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Theorem itgsplitioo 25202

Description: The ∫ integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)

**Hypotheses**
Ref	Expression
itgsplitioo.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
itgsplitioo.2	⊢ (𝜑 → 𝐶 ∈ ℝ)
itgsplitioo.3	⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
itgsplitioo.4	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
itgsplitioo.5	⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
itgsplitioo.6	⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)

**Assertion**
Ref	Expression
itgsplitioo	⊢ (𝜑 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))

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

**Proof of Theorem itgsplitioo**
Step	Hyp	Ref	Expression
1		itgsplitioo.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))
2		itgsplitioo.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		itgsplitioo.2	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4		elicc2 13329	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)))
5	2, 3, 4	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)))
6	1, 5	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))
7	6	simp2d 1143	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
8	6	simp1d 1142	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	2, 8	leloed 11298	. . . . 5 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
10	7, 9	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
11	10	ord 862	. . 3 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵))
12	2	rexrd 11205	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
13		iooss1 13299	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
14	12, 7, 13	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
15	14	sselda 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴(,)𝐶))
16		itgsplitioo.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
17	15, 16	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐷 ∈ ℂ)
18		itgsplitioo.6	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
19	17, 18	itgcl 25148	. . . . . 6 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 ∈ ℂ)
20	19	addid2d 11356	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = ∫(𝐵(,)𝐶)𝐷 d𝑥)
21	20	eqcomd 2742	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
22		oveq1 7364	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐶) = (𝐵(,)𝐶))
23		itgeq1 25137	. . . . . 6 ⊢ ((𝐴(,)𝐶) = (𝐵(,)𝐶) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
24	22, 23	syl 17	. . . . 5 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
25		oveq1 7364	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵))
26		iooid 13292	. . . . . . . . 9 ⊢ (𝐵(,)𝐵) = ∅
27	25, 26	eqtrdi 2792	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅)
28		itgeq1 25137	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) = ∅ → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫∅𝐷 d𝑥)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫∅𝐷 d𝑥)
30		itg0 25144	. . . . . . 7 ⊢ ∫∅𝐷 d𝑥 = 0
31	29, 30	eqtrdi 2792	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = 0)
32	31	oveq1d 7372	. . . . 5 ⊢ (𝐴 = 𝐵 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
33	24, 32	eqeq12d 2752	. . . 4 ⊢ (𝐴 = 𝐵 → (∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) ↔ ∫(𝐵(,)𝐶)𝐷 d𝑥 = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
34	21, 33	syl5ibrcom 246	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
35	11, 34	syld 47	. 2 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
36	6	simp3d 1144	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
37	8, 3	leloed 11298	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
38	36, 37	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))
39	38	ord 862	. . 3 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → 𝐵 = 𝐶))
40	3	rexrd 11205	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
41		iooss2 13300	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
42	40, 36, 41	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
43	42	sselda 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐶))
44	43, 16	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 ∈ ℂ)
45		itgsplitioo.5	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
46	44, 45	itgcl 25148	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 ∈ ℂ)
47	46	addid1d 11355	. . . . 5 ⊢ (𝜑 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = ∫(𝐴(,)𝐵)𝐷 d𝑥)
48	47	eqcomd 2742	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0))
49		oveq2 7365	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴(,)𝐵) = (𝐴(,)𝐶))
50		itgeq1 25137	. . . . . 6 ⊢ ((𝐴(,)𝐵) = (𝐴(,)𝐶) → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫(𝐴(,)𝐶)𝐷 d𝑥)
51	49, 50	syl 17	. . . . 5 ⊢ (𝐵 = 𝐶 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫(𝐴(,)𝐶)𝐷 d𝑥)
52		oveq2 7365	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (𝐵(,)𝐵) = (𝐵(,)𝐶))
53	26, 52	eqtr3id 2790	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ∅ = (𝐵(,)𝐶))
54		itgeq1 25137	. . . . . . . 8 ⊢ (∅ = (𝐵(,)𝐶) → ∫∅𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
55	53, 54	syl 17	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ∫∅𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
56	30, 55	eqtr3id 2790	. . . . . 6 ⊢ (𝐵 = 𝐶 → 0 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
57	56	oveq2d 7373	. . . . 5 ⊢ (𝐵 = 𝐶 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
58	51, 57	eqeq12d 2752	. . . 4 ⊢ (𝐵 = 𝐶 → (∫(𝐴(,)𝐵)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) ↔ ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
59	48, 58	syl5ibcom 244	. . 3 ⊢ (𝜑 → (𝐵 = 𝐶 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
60	39, 59	syld 47	. 2 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
