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Theorem itgsplitioo 25978

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 13439	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)))
5	2, 3, 4	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)))
6	1, 5	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶))
7	6	simp2d 1161	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
8	6	simp1d 1160	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	2, 8	leloed 11354	. . . . 5 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
10	7, 9	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
11	10	ord 877	. . 3 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵))
12	2	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
13		iooss1 13408	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
14	12, 7, 13	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
15	14	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴(,)𝐶))
16		itgsplitioo.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
17	15, 16	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐷 ∈ ℂ)
18		itgsplitioo.6	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
19	17, 18	itgcl 25924	. . . . . 6 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 ∈ ℂ)
20	19	addlidd 11412	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = ∫(𝐵(,)𝐶)𝐷 d𝑥)
21	20	eqcomd 2769	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
22		oveq1 7419	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐶) = (𝐵(,)𝐶))
23		itgeq1 25913	. . . . . 6 ⊢ ((𝐴(,)𝐶) = (𝐵(,)𝐶) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
24	22, 23	syl 18	. . . . 5 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
25		oveq1 7419	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵))
26		iooid 13401	. . . . . . . . 9 ⊢ (𝐵(,)𝐵) = ∅
27	25, 26	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅)
28		itgeq1 25913	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) = ∅ → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫∅𝐷 d𝑥)
29	27, 28	syl 18	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫∅𝐷 d𝑥)
30		itg0 25920	. . . . . . 7 ⊢ ∫∅𝐷 d𝑥 = 0
31	29, 30	eqtrdi 2814	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = 0)
32	31	oveq1d 7427	. . . . 5 ⊢ (𝐴 = 𝐵 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
33	24, 32	eqeq12d 2779	. . . 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 250	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
35	11, 34	syld 48	. 2 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
36	6	simp3d 1162	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
37	8, 3	leloed 11354	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
38	36, 37	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))
39	38	ord 877	. . 3 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → 𝐵 = 𝐶))
40	3	rexrd 11260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
41		iooss2 13409	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
42	40, 36, 41	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
43	42	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐶))
44	43, 16	syldan 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 ∈ ℂ)
45		itgsplitioo.5	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
46	44, 45	itgcl 25924	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 ∈ ℂ)
47	46	addridd 11411	. . . . 5 ⊢ (𝜑 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = ∫(𝐴(,)𝐵)𝐷 d𝑥)
48	47	eqcomd 2769	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0))
49		oveq2 7420	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴(,)𝐵) = (𝐴(,)𝐶))
50		itgeq1 25913	. . . . . 6 ⊢ ((𝐴(,)𝐵) = (𝐴(,)𝐶) → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫(𝐴(,)𝐶)𝐷 d𝑥)
51	49, 50	syl 18	. . . . 5 ⊢ (𝐵 = 𝐶 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫(𝐴(,)𝐶)𝐷 d𝑥)
52		oveq2 7420	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (𝐵(,)𝐵) = (𝐵(,)𝐶))
53	26, 52	eqtr3id 2812	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ∅ = (𝐵(,)𝐶))
54		itgeq1 25913	. . . . . . . 8 ⊢ (∅ = (𝐵(,)𝐶) → ∫∅𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
55	53, 54	syl 18	. . . . . . 7 ⊢ (𝐵 = 𝐶 → ∫∅𝐷 d𝑥 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
56	30, 55	eqtr3id 2812	. . . . . 6 ⊢ (𝐵 = 𝐶 → 0 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
57	56	oveq2d 7428	. . . . 5 ⊢ (𝐵 = 𝐶 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
58	51, 57	eqeq12d 2779	. . . 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 248	. . 3 ⊢ (𝜑 → (𝐵 = 𝐶 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
60	39, 59	syld 48	. 2 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
61		indir 4240	. . . . . . . 8 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶)))
62	8	rexrd 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
63	12, 62	jca 520	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
64	63	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
65	62, 40	jca 520	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
66	65	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
67	8	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
68	67	leidd 11781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
69		ioodisj 13510	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐶 ∈ ℝ^*)) ∧ 𝐵 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
70	64, 66, 68, 69	syl21anc 850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
71		incom 4163	. . . . . . . . . . 11 ⊢ ({𝐵} ∩ (𝐵(,)𝐶)) = ((𝐵(,)𝐶) ∩ {𝐵})
72	67	ltnrd 11345	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐵)
73		eliooord 13433	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐵 ∧ 𝐵 < 𝐶))
74	73	simpld 499	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → 𝐵 < 𝐵)
75	72, 74	nsyl 141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 ∈ (𝐵(,)𝐶))
76		disjsn 4678	. . . . . . . . . . . 12 ⊢ (((𝐵(,)𝐶) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (𝐵(,)𝐶))
77	75, 76	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐵(,)𝐶) ∩ {𝐵}) = ∅)
78	71, 77	eqtrid 2810	. . . . . . . . . 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 4352	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
81	79, 80	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = ∅)
82	61, 81	eqtrid 2810	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = ∅)
83	82	fveq2d 6887	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = (vol‘∅))
84		ovol0 25633	. . . . . 6 ⊢ (vol*‘∅) = 0
85	83, 84	eqtrdi 2814	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = 0)
86	12, 62, 40	3jca 1146	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
87		ioojoin 13511	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
88	86, 87	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
89	88	eqcomd 2769	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐶) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)))
90	16	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
91	45	adantr 485	. . . . . 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 13435	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
95	94	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ℝ)
96	67	snssd 4753	. . . . . . . . 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 4443	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
101	99, 100	eqtri 2786	. . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
102		difss 4091	. . . . . . . . . 10 ⊢ ({𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
103	101, 102	eqsstri 3984	. . . . . . . . 9 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
104		ovolsn 25635	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0)
105	67, 104	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘{𝐵}) = 0)
106		ovolssnul 25627	. . . . . . . . 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 1495	. . . . . . . 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 3980	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (𝐴(,)𝐶))
110	109	sselda 3938	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝑥 ∈ (𝐴(,)𝐶))
111	110, 90	syldan 602	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝐷 ∈ ℂ)
112	93, 97, 107, 111	itgss3 25955	. . . . . . 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 499	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹ ↔ (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹))
114	91, 113	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹)
115	18	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
116	85, 89, 90, 114, 115	itgsplit 25976	. . . 4 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
117	112	simprd 500	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥)
118	117	oveq1d 7427	. . . 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 2801	. . 3 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
120	119	ex 417	. 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
121	35, 60, 120	ecased 1051	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 + caddc 11104 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 (,)cioo 13373 [,]cicc 13376 vol*covol 25602 𝐿¹cibl 25757 ∫citg 25758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4207 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-disj 5078 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-rest 17476 df-topgen 17497 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-top 23032 df-topon 23049 df-bases 23084 df-cmp 23525 df-ovol 25604 df-vol 25605 df-mbf 25759 df-itg1 25760 df-itg2 25761 df-ibl 25762 df-itg 25763 df-0p 25810

This theorem is referenced by: ditgsplitlem 26000 ftc1lem1 26175 ftc1anc 38330 fourierdlem103 46903 fourierdlem104 46904 fourierdlem111 46911 sqwvfoura 46922 sqwvfourb 46923

W3C validator