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Theorem ovncvrrp 45852
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 (𝜑𝑋 ∈ Fin)
ovncvrrp.n0 (𝜑𝑋 ≠ ∅)
ovncvrrp.a (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
ovncvrrp.e (𝜑𝐸 ∈ ℝ+)
ovncvrrp.c 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
ovncvrrp.l 𝐿 = ( ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
ovncvrrp.d 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
Assertion
Ref Expression
ovncvrrp (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷𝐴)‘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   ,𝑋,𝑘,𝑖,𝑗   𝑋,𝑙   𝑘,𝑎   𝑗,𝑙,𝑘   𝜑,𝑎,𝑒,𝑖,𝑗   𝜑,𝑘
Allowed substitution hints:   𝜑(,𝑙)   𝐴(,𝑗,𝑘)   𝐶(,𝑗,𝑘,𝑎,𝑙)   𝐷(𝑒,,𝑖,𝑗,𝑘,𝑎,𝑙)   𝐸(,𝑗,𝑘,𝑎,𝑙)   𝐿(,𝑖,𝑗,𝑘,𝑙)

Proof of Theorem ovncvrrp
Dummy variables 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovncvrrp.x . . . 4 (𝜑𝑋 ∈ Fin)
2 ovncvrrp.n0 . . . 4 (𝜑𝑋 ≠ ∅)
3 ovncvrrp.a . . . 4 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
4 ovncvrrp.e . . . 4 (𝜑𝐸 ∈ ℝ+)
5 eqid 2726 . . . 4 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
61, 2, 3, 4, 5ovnlerp 45850 . . 3 (𝜑 → ∃𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
7 simp1 1133 . . . . . 6 ((𝜑𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝜑)
8 simp3 1135 . . . . . 6 ((𝜑𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
9 rabid 3446 . . . . . . . . . 10 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ↔ (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
109biimpi 215 . . . . . . . . 9 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} → (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
1110simprd 495 . . . . . . . 8 (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
1211adantr 480 . . . . . . 7 ((𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
13123adant1 1127 . . . . . 6 ((𝜑𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
14 nfv 1909 . . . . . . . 8 𝑖(𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15 nfe1 2139 . . . . . . . 8 𝑖𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
16 simp1l 1194 . . . . . . . . . . . 12 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝜑)
17 simp2 1134 . . . . . . . . . . . 12 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
18 simp3l 1198 . . . . . . . . . . . 12 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
19 id 22 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
20 fveq1 6884 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑖 → (𝑙𝑗) = (𝑖𝑗))
2120coeq2d 5856 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑖 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝑖𝑗)))
2221fveq1d 6887 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑖 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
2322ixpeq2dv 8909 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑖X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
2423iuneq2d 5019 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
2524sseq2d 4009 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
2625elrab 3678 . . . . . . . . . . . . . . 15 (𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ↔ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
2719, 26sylibr 233 . . . . . . . . . . . . . 14 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
28273adant1 1127 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝑖 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
29 ovncvrrp.c . . . . . . . . . . . . . . . 16 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
30 sseq1 4002 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
3130rabbidv 3434 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐴 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
32 ovexd 7440 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
3332, 3ssexd 5317 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
34 elpwg 4600 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
3533, 34syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
363, 35mpbird 257 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
37 ovex 7438 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
3837rabex 5325 . . . . . . . . . . . . . . . . 17 {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ∈ V
3938a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ∈ V)
4029, 31, 36, 39fvmptd3 7015 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
4140eqcomd 2732 . . . . . . . . . . . . . 14 (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = (𝐶𝐴))
42413ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = (𝐶𝐴))
4328, 42eleqtrd 2829 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝑖 ∈ (𝐶𝐴))
4416, 17, 18, 43