Theorem ovoliun2 23274

Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 23273.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun2.t	⊢ (𝜑 → 𝑇 ∈ dom ⇝ )

Assertion

Ref	Expression
ovoliun2	⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Distinct variable group:   𝜑,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun2

Dummy variables 𝑘 𝑚 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovoliun.t	. . 3 ⊢ 𝑇 = seq1( + , 𝐺)
2		ovoliun.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
3		ovoliun.a	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4		ovoliun.v	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
5	1, 2, 3, 4	ovoliun 23273	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
7		1zzd 11408	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
8		fvex 6201	. . . . . . . . . . 11 ⊢ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V
9		nfcv 2764	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(vol*‘𝐴)
10		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
11		nfcsb1v 3549	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
12	10, 11	nffv 6198	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
13		csbeq1a 3542	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
14	13	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
15	9, 12, 14	cbvmpt 4749	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
16	2, 15	eqtri 2644	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
17	16	fvmpt2 6291	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
18	8, 17	mpan2 707	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
19	18	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
20	4	ralrimiva 2966	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
21	9	nfel1 2779	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
22	12	nfel1 2779	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
23	14	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
24	21, 22, 23	cbvral 3167	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
25	20, 24	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
26	25	r19.21bi 2932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
27	19, 26	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
28	6, 7, 27	serfre 12830	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
29	1	feq1i 6036	. . . . . . 7 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
30	28, 29	sylibr 224	. . . . . 6 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
31		frn 6053	. . . . . 6 ⊢ (𝑇:ℕ⟶ℝ → ran 𝑇 ⊆ ℝ)
32	30, 31	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
33		1nn 11031	. . . . . . . 8 ⊢ 1 ∈ ℕ
34		fdm 6051	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ → dom 𝑇 = ℕ)
35	30, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = ℕ)
36	33, 35	syl5eleqr 2708	. . . . . . 7 ⊢ (𝜑 → 1 ∈ dom 𝑇)
37		ne0i 3921	. . . . . . 7 ⊢ (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
39		dm0rn0 5342	. . . . . . 7 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
40	39	necon3bii 2846	. . . . . 6 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
41	38, 40	sylib 208	. . . . 5 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
42		ovoliun2.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ dom ⇝ )
43	1, 42	syl5eqelr 2706	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
44	6, 7, 19, 26, 43	isumrecl 14496	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
45		elfznn 12370	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
46	45	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → 𝑚 ∈ ℕ)
47	46, 18	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
48		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
49	48, 6	syl6eleq 2711	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
50		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
51	50, 45, 26	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
52	51	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℂ)
53	47, 49, 52	fsumser 14461	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (seq1( + , 𝐺)‘𝑘))
54	1	fveq1i 6192	. . . . . . . . . 10 ⊢ (𝑇‘𝑘) = (seq1( + , 𝐺)‘𝑘)
55	53, 54	syl6eqr 2674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (𝑇‘𝑘))
56		fzfid 12772	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ∈ Fin)
57		elfznn 12370	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57	ssriv 3607	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ ℕ
59	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ⊆ ℕ)
60	3	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
61		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
62		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛ℝ
63	11, 62	nfss 3596	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
64	13	sseq1d 3632	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
65	61, 63, 64	cbvral 3167	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
66	60, 65	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
67	66	r19.21bi 2932	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
68		ovolge0 23249	. . . . . . . . . . . 12 ⊢ (⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ → 0 ≤ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
69	67, 68	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 0 ≤ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
70	6, 7, 56, 59, 19, 26, 69, 43	isumless 14577	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
71	70	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
72	55, 71	eqbrtrrd 4677	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
73	72	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
74		breq2 4657	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) → ((𝑇‘𝑘) ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)))
75	74	ralbidv 2986	. . . . . . . 8 ⊢ (𝑥 = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) → (∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)))
76	75	rspcev 3309	. . . . . . 7 ⊢ ((Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
77	44, 73, 76	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
78		ffn 6045	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ → 𝑇 Fn ℕ)
79	30, 78	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
80		breq1 4656	. . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ 𝑥))
81	80	ralrn 6362	. . . . . . . 8 ⊢ (𝑇 Fn ℕ → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
82	79, 81	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
83	82	rexbidv 3052	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
84	77, 83	mpbird 247	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥)
85		supxrre 12157	. . . . 5 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
86	32, 41, 84, 85	syl3anc 1326	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
87	6, 1, 7, 19, 26, 69, 77	isumsup 14579	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) = sup(ran 𝑇, ℝ, < ))
88	86, 87	eqtr4d 2659	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
89	9, 12, 14	cbvsumi 14427	. . 3 ⊢ Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)
90	88, 89	syl6eqr 2674	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
91	5, 90	breqtrd 4679	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ⦋csb 3533   ⊆ wss 3574  ∅c0 3915  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  supcsup 8346  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℕcn 11020  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801   ⇝ cli 14215  Σcsu 14416  vol*covol 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-ovol 23233

This theorem is referenced by:  ovoliunnul  23275  vitalilem5  23381