Theorem ovncvrrp 46485

Description: The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovncvrrp.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovncvrrp.n0	⊢ (𝜑 → 𝑋 ≠ ∅)
ovncvrrp.a	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
ovncvrrp.e	⊢ (𝜑 → 𝐸 ∈ ℝ+)
ovncvrrp.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
ovncvrrp.l	⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
ovncvrrp.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
ovncvrrp	⊢ (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸))

Distinct variable groups:   𝐴,𝑎,𝑒,𝑖   𝐴,𝑙,𝑎,𝑖   𝐶,𝑒,𝑖   𝑒,𝐸,𝑖   𝐿,𝑎,𝑒   𝑋,𝑎,𝑒,𝑖,𝑗   ℎ,𝑋,𝑘,𝑖,𝑗   𝑋,𝑙   𝑘,𝑎   𝑗,𝑙,𝑘   𝜑,𝑎,𝑒,𝑖,𝑗   𝜑,𝑘

Allowed substitution hints:   𝜑(ℎ,𝑙)   𝐴(ℎ,𝑗,𝑘)   𝐶(ℎ,𝑗,𝑘,𝑎,𝑙)   𝐷(𝑒,ℎ,𝑖,𝑗,𝑘,𝑎,𝑙)   𝐸(ℎ,𝑗,𝑘,𝑎,𝑙)   𝐿(ℎ,𝑖,𝑗,𝑘,𝑙)

