ovnsubaddlem2 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem2 42860

Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
ovnsubaddlem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnsubaddlem2.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnsubaddlem2.a	⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
ovnsubaddlem2.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ovnsubaddlem2.z	⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
ovnsubaddlem2.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
ovnsubaddlem2.l	⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
ovnsubaddlem2.d	⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))

**Assertion**
Ref	Expression
ovnsubaddlem2	⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Distinct variable groups: 𝐴,𝑎,𝑒,𝑖,𝑗,𝑛 𝐴,𝑘,𝑙,𝑎,𝑖,𝑗,𝑛 𝑧,𝐴,𝑎,𝑖,𝑗,𝑘,𝑛 𝐶,𝑎,𝑒,𝑖 𝐷,𝑎,𝑒,𝑖,𝑗,𝑛 𝐷,𝑘 𝐸,𝑎,𝑒,𝑖,𝑗,𝑛 𝑘,𝐸 𝐿,𝑎,𝑒,𝑖,𝑗,𝑛 𝑋,𝑎,𝑒,𝑖,𝑗,𝑛 ℎ,𝑋,𝑘,𝑖,𝑗 𝑋,𝑙 𝑧,𝑋 𝜑,𝑎,𝑒,𝑖,𝑗,𝑛 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑧,ℎ,𝑙) 𝐴(ℎ) 𝐶(𝑧,ℎ,𝑗,𝑘,𝑛,𝑙) 𝐷(𝑧,ℎ,𝑙) 𝐸(𝑧,ℎ,𝑙) 𝐿(𝑧,ℎ,𝑘,𝑙) 𝑍(𝑧,𝑒,ℎ,𝑖,𝑗,𝑘,𝑛,𝑎,𝑙)

Proof of Theorem ovnsubaddlem2 Dummy variables 𝑓 𝑔 𝑚 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6685	. . . 4 ⊢ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∈ V
2		nnenom 13351	. . . 4 ⊢ ℕ ≈ ω
3	1, 2	axcc3 9862	. . 3 ⊢ ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
4		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → 𝑔 Fn ℕ)
5		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑛𝜑
6		nfra1 3221	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
7	5, 6	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
8		rspa 3208	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
9	8	adantll 712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
10		ovnsubaddlem2.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ Fin)
11	10	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
12		ovnsubaddlem2.n0	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13	12	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
14		ovnsubaddlem2.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
15	14	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
16		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
17	15, 16	ffvelrnd 6854	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
18		elpwi 4550	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
19	17, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑_m 𝑋))
20		ovnsubaddlem2.e	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
21	20	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
22		nnnn0 11907	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
23		2nn 11713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ)
25		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
26		nnexpcl 13445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
27	24, 25, 26	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℕ)
28		nnrp 12403	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2↑𝑛) ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℝ⁺)
30	22, 29	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℝ⁺)
31	30	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
32	21, 31	rpdivcld 12451	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈ ℝ⁺)
33		ovnsubaddlem2.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
34		ovnsubaddlem2.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
35		ovnsubaddlem2.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ (𝑒 ∈ ℝ⁺ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +_𝑒 𝑒)}))
36	11, 13, 19, 32, 33, 34, 35	ovncvrrp 42853	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
37		n0 4312	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
38	36, 37	sylibr 236	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
39	38	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
40		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
41	39, 40	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
42	41	ex 415	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
43	42	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
44	9, 43	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
45	44	ex 415	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑛 ∈ ℕ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
46	7, 45	ralrimi 3218	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
47	46	adantrl 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
48	4, 47	jca 514	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
49	48	ex 415	. . . 4 ⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))))
50	49	eximdv 1918	. . 3 ⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))))
51	3, 50	mpi 20	. 2 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
52		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝜑)
53		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝑔 Fn ℕ)
54		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
55		nnf1oxpnn 41464	. . . . . 6 ⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)
56		simpl1 1187	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝜑)
57		simpl2 1188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑔 Fn ℕ)
58		fveq2 6672	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → (𝑔‘𝑞) = (𝑔‘𝑛))
59		2fveq3 6677	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐷‘(𝐴‘𝑞)) = (𝐷‘(𝐴‘𝑛)))
60		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑛 → (2↑𝑞) = (2↑𝑛))
61	60	oveq2d 7174	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐸 / (2↑𝑞)) = (𝐸 / (2↑𝑛)))
62	59, 61	fveq12d 6679	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) = ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
63	58, 62	eleq12d 2909	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑛 → ((𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
64	63	cbvralvw 3451	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
65	64	biimpri 230	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
66	65	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
67	66	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
68		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ))
69	10	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
70	69	3ad2antl1 1181	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
71	12	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
72	71	3ad2antl1 1181	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
73	14	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
74	73	3ad2antl1 1181	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
75	20	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
76	75	3ad2antl1 1181	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
77		ovnsubaddlem2.z	. . . . . . . . . 10 ⊢ 𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑_m 𝑋) ↦ {𝑧 ∈ ℝ^* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
78		coeq2 5731	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑖 → ([,) ∘ ℎ) = ([,) ∘ 𝑖))
79	78	fveq1d 6674	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑖 → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ 𝑖)‘𝑘))
80	79	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑖 → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ 𝑖)‘𝑘)))
81	80	prodeq2ad 41880	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
82	81	cbvmptv 5171	. . . . . . . . . . 11 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
83	34, 82	eqtri 2846	. . . . . . . . . 10 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
84	64	biimpi 218	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
85	84	3ad2ant3 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
86	85	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
87		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
88		rspa 3208	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
89	86, 87, 88	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
90		simpr 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ))
91		2fveq3 6677	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑚 → (1^st ‘(𝑓‘𝑞)) = (1^st ‘(𝑓‘𝑚)))
92	91	fveq2d 6676	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (𝑔‘(1^st ‘(𝑓‘𝑞))) = (𝑔‘(1^st ‘(𝑓‘𝑚))))
93		2fveq3 6677	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (2^nd ‘(𝑓‘𝑞)) = (2^nd ‘(𝑓‘𝑚)))
94	92, 93	fveq12d 6679	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑚 → ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞))) = ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
95	94	cbvmptv 5171	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞)))) = (𝑚 ∈ ℕ ↦ ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
96	70, 72, 74, 76, 77, 33, 83, 35, 89, 90, 95	ovnsubaddlem1 42859	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
97	56, 57, 67, 68, 96	syl31anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
98	97	ex 415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑓:ℕ–1-1-onto→(ℕ × ℕ) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
99	98	exlimdv 1934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
100	55, 99	mpi 20	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
101	52, 53, 54, 100	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))
102	101	ex 415	. . 3 ⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
103	102	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸)))
104	51, 103	mpd 15	1 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ ((Σ^{^}‘(𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐴‘𝑛)))) +_𝑒 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 ↑_m cmap 8408 Xcixp 8463 Fincfn 8511 ℝcr 10538 ℝ^*cxr 10676 ≤ cle 10678 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ₀cn0 11900 ℝ⁺crp 12392 +_𝑒 cxad 12508 [,)cico 12743 ↑cexp 13432 ∏cprod 15261 volcvol 24066 Σ^{^}csumge0 42651 voln*covoln 42825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-prod 15262 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-bases 21556 df-cmp 21997 df-ovol 24067 df-vol 24068 df-sumge0 42652 df-ovoln 42826

This theorem is referenced by: ovnsubadd 42861

W3C validator