 11572	. . . . . . . . . . . . . 14 ⊢ i ≠ 0
40		elfzelz 13440	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
41	40	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝑘 ∈ ℤ)
42		expne0i 14017	. . . . . . . . . . . . . 14 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
43	33, 39, 41, 42	mp3an12i 1467	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (i↑𝑘) ≠ 0)
44	43	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (i↑𝑘) ≠ 0)
45	32, 38, 44	divcld 11917	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (𝐶 / (i↑𝑘)) ∈ ℂ)
46	45	recld 15117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
47		0re 11134	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
48		ifcl 4525	. . . . . . . . . 10 ⊢ (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
49	46, 47, 48	sylancl 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
50	49	rexrd 11182	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ^*)
51		max1 13100	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
52	47, 46, 51	sylancr 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
53		elxrge0 13373	. . . . . . . 8 ⊢ (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
54	50, 52, 53	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝑈) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
55		ifan 4533	. . . . . . . 8 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
56	55	mpteq2i 5194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
57		ifan 4533	. . . . . . . 8 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
58	57	mpteq2i 5194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
59		ifan 4533	. . . . . . . 8 ⊢ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
60	59	mpteq2i 5194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0))
61		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
62		eqidd 2737	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
63	61, 62, 1, 9	iblitg 25725	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
64	40, 63	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
65		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
66		eqidd 2737	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
67	65, 66, 12, 18	iblitg 25725	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
68	40, 67	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)
69	11, 20, 22, 23, 54, 56, 58, 60, 64, 68	itg2split 25706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))))
70	69	oveq2d 7374	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = ((i↑𝑘) · ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
71	63	recnd 11160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℂ)
72	40, 71	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℂ)
73	68	recnd 11160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℂ)
74	37, 72, 73	adddid 11156	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))) = (((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
75	70, 74	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = (((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
76	75	sumeq2dv 15625	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = Σ𝑘 ∈ (0...3)(((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
77		fzfid 13896	. . . 4 ⊢ (𝜑 → (0...3) ∈ Fin)
78	37, 72	mulcld 11152	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) ∈ ℂ)
79	37, 73	mulcld 11152	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) ∈ ℂ)
80	77, 78, 79	fsumadd 15663	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)(((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + ((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))) = (Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
81	76, 80	eqtrd 2771	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = (Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))))
82		eqid 2736	. . 3 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
83	82	dfitg 25726	. 2 ⊢ ∫𝑈𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝑈 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
84	82	dfitg 25726	. . 3 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
85	82	dfitg 25726	. . 3 ⊢ ∫𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
86	84, 85	oveq12i 7370	. 2 ⊢ (∫𝐴𝐶 d𝑥 + ∫𝐵𝐶 d𝑥) = (Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) + Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))))
87	81, 83, 86	3eqtr4g 2796	1 ⊢ (𝜑 → ∫𝑈𝐶 d𝑥 = (∫𝐴𝐶 d𝑥 + ∫𝐵𝐶 d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∪ cun 3899 ∩ cin 3900 ifcif 4479 class class class wbr 5098 ↦ cmpt 5179 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 ici 11028 + caddc 11029 · cmul 11031 +∞cpnf 11163 ℝ^*cxr 11165 ≤ cle 11167 / cdiv 11794 3c3 12201 ℕ₀cn0 12401 ℤcz 12488 [,]cicc 13264 ...cfz 13423 ↑cexp 13984 ℜcre 15020 Σcsu 15609 vol*covol 25419 volcvol 25420 MblFncmbf 25571 ∫₂citg2 25573 𝐿¹cibl 25574 ∫citg 25575

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-rest 17342 df-topgen 17363 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-top 22838 df-topon 22855 df-bases 22890 df-cmp 23331 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-itg 25580

This theorem is referenced by: itgspliticc 25794 itgsplitioo 25795

W3C validator