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Theorem itgsplitioo 25739

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 13372	. . . . . . . 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 232	. . . . . 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 11317	. . . . 5 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
10	7, 9	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
11	10	ord 864	. . 3 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵))
12	2	rexrd 11224	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
13		iooss1 13341	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
14	12, 7, 13	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
15	14	sselda 3946	. . . . . . . 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 25685	. . . . . 6 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 ∈ ℂ)
20	19	addlidd 11375	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = ∫(𝐵(,)𝐶)𝐷 d𝑥)
21	20	eqcomd 2735	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
22		oveq1 7394	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐶) = (𝐵(,)𝐶))
23		itgeq1 25674	. . . . . 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 7394	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵))
26		iooid 13334	. . . . . . . . 9 ⊢ (𝐵(,)𝐵) = ∅
27	25, 26	eqtrdi 2780	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅)
28		itgeq1 25674	. . . . . . . 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 25681	. . . . . . 7 ⊢ ∫∅𝐷 d𝑥 = 0
31	29, 30	eqtrdi 2780	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = 0)
32	31	oveq1d 7402	. . . . 5 ⊢ (𝐴 = 𝐵 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
33	24, 32	eqeq12d 2745	. . . 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 247	. . 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 11317	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
38	36, 37	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))
39	38	ord 864	. . 3 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → 𝐵 = 𝐶))
40	3	rexrd 11224	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
41		iooss2 13342	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
42	40, 36, 41	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
43	42	sselda 3946	. . . . . . . 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 25685	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 ∈ ℂ)
47	46	addridd 11374	. . . . 5 ⊢ (𝜑 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = ∫(𝐴(,)𝐵)𝐷 d𝑥)
48	47	eqcomd 2735	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0))
49		oveq2 7395	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴(,)𝐵) = (𝐴(,)𝐶))
50		itgeq1 25674	. . . . . 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 7395	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (𝐵(,)𝐵) = (𝐵(,)𝐶))
53	26, 52	eqtr3id 2778	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ∅ = (𝐵(,)𝐶))
54		itgeq1 25674	. . . . . . . 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 2778	. . . . . 6 ⊢ (𝐵 = 𝐶 → 0 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
57	56	oveq2d 7403	. . . . 5 ⊢ (𝐵 = 𝐶 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
58	51, 57	eqeq12d 2745	. . . 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 245	. . 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 4249	. . . . . . . 8 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶)))
62	8	rexrd 11224	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
63	12, 62	jca 511	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
64	63	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
65	62, 40	jca 511	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
66	65	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
67	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)
68	67	leidd 11744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
69		ioodisj 13443	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐶 ∈ ℝ^*)) ∧ 𝐵 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
70	64, 66, 68, 69	syl21anc 837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
71		incom 4172	. . . . . . . . . . 11 ⊢ ({𝐵} ∩ (𝐵(,)𝐶)) = ((𝐵(,)𝐶) ∩ {𝐵})
72	67	ltnrd 11308	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐵)
73		eliooord 13366	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐵 ∧ 𝐵 < 𝐶))
74	73	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (𝐵(,)𝐶) → 𝐵 < 𝐵)
75	72, 74	nsyl 140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 ∈ (𝐵(,)𝐶))
76		disjsn 4675	. . . . . . . . . . . 12 ⊢ (((𝐵(,)𝐶) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (𝐵(,)𝐶))
77	75, 76	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐵(,)𝐶) ∩ {𝐵}) = ∅)
78	71, 77	eqtrid 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∩ (𝐵(,)𝐶)) = ∅)
79	70, 78	uneq12d 4132	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = (∅ ∪ ∅))
80		un0 4357	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
81	79, 80	eqtrdi 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = ∅)
82	61, 81	eqtrid 2776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = ∅)
83	82	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = (vol‘∅))
84		ovol0 25394	. . . . . 6 ⊢ (vol*‘∅) = 0
85	83, 84	eqtrdi 2780	. . . . 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 13444	. . . . . . 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 2735	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐶) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)))
90	16	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
91	45	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
92		ssun1 4141	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵})
93	92	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵}))
94		ioossre 13368	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
95	94	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ℝ)
96	67	snssd 4773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ℝ)
97	95, 96	unssd 4155	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ)
98		uncom 4121	. . . . . . . . . . . 12 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) = ({𝐵} ∪ (𝐴(,)𝐵))
99	98	difeq1i 4085	. . . . . . . . . . 11 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))
100		difun2 4444	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
101	99, 100	eqtri 2752	. . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
102		difss 4099	. . . . . . . . . 10 ⊢ ({𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
103	101, 102	eqsstri 3993	. . . . . . . . 9 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
104		ovolsn 25396	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0)
105	67, 104	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘{𝐵}) = 0)
106		ovolssnul 25388	. . . . . . . . 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 1468	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵))) = 0)
108		ssun1 4141	. . . . . . . . . . 11 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶))
109	108, 88	sseqtrid 3989	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (𝐴(,)𝐶))
110	109	sselda 3946	. . . . . . . . 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 25716	. . . . . . 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 494	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹ ↔ (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹))
114	91, 113	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵}) ↦ 𝐷) ∈ 𝐿¹)
115	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
116	85, 89, 90, 114, 115	itgsplit 25737	. . . 4 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
117	112	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐵)𝐷 d𝑥 = ∫((𝐴(,)𝐵) ∪ {𝐵})𝐷 d𝑥)
118	117	oveq1d 7402	. . . 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 2767	. . 3 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
120	119	ex 412	. 2 ⊢ (𝜑 → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥)))
121	35, 60, 120	ecased 1035	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 + caddc 11071 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,)cioo 13306 [,]cicc 13309 vol*covol 25363 𝐿¹cibl 25518 ∫citg 25519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-symdif 4216 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-rest 17385 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-top 22781 df-topon 22798 df-bases 22833 df-cmp 23274 df-ovol 25365 df-vol 25366 df-mbf 25520 df-itg1 25521 df-itg2 25522 df-ibl 25523 df-itg 25524 df-0p 25571

This theorem is referenced by: ditgsplitlem 25761 ftc1lem1 25942 ftc1anc 37695 fourierdlem103 46207 fourierdlem104 46208 fourierdlem111 46215 sqwvfoura 46226 sqwvfourb 46227

W3C validator