Theorem ovncvrrp 46992

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 2736	. . . 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 46990	. . 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 1137	. . . . . 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 1139	. . . . . 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 3410	. . . . . . . . . 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 1131	. . . . . 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 1916	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15		nfe1 2156	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑖(𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
16		simp1l 1199	. . . . . . . . . . . 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 1138	. . . . . . . . . . . 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 1203	. . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑖 → (𝑙‘𝑗) = (𝑖‘𝑗))
21	20	coeq2d 5817	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑖 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
22	21	fveq1d 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑖 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
23	22	ixpeq2dv 8861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑖 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
24	23	iuneq2d 4964	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
25	24	sseq2d 3954	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑖 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
26	25	elrab 3634	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
31	30	rabbidv 3396	. . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
33	32, 3	ssexd 5265	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
34		elpwg 4544	. . . . . . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
38	37	rabex 5280	. . . . . . . . . . . . . . . . 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 6971	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
41	40	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = (𝐶‘𝐴))
42	41	3ad2ant1 1134	. . . . . . . . . . . . 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 2838	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 5813	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑖‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝑖‘𝑗)))
47	46	fveq1d 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑖‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
48	47	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑖‘𝑗) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
49	48	prodeq2ad 46022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑖‘𝑗) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
50		elmapi 8796	. . . . . . . . . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
54		prodex 15870	. . . . . . . . . . . . . . . . . . . . 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 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝑖‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
57	56	mpteq2dva 5178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
58	57	fveq2d 6844	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . 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 1178	. . . . . . . . . . . 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 1182	. . . . . . . . . . 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 2190	. . . . . . . . 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 1120	. . . . . . . 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 3244	. . . . . . 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 838	. . . . 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 1120	. . . 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 3136	. . 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 3410	. . . . . . . 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 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
84		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑎ℝ+
85		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)
86		nfmpt1 5184	. . . . . . . . . . . . . 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 2896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝐶
88		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎𝑏
89	87, 88	nffv 6850	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎(𝐶‘𝑏)
90	85, 89	nfrabw 3426	. . . . . . . . . . 11 ⊢ Ⅎ𝑎{𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}
91	84, 90	nfmpt 5183	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
92		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝐶‘𝑎) = (𝐶‘𝑏))
93	92	eleq2d 2822	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝑏)))
94		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝑏))
95	94	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))
96	95	breq2d 5097	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 3391	. . . . . . . . . . 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 5179	. . . . . . . . . 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 5187	. . . . . . . . 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 2759	. . . . . . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (𝐶‘𝑏) = (𝐶‘𝐴))
103	102	eleq2d 2822	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝑖 ∈ (𝐶‘𝑏) ↔ 𝑖 ∈ (𝐶‘𝐴)))
104		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐴 → ((voln*‘𝑋)‘𝑏) = ((voln*‘𝑋)‘𝐴))
105	104	oveq1d 7382	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐴 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))
106	105	breq2d 5097	. . . . . . . . . . 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 633	. . . . . . . . . 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 3391	. . . . . . . . 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 5179	. . . . . . . 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 45776	. . . . . . . . . 10 ⊢ ℝ+ ∈ V
112	111	mptex 7178	. . . . . . . . 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 6971	. . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑒 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
116	115	breq2d 5097	. . . . . . . . 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 3396	. . . . . . . 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 6853	. . . . . . . . 9 ⊢ (𝐶‘𝐴) ∈ V
121	120	rabex 5280	. . . . . . . 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 6955	. . . . . 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 2742	. . . . 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 2838	. . . 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 1919	. 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 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  {crab 3389  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  Xcixp 8845  Fincfn 8893  ℝcr 11037  ℝ^*cxr 11178   ≤ cle 11180  ℕcn 12174  ℝ⁺crp 12942   +_𝑒 cxad 13061  [,)cico 13300  ∏cprod 15868  volcvol 25430  Σ^{^}csumge0 46790  voln*covoln 46964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-ico 13304  df-icc 13305  df-fz 13462  df-fzo 13609  df-fl 13751  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-rlim 15451  df-sum 15649  df-prod 15869  df-rest 17385  df-topgen 17406  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-top 22859  df-topon 22876  df-bases 22911  df-cmp 23352  df-ovol 25431  df-vol 25432  df-sumge0 46791  df-ovoln 46965

This theorem is referenced by:  ovnsubaddlem2  46999  hspmbllem3  47056