ovnsubaddlem2 - Mathbox for Glauco Siliprandi

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Theorem ovnsubaddlem2 46102

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 6909	. . . 4 ⊢ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∈ V
2		nnenom 13986	. . . 4 ⊢ ℕ ≈ ω
3	1, 2	axcc3 10468	. . 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 1909	. . . . . . . . 9 ⊢ Ⅎ𝑛𝜑
6		nfra1 3271	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
7	5, 6	nfan 1894	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
8		rspa 3235	. . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
12		ovnsubaddlem2.n0	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13	12	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
14		ovnsubaddlem2.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
15	14	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
16		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
17	15, 16	ffvelcdmd 7094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
18		elpwi 4611	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
22		nnnn0 12517	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
23		2nn 12323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ)
25		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
26		nnexpcl 14080	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
27	24, 25, 26	syl2anc 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℕ)
28		nnrp 13025	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
32	21, 31	rpdivcld 13073	. . . . . . . . . . . . . . . 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 46095	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
37		n0 4346	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
38	36, 37	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
39	38	adantr 479	. . . . . . . . . . . . 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 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))) ≠ ∅ → (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 411	. . . . . . . . . . 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 411	. . . . . . . 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 3244	. . . . . . 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 510	. . . . 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 411	. . . 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 1912	. . 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 481	. . . . 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 44712	. . . . . 6 ⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)
56		simpl1 1188	. . . . . . . . 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 1189	. . . . . . . . 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 6896	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → (𝑔‘𝑞) = (𝑔‘𝑛))
59		2fveq3 6901	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐷‘(𝐴‘𝑞)) = (𝐷‘(𝐴‘𝑛)))
60		oveq2 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑛 → (2↑𝑞) = (2↑𝑛))
61	60	oveq2d 7435	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐸 / (2↑𝑞)) = (𝐸 / (2↑𝑛)))
62	59, 61	fveq12d 6903	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) = ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
63	58, 62	eleq12d 2819	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑛 → ((𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
64	63	cbvralvw 3224	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
65	64	biimpri 227	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
66	65	3ad2ant3 1132	. . . . . . . . . 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 479	. . . . . . . . 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 483	. . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
70	69	3ad2antl1 1182	. . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
72	71	3ad2antl1 1182	. . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
74	73	3ad2antl1 1182	. . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
76	75	3ad2antl1 1182	. . . . . . . . . 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 5861	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑖 → ([,) ∘ ℎ) = ([,) ∘ 𝑖))
79	78	fveq1d 6898	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑖 → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ 𝑖)‘𝑘))
80	79	fveq2d 6900	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑖 → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ 𝑖)‘𝑘)))
81	80	prodeq2ad 45123	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
82	81	cbvmptv 5262	. . . . . . . . . . 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 2753	. . . . . . . . . 10 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
84	64	biimpi 215	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
85	84	3ad2ant3 1132	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 3235	. . . . . . . . . . 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 582	. . . . . . . . . 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 483	. . . . . . . . . 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 6901	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑚 → (1^st ‘(𝑓‘𝑞)) = (1^st ‘(𝑓‘𝑚)))
92	91	fveq2d 6900	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (𝑔‘(1^st ‘(𝑓‘𝑞))) = (𝑔‘(1^st ‘(𝑓‘𝑚))))
93		2fveq3 6901	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (2^nd ‘(𝑓‘𝑞)) = (2^nd ‘(𝑓‘𝑚)))
94	92, 93	fveq12d 6903	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑚 → ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞))) = ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
95	94	cbvmptv 5262	. . . . . . . . . 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 46101	. . . . . . . . 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 1370	. . . . . . . 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 411	. . . . . . 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 1928	. . . . . 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 1368	. . . 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 411	. . 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 1928	. 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 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 {crab 3418 ⊆ wss 3944 ∅c0 4322 𝒫 cpw 4604 ∪ ciun 4997 class class class wbr 5149 ↦ cmpt 5232 × cxp 5676 ∘ ccom 5682 Fn wfn 6544 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 1^st c1st 7992 2^nd c2nd 7993 ↑_m cmap 8845 Xcixp 8916 Fincfn 8964 ℝcr 11144 ℝ^*cxr 11284 ≤ cle 11286 / cdiv 11908 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℝ⁺crp 13014 +_𝑒 cxad 13130 [,)cico 13366 ↑cexp 14067 ∏cprod 15893 volcvol 25453 Σ^{^}csumge0 45893 voln*covoln 46067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-ac2 10493 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-dju 9931 df-card 9969 df-acn 9972 df-ac 10146 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-rlim 15477 df-sum 15677 df-prod 15894 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-rest 17423 df-0g 17442 df-topgen 17444 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-subg 19103 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-drng 20655 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22857 df-topon 22874 df-bases 22910 df-cmp 23352 df-ovol 25454 df-vol 25455 df-sumge0 45894 df-ovoln 46068

This theorem is referenced by: ovnsubadd 46103

W3C validator