Theorem ovoliun2 23081

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 23080.) (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 23080	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
6		nnuz 11599	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
7		1zzd 11285	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
8		fvex 6113	. . . . . . . . . . 11 ⊢ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V
9		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(vol*‘𝐴)
10		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
11		nfcsb1v 3515	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
12	10, 11	nffv 6110	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
13		csbeq1a 3508	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
14	13	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
15	9, 12, 14	cbvmpt 4677	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
16	2, 15	eqtri 2632	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
17	16	fvmpt2 6200	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ V) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
18	8, 17	mpan2 703	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
20	4	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
21	9	nfel1 2765	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
22	12	nfel1 2765	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
23	14	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
24	21, 22, 23	cbvral 3143	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
25	20, 24	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
26	25	r19.21bi 2916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
27	19, 26	eqeltrd 2688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℝ)
28	6, 7, 27	serfre 12692	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
29	1	feq1i 5949	. . . . . . 7 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
30	28, 29	sylibr 223	. . . . . 6 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
31		frn 5966	. . . . . 6 ⊢ (𝑇:ℕ⟶ℝ → ran 𝑇 ⊆ ℝ)
32	30, 31	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
33		1nn 10908	. . . . . . . 8 ⊢ 1 ∈ ℕ
34		fdm 5964	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ → dom 𝑇 = ℕ)
35	30, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑇 = ℕ)
36	33, 35	syl5eleqr 2695	. . . . . . 7 ⊢ (𝜑 → 1 ∈ dom 𝑇)
37		ne0i 3880	. . . . . . 7 ⊢ (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
38	36, 37	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
39		dm0rn0 5263	. . . . . . 7 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
40	39	necon3bii 2834	. . . . . 6 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
41	38, 40	sylib 207	. . . . 5 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
42		ovoliun2.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ dom ⇝ )
43	1, 42	syl5eqelr 2693	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
44	6, 7, 19, 26, 43	isumrecl 14338	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
45		elfznn 12241	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ)
46	45	adantl 481	. . . . . . . . . . . 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 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
49	48, 6	syl6eleq 2698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
50		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝜑)
51	50, 45, 26	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
52	51	recnd 9947	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℂ)
53	47, 49, 52	fsumser 14308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (seq1( + , 𝐺)‘𝑘))
54	1	fveq1i 6104	. . . . . . . . . 10 ⊢ (𝑇‘𝑘) = (seq1( + , 𝐺)‘𝑘)
55	53, 54	syl6eqr 2662	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) = (𝑇‘𝑘))
56		fzfid 12634	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ∈ Fin)
57		elfznn 12241	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
58	57	ssriv 3572	. . . . . . . . . . . 12 ⊢ (1...𝑘) ⊆ ℕ
59	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑘) ⊆ ℕ)
60	3	ralrimiva 2949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
61		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
62		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛ℝ
63	11, 62	nfss 3561	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
64	13	sseq1d 3595	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
65	61, 63, 64	cbvral 3143	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
66	60, 65	sylib 207	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
67	66	r19.21bi 2916	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
68		ovolge0 23056	. . . . . . . . . . . 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 14416	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
71	70	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(vol*‘⦋𝑚 / 𝑛⦌𝐴) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
72	55, 71	eqbrtrrd 4607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
73	72	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
74		breq2 4587	. . . . . . . . 9 ⊢ (𝑥 = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) → ((𝑇‘𝑘) ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)))
75	74	ralbidv 2969	. . . . . . . 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 3282	. . . . . . 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 691	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥)
78		ffn 5958	. . . . . . . . 9 ⊢ (𝑇:ℕ⟶ℝ → 𝑇 Fn ℕ)
79	30, 78	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn ℕ)
80		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = (𝑇‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑘) ≤ 𝑥))
81	80	ralrn 6270	. . . . . . . 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 3034	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ 𝑥))
84	77, 83	mpbird 246	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥)
85		supxrre 12029	. . . . 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 1318	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
87	6, 1, 7, 19, 26, 69, 77	isumsup 14418	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) = sup(ran 𝑇, ℝ, < ))
88	86, 87	eqtr4d 2647	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
89	9, 12, 14	cbvsumi 14275	. . 3 ⊢ Σ𝑛 ∈ ℕ (vol*‘𝐴) = Σ𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴)
90	88, 89	syl6eqr 2662	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ ℕ (vol*‘𝐴))
91	5, 90	breqtrd 4609	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ Σ𝑛 ∈ ℕ (vol*‘𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ⦋csb 3499   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  ℤ_≥cuz 11563  ...cfz 12197  seqcseq 12663   ⇝ cli 14063  Σcsu 14264  vol*covol 23038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-ovol 23040

This theorem is referenced by:  ovoliunnul  23082  vitalilem5  23187