61		indir 4235	. . . . . . . 8 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶)))
62	8	rexrd 11205	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
63	12, 62	jca 512	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
64	63	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
65	62, 40	jca 512	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
66	65	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
67	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
68	67	leidd 11721	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
69		ioodisj 13399	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐶 ∈ ℝ^*)) ∧ 𝐵 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
70	64, 66, 68, 69	syl21anc 836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
71		incom 4161	. . . . . . . . . . 11 ⊢ ({𝐵} ∩ (𝐵(,)𝐶)) = ((𝐵(,)𝐶) ∩ {𝐵})
72	67	ltnrd 11289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐵)
73		eliooord 13323	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐵 ∧ 𝐵 < 𝐶))
74	73	simpld 495	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → 𝐵 < 𝐵)
75	72, 74	nsyl 140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 ∈ (𝐵(,)𝐶))
76		disjsn 4672	. . . . . . . . . . . 12 ⊢ (((𝐵(,)𝐶) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (𝐵(,)𝐶))
77	75, 76	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐵(,)𝐶) ∩ {𝐵}) = ∅)
78	71, 77	eqtrid 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∩ (𝐵(,)𝐶)) = ∅)
79	70, 78	uneq12d 4124	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = (∅ ∪ ∅))
80		un0 4350	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
81	79, 80	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = ∅)
82	61, 81	eqtrid 2788	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = ∅)
83	82	fveq2d 6846	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = (vol‘∅))
84		ovol0 24857	. . . . . 6 ⊢ (vol*‘∅) = 0
85	83, 84	eqtrdi 2792	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = 0)
86	12, 62, 40	3jca 1128	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
87		ioojoin 13400	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
88	86, 87	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
89	88	eqcomd 2742	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐶) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)))
90	16	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
91	45	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
92		ssun1 4132	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵})
93	92	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵}))
94		ioossre 13325	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
95	94	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ℝ)
96	67	snssd 4769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ℝ)
97	95, 96	unssd 4146	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ)
98		uncom 4113	. . . . . . . . . . . 12 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) = ({𝐵} ∪ (𝐴(,)𝐵))
99	98	difeq1i 4078	. . . . . . . . . . 11 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))
100		difun2 4440	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
101	99, 100	eqtri 2764	. . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
102		difss 4091	. . . . . . . . . 10 ⊢ ({𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
103	101, 102	eqsstri 3978	. . . . . . . . 9 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
104		ovolsn 24859	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0)
105	67, 104	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘{𝐵}) = 0)
106		ovolssnul 24851	. . . . . . . . 9 ⊢ (((((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) ⊆ {𝐵} ∧ {𝐵} ⊆ ℝ ∧ (vol‘{𝐵}) = 0) → (vol‘(((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵))) = 0)
107	103, 96, 105, 106	mp3an2i 1466	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵))) = 0)
108		ssun1 4132	. . . . . . . . . . 11 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶))
109	108, 88	sseqtrid 3996	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (𝐴(,)𝐶))
110	109	sselda 3944	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝑥 ∈ (𝐴(,)𝐶))
111	110, 90	syldan 591	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝐷 ∈ ℂ)
112	93, 97, 107, 111	itgss3 25179	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹ ↔ (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹) ∧ ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥))
113	112	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹ ↔ (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹))
114	91, 113	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹)
115	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
116	85, 89, 90, 114, 115	itgsplit 25200	. . . 4 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
117	112	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥)
118	117	oveq1d 7372	. . . 4 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = (∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
119	116, 118	eqtr4d 2779	. . 3 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
120	119	ex 413	. 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
121	35, 60, 120	ecased 1033	1 ⊢ (𝜑 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 (,)cioo 13264 [,]cicc 13267 vol*covol 24826 𝐿¹cibl 24981 ∫citg 24982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-rest 17304 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296 df-cmp 22738 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034

This theorem is referenced by: ditgsplitlem 25224 ftc1lem1 25399 ftc1anc 36159 fourierdlem103 44440 fourierdlem104 44441 fourierdlem111 44448 sqwvfoura 44459 sqwvfourb 44460

W3C validator