ovnsubaddlem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnsubaddlem2		Structured version Visualization version GIF version

Theorem ovnsubaddlem2 46492

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 6933	. . . 4 ⊢ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∈ V
2		nnenom 14031	. . . 4 ⊢ ℕ ≈ ω
3	1, 2	axcc3 10507	. . 3 ⊢ ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
4		simprl 770	. . . . . 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 1913	. . . . . . . . 9 ⊢ Ⅎ𝑛𝜑
6		nfra1 3290	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
7	5, 6	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
8		rspa 3254	. . . . . . . . . . 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 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)))))
10		ovnsubaddlem2.x	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ Fin)
11	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
12		ovnsubaddlem2.n0	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
13	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
14		ovnsubaddlem2.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
15	14	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
16		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
17	15, 16	ffvelcdmd 7119	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ ↑_m 𝑋))
18		elpwi 4629	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
22		nnnn0 12560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
23		2nn 12366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ)
25		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
26		nnexpcl 14125	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
27	24, 25, 26	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → (2↑𝑛) ∈ ℕ)
28		nnrp 13068	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ⁺)
32	21, 31	rpdivcld 13116	. . . . . . . . . . . . . . . 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 46485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
37		n0 4376	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
38	36, 37	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅)
39	38	adantr 480	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 714	. . . . . . . . . 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 412	. . . . . . . 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 3263	. . . . . . 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 715	. . . . . 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 511	. . . . 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 412	. . . 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 1916	. . 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 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝜑)
53		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝑔 Fn ℕ)
54		simprr 772	. . . . 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 45102	. . . . . 6 ⊢ ∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ)
56		simpl1 1191	. . . . . . . . 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 1192	. . . . . . . . 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 6920	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → (𝑔‘𝑞) = (𝑔‘𝑛))
59		2fveq3 6925	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐷‘(𝐴‘𝑞)) = (𝐷‘(𝐴‘𝑛)))
60		oveq2 7456	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑛 → (2↑𝑞) = (2↑𝑛))
61	60	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑛 → (𝐸 / (2↑𝑞)) = (𝐸 / (2↑𝑛)))
62	59, 61	fveq12d 6927	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑛 → ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) = ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
63	58, 62	eleq12d 2838	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑛 → ((𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))
64	63	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
65	64	biimpri 228	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))))
66	65	3ad2ant3 1135	. . . . . . . . . 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 480	. . . . . . . . 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 484	. . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin)
70	69	3ad2antl1 1185	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅)
72	71	3ad2antl1 1185	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫 (ℝ ↑_m 𝑋))
74	73	3ad2antl1 1185	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈ ℝ⁺)
76	75	3ad2antl1 1185	. . . . . . . . . 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 5883	. . . . . . . . . . . . . . 15 ⊢ (ℎ = 𝑖 → ([,) ∘ ℎ) = ([,) ∘ 𝑖))
79	78	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ (ℎ = 𝑖 → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ 𝑖)‘𝑘))
80	79	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑖 → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ 𝑖)‘𝑘)))
81	80	prodeq2ad 45513	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
82	81	cbvmptv 5279	. . . . . . . . . . 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 2768	. . . . . . . . . 10 ⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑_m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
84	64	biimpi 216	. . . . . . . . . . . . 13 ⊢ (∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))
85	84	3ad2ant3 1135	. . . . . . . . . . . 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 725	. . . . . . . . . . 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 484	. . . . . . . . . . 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 3254	. . . . . . . . . . 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 583	. . . . . . . . . 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 484	. . . . . . . . . 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 6925	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑚 → (1^st ‘(𝑓‘𝑞)) = (1^st ‘(𝑓‘𝑚)))
92	91	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (𝑔‘(1^st ‘(𝑓‘𝑞))) = (𝑔‘(1^st ‘(𝑓‘𝑚))))
93		2fveq3 6925	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑚 → (2^nd ‘(𝑓‘𝑞)) = (2^nd ‘(𝑓‘𝑚)))
94	92, 93	fveq12d 6927	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑚 → ((𝑔‘(1^st ‘(𝑓‘𝑞)))‘(2^nd ‘(𝑓‘𝑞))) = ((𝑔‘(1^st ‘(𝑓‘𝑚)))‘(2^nd ‘(𝑓‘𝑚))))
95	94	cbvmptv 5279	. . . . . . . . . 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 46491	. . . . . . . . 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 1373	. . . . . . . 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 412	. . . . . . 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 1932	. . . . . 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 1371	. . . 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 412	. . 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 1932	. 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 395 ∧ w3a 1087 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 ∪ ciun 5015 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ↑_m cmap 8884 Xcixp 8955 Fincfn 9003 ℝcr 11183 ℝ^*cxr 11323 ≤ cle 11325 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℝ⁺crp 13057 +_𝑒 cxad 13173 [,)cico 13409 ↑cexp 14112 ∏cprod 15951 volcvol 25517 Σ^{^}csumge0 46283 voln*covoln 46457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-prod 15952 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17482 df-0g 17501 df-topgen 17503 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-subg 19163 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-bases 22974 df-cmp 23416 df-ovol 25518 df-vol 25519 df-sumge0 46284 df-ovoln 46458

This theorem is referenced by: ovnsubadd 46493

W3C validator