Proof of Theorem ovncvrrp

Dummy variables 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovncvrrp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovncvrrp.n0	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
3		ovncvrrp.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
4		ovncvrrp.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
5		eqid 2740	. . . 4 ⊢ {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)))))} = {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)))))}
6	1, 2, 3, 4, 5	ovnlerp 46483	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ {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)))))}𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
7		simp1 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {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)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝜑)
8		simp3 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {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)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
9		rabid 3465	. . . . . . . . . 10 ⊢ (𝑧 ∈ {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)))))} ↔ (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)))))))
10	9	biimpi 216	. . . . . . . . 9 ⊢ (𝑧 ∈ {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)))))} → (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)))))))
11	10	simprd 495	. . . . . . . 8 ⊢ (𝑧 ∈ {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)))))} → ∃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))))))
12	11	adantr 480	. . . . . . 7 ⊢ ((𝑧 ∈ {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)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃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))))))
13	12	3adant1 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ {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)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃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))))))
14		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15		nfe1 2151	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
16		simp1l 1197	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → 𝜑)
17		simp2 1137	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
18		simp3l 1201	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
19		id 22	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
20		fveq1 6919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑖 → (𝑙‘𝑗) = (𝑖‘𝑗))
21	20	coeq2d 5887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑖 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
22	21	fveq1d 6922	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑖 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
23	22	ixpeq2dv 8971	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑖 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
24	23	iuneq2d 5045	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
25	24	sseq2d 4041	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑖 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
26	25	elrab 3708	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ↔ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
27	19, 26	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
28	27	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
29		ovncvrrp.c	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
30		sseq1 4034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
31	30	rabbidv 3451	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐴 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
32		ovexd 7483	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
33	32, 3	ssexd 5342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
34		elpwg 4625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
36	3, 35	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
37		ovex 7481	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
38	37	rabex 5357	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V
39	38	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V)
40	29, 31, 36, 39	fvmptd3 7052	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
41	40	eqcomd 2746	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = (𝐶‘𝐴))
42	41	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = (𝐶‘𝐴))
43	28, 42	eleqtrd 2846	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝑖 ∈ (𝐶‘𝐴))
44	16, 17, 18, 43	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → 𝑖 ∈ (𝐶‘𝐴))
45		ovncvrrp.l	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
46		coeq2 5883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑖‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝑖‘𝑗)))
47	46	fveq1d 6922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑖‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
48	47	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑖‘𝑗) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
49	48	prodeq2ad 45513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑖‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
50		elmapi 8907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
51	50	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
52		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
53	51, 52	ffvelcdmd 7119	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
54		prodex 15953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) ∈ V
55	54	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) ∈ V)
56	45, 49, 53, 55	fvmptd3 7052	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝑖‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
57	56	mpteq2dva 5266	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
58	57	fveq2d 6924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
60		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
61	60	eqcomd 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = 𝑧)
62	61	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = 𝑧)
63	59, 62	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = 𝑧)
64	63	3adant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = 𝑧)
65		simp1 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
66	64, 65	eqbrtrd 5188	. . . . . . . . . . . . 13 ⊢ ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
67	66	3adant1l 1176	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
68	67	3adant3l 1180	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
69	44, 68	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
70	69	19.8ad 2183	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 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)))))) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
71	70	3exp 1119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → (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))))) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))))
72	14, 15, 71	rexlimd 3272	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → (∃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))))) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))))
73	72	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ ∃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)))))) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
74	7, 8, 13, 73	syl21anc 837	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ {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)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
75	74	3exp 1119	. . . 4 ⊢ (𝜑 → (𝑧 ∈ {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)))))} → (𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))))
76	75	rexlimdv 3159	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ {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)))))}𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))))
77	6, 76	mpd 15	. 2 ⊢ (𝜑 → ∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
78		rabid 3465	. . . . . . . 8 ⊢ (𝑖 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ↔ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
79	78	bicomi 224	. . . . . . 7 ⊢ ((𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ↔ 𝑖 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
80	79	biimpi 216	. . . . . 6 ⊢ ((𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
81	80	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
82		ovncvrrp.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
83		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
84		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑎ℝ+
85		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)
86		nfmpt1 5274	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
87	29, 86	nfcxfr 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝐶
88		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝑏
89	87, 88	nffv 6930	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝐶‘𝑏)
90	85, 89	nfrabw 3483	. . . . . . . . . . 11 ⊢ Ⅎ𝑎{𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}
91	84, 90	nfmpt 5273	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
92		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝐶‘𝑎) = (𝐶‘𝑏))
93	92	eleq2d 2830	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝑏)))
94		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝑏))
95	94	oveq1d 7463	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))
96	95	breq2d 5178	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)))
97	93, 96	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ((𝑖 ∈ (𝐶‘𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))))
98	97	rabbidva2 3445	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
99	98	mpteq2dv 5268	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
100	83, 91, 99	cbvmpt 5277	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
101	82, 100	eqtri 2768	. . . . . . . 8 ⊢ 𝐷 = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
102		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (𝐶‘𝑏) = (𝐶‘𝐴))
103	102	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝑖 ∈ (𝐶‘𝑏) ↔ 𝑖 ∈ (𝐶‘𝐴)))
104		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐴 → ((voln*‘𝑋)‘𝑏) = ((voln*‘𝑋)‘𝐴))
105	104	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))
106	105	breq2d 5178	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)))
107	103, 106	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → ((𝑖 ∈ (𝐶‘𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))))
108	107	rabbidva2 3445	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)})
109	108	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}))
110	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
111		rpex 45261	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
112	111	mptex 7260	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}) ∈ V
113	112	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}) ∈ V)
114	101, 109, 110, 113	fvmptd3 7052	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → (𝐷‘𝐴) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}))
115		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
116	115	breq2d 5178	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
117	116	rabbidv 3451	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
118	117	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) ∧ 𝑒 = 𝐸) → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
119	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐸 ∈ ℝ+)
120		fvex 6933	. . . . . . . . 9 ⊢ (𝐶‘𝐴) ∈ V
121	120	rabex 5357	. . . . . . . 8 ⊢ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
122	121	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V)
123	114, 118, 119, 122	fvmptd 7036	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → ((𝐷‘𝐴)‘𝐸) = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
124	123	eqcomd 2746	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} = ((𝐷‘𝐴)‘𝐸))
125	81, 124	eleqtrd 2846	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ ((𝐷‘𝐴)‘𝐸))
126	125	ex 412	. . 3 ⊢ (𝜑 → ((𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ ((𝐷‘𝐴)‘𝐸)))
127	126	eximdv 1916	. 2 ⊢ (𝜑 → (∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸)))
128	77, 127	mpd 15	1 ⊢ (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Xcixp 8955  Fincfn 9003  ℝcr 11183  ℝ^*cxr 11323   ≤ cle 11325  ℕcn 12293  ℝ⁺crp 13057   +_𝑒 cxad 13173  [,)cico 13409  ∏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-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-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-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:  ovnsubaddlem2  46492  hspmbllem3  46549