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Theorem itgsplitioo 24907

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 13073	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐶) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)))
5	2, 3, 4	syl2anc 583	. . . . . . 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 1141	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
8	6	simp1d 1140	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	2, 8	leloed 11048	. . . . 5 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
10	7, 9	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
11	10	ord 860	. . 3 ⊢ (𝜑 → (¬ 𝐴 < 𝐵 → 𝐴 = 𝐵))
12	2	rexrd 10956	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
13		iooss1 13043	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
14	12, 7, 13	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))
15	14	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴(,)𝐶))
16		itgsplitioo.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)
17	15, 16	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐷 ∈ ℂ)
18		itgsplitioo.6	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿¹)
19	17, 18	itgcl 24853	. . . . . 6 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 ∈ ℂ)
20	19	addid2d 11106	. . . . 5 ⊢ (𝜑 → (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = ∫(𝐵(,)𝐶)𝐷 d𝑥)
21	20	eqcomd 2744	. . . 4 ⊢ (𝜑 → ∫(𝐵(,)𝐶)𝐷 d𝑥 = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
22		oveq1 7262	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐶) = (𝐵(,)𝐶))
23		itgeq1 24842	. . . . . 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 7262	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = (𝐵(,)𝐵))
26		iooid 13036	. . . . . . . . 9 ⊢ (𝐵(,)𝐵) = ∅
27	25, 26	eqtrdi 2795	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴(,)𝐵) = ∅)
28		itgeq1 24842	. . . . . . . 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 24849	. . . . . . 7 ⊢ ∫∅𝐷 d𝑥 = 0
31	29, 30	eqtrdi 2795	. . . . . 6 ⊢ (𝐴 = 𝐵 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = 0)
32	31	oveq1d 7270	. . . . 5 ⊢ (𝐴 = 𝐵 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥) = (0 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
33	24, 32	eqeq12d 2754	. . . 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 1142	. . . . 5 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
37	8, 3	leloed 11048	. . . . 5 ⊢ (𝜑 → (𝐵 ≤ 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐵 = 𝐶)))
38	36, 37	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ∨ 𝐵 = 𝐶))
39	38	ord 860	. . 3 ⊢ (𝜑 → (¬ 𝐵 < 𝐶 → 𝐵 = 𝐶))
40	3	rexrd 10956	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
41		iooss2 13044	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
42	40, 36, 41	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
43	42	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐶))
44	43, 16	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 ∈ ℂ)
45		itgsplitioo.5	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿¹)
46	44, 45	itgcl 24853	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 ∈ ℂ)
47	46	addid1d 11105	. . . . 5 ⊢ (𝜑 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = ∫(𝐴(,)𝐵)𝐷 d𝑥)
48	47	eqcomd 2744	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0))
49		oveq2 7263	. . . . . 6 ⊢ (𝐵 = 𝐶 → (𝐴(,)𝐵) = (𝐴(,)𝐶))
50		itgeq1 24842	. . . . . 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 7263	. . . . . . . . 9 ⊢ (𝐵 = 𝐶 → (𝐵(,)𝐵) = (𝐵(,)𝐶))
53	26, 52	eqtr3id 2793	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → ∅ = (𝐵(,)𝐶))
54		itgeq1 24842	. . . . . . . 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 2793	. . . . . 6 ⊢ (𝐵 = 𝐶 → 0 = ∫(𝐵(,)𝐶)𝐷 d𝑥)
57	56	oveq2d 7271	. . . . 5 ⊢ (𝐵 = 𝐶 → (∫(𝐴(,)𝐵)𝐷 d𝑥 + 0) = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
58	51, 57	eqeq12d 2754	. . . 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 4206	. . . . . . . 8 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶)))
62	8	rexrd 10956	. . . . . . . . . . . . 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 11471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
69		ioodisj 13143	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐵 ∈ ℝ^ ∧ 𝐶 ∈ ℝ^*)) ∧ 𝐵 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
70	64, 66, 68, 69	syl21anc 834	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) = ∅)
71		incom 4131	. . . . . . . . . . 11 ⊢ ({𝐵} ∩ (𝐵(,)𝐶)) = ((𝐵(,)𝐶) ∩ {𝐵})
72	67	ltnrd 11039	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐵)
73		eliooord 13067	. . . . . . . . . . . . . 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 4644	. . . . . . . . . . . 12 ⊢ (((𝐵(,)𝐶) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (𝐵(,)𝐶))
77	75, 76	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐵(,)𝐶) ∩ {𝐵}) = ∅)
78	71, 77	syl5eq 2791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ({𝐵} ∩ (𝐵(,)𝐶)) = ∅)
79	70, 78	uneq12d 4094	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = (∅ ∪ ∅))
80		un0 4321	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
81	79, 80	eqtrdi 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵(,)𝐶)) ∪ ({𝐵} ∩ (𝐵(,)𝐶))) = ∅)
82	61, 81	syl5eq 2791	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶)) = ∅)
83	82	fveq2d 6760	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = (vol‘∅))
84		ovol0 24562	. . . . . 6 ⊢ (vol*‘∅) = 0
85	83, 84	eqtrdi 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∩ (𝐵(,)𝐶))) = 0)
86	12, 62, 40	3jca 1126	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
87		ioojoin 13144	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
88	86, 87	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))
89	88	eqcomd 2744	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐶) = (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)))
90	16	adantlr 711	. . . . 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 4102	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵})
93	92	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ((𝐴(,)𝐵) ∪ {𝐵}))
94		ioossre 13069	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
95	94	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴(,)𝐵) ⊆ ℝ)
96	67	snssd 4739	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ℝ)
97	95, 96	unssd 4116	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ)
98		uncom 4083	. . . . . . . . . . . 12 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) = ({𝐵} ∪ (𝐴(,)𝐵))
99	98	difeq1i 4049	. . . . . . . . . . 11 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵))
100		difun2 4411	. . . . . . . . . . 11 ⊢ (({𝐵} ∪ (𝐴(,)𝐵)) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
101	99, 100	eqtri 2766	. . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) = ({𝐵} ∖ (𝐴(,)𝐵))
102		difss 4062	. . . . . . . . . 10 ⊢ ({𝐵} ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
103	101, 102	eqsstri 3951	. . . . . . . . 9 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵)) ⊆ {𝐵}
104		ovolsn 24564	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (vol*‘{𝐵}) = 0)
105	67, 104	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘{𝐵}) = 0)
106		ovolssnul 24556	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (vol*‘(((𝐴(,)𝐵) ∪ {𝐵}) ∖ (𝐴(,)𝐵))) = 0)
108		ssun1 4102	. . . . . . . . . . 11 ⊢ ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶))
109	108, 88	sseqtrid 3969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ (𝐴(,)𝐶))
110	109	sselda 3917	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝑥 ∈ (𝐴(,)𝐶))
111	110, 90	syldan 590	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) ∧ 𝑥 ∈ ((𝐴(,)𝐵) ∪ {𝐵})) → 𝐷 ∈ ℂ)
112	93, 97, 107, 111	itgss3 24884	. . . . . . 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 231	. . . . 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 24905	. . . 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 7270	. . . 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 2781	. . 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 1031	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 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 (,)cioo 13008 [,]cicc 13011 vol*covol 24531 𝐿¹cibl 24686 ∫citg 24687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739

This theorem is referenced by: ditgsplitlem 24929 ftc1lem1 25104 ftc1anc 35785 fourierdlem103 43640 fourierdlem104 43641 fourierdlem111 43648 sqwvfoura 43659 sqwvfourb 43660

W3C validator