Theorem ovncvrrp 47102

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 2761	. . . 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 47100	. . 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 1148	. . . . . 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 1150	. . . . . 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 3434	. . . . . . . . . 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 218	. . . . . . . . 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 499	. . . . . . . 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 484	. . . . . . 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 1142	. . . . . 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 1933	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15		nfe1 2183	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
16		simp1l 1210	. . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 1214	. . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑖 → (𝑙‘𝑗) = (𝑖‘𝑗))
21	20	coeq2d 5832	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑖 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
22	21	fveq1d 6865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑖 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
23	22	ixpeq2dv 8891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑖 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
24	23	iuneq2d 4979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
25	24	sseq2d 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑖 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
26	25	elrab 3650	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
31	30	rabbidv 3420	. . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
33	32, 3	ssexd 5279	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
34		elpwg 4557	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
37		ovex 7425	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
38	37	rabex 5294	. . . . . . . . . . . . . . . . 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 6995	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
41	40	eqcomd 2767	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = (𝐶‘𝐴))
42	41	3ad2ant1 1145	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . 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 1389	. . . . . . . . . . 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 5828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑖‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝑖‘𝑗)))
47	46	fveq1d 6865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑖‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
48	47	fveq2d 6867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑖‘𝑗) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
49	48	prodeq2ad 46132	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑖‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
50		elmapi 8826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
51	50	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
52		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
53	51, 52	ffvelcdmd 7062	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
54		prodex 15918	. . . . . . . . . . . . . . . . . . . . 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 6995	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝑖‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
57	56	mpteq2dva 5192	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
58	57	fveq2d 6867	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = 𝑧)
62	61	adantl 485	. . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . 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 1148	. . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . 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 1189	. . . . . . . . . . . 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 1193	. . . . . . . . . . 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 519	. . . . . . . . . 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 2216	. . . . . . . . 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 1131	. . . . . . . 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 3268	. . . . . . 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 410	. . . . . 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 848	. . . . 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 1131	. . . 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 3160	. . 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 3434	. . . . . 6 ⊢ (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	bilanri 510	. . . . 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)})
80		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)}))
81		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
82		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑎ℝ+
83		nfv 1933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)
84		nfmpt1 5198	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
85	29, 84	nfcxfr 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝐶
86		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝑏
87	85, 86	nffv 6873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝐶‘𝑏)
88	83, 87	nfrabw 3450	. . . . . . . . . . 11 ⊢ Ⅎ𝑎{𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}
89	82, 88	nfmpt 5197	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
90		fveq2 6863	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝐶‘𝑎) = (𝐶‘𝑏))
91	90	eleq2d 2847	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝑏)))
92		fveq2 6863	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝑏))
93	92	oveq1d 7407	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))
94	93	breq2d 5111	. . . . . . . . . . . . 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)))
95	91, 94	anbi12d 641	. . . . . . . . . . . 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))))
96	95	rabbidva2 3415	. . . . . . . . . . 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)})
97	96	mpteq2dv 5193	. . . . . . . . . 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)}))
98	81, 89, 97	cbvmpt 5201	. . . . . . . . 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)}))
99	80, 98	eqtri 2784	. . . . . . . 8 ⊢ 𝐷 = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
100		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (𝐶‘𝑏) = (𝐶‘𝐴))
101	100	eleq2d 2847	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝑖 ∈ (𝐶‘𝑏) ↔ 𝑖 ∈ (𝐶‘𝐴)))
102		fveq2 6863	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐴 → ((voln*‘𝑋)‘𝑏) = ((voln*‘𝑋)‘𝐴))
103	102	oveq1d 7407	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))
104	103	breq2d 5111	. . . . . . . . . . 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)))
105	101, 104	anbi12d 641	. . . . . . . . . 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))))
106	105	rabbidva2 3415	. . . . . . . . 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)})
107	106	mpteq2dv 5193	. . . . . . . 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)}))
108	36	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
109		rpex 45886	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
110	109	mptex 7203	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}) ∈ V
111	110	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)
112	99, 107, 108, 111	fvmptd3 6995	. . . . . . 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)}))
113		oveq2 7400	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
114	113	breq2d 5111	. . . . . . . . 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)))
115	114	rabbidv 3420	. . . . . . . 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)})
116	115	adantl 485	. . . . . . 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)})
117	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐸 ∈ ℝ+)
118		fvex 6876	. . . . . . . . 9 ⊢ (𝐶‘𝐴) ∈ V
119	118	rabex 5294	. . . . . . . 8 ⊢ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
120	119	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)
121	112, 116, 117, 120	fvmptd 6979	. . . . . 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)})
122	121	eqcomd 2767	. . . . 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))
123	79, 122	eleqtrd 2863	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ ((𝐷‘𝐴)‘𝐸))
124	123	ex 416	. . 3 ⊢ (𝜑 → ((𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ ((𝐷‘𝐴)‘𝐸)))
125	124	eximdv 1936	. 2 ⊢ (𝜑 → (∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸)))
126	77, 125	mpd 15	1 ⊢ (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷‘𝐴)‘𝐸))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  ∪ ciun 4948   class class class wbr 5099   ↦ cmpt 5180   × cxp 5643   ∘ ccom 5649  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803  Xcixp 8875  Fincfn 8923  ℝcr 11069  ℝ^*cxr 11212   ≤ cle 11214  ℕcn 12207  ℝ⁺crp 12990   +_𝑒 cxad 13109  [,)cico 13348  ∏cprod 15916  volcvol 25505  Σ^{^}csumge0 46900  voln*covoln 47074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-map 8805  df-pm 8806  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9455  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-q 12947  df-rp 12991  df-xneg 13111  df-xadd 13112  df-xmul 13113  df-ioo 13350  df-ico 13352  df-icc 13353  df-fz 13510  df-fzo 13657  df-fl 13799  df-seq 14012  df-exp 14072  df-hash 14341  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-rlim 15499  df-sum 15697  df-prod 15917  df-rest 17434  df-topgen 17455  df-psmet 21396  df-xmet 21397  df-met 21398  df-bl 21399  df-mopn 21400  df-top 22934  df-topon 22951  df-bases 22986  df-cmp 23427  df-ovol 25506  df-vol 25507  df-sumge0 46901  df-ovoln 47075

This theorem is referenced by:  ovnsubaddlem2  47109  hspmbllem3  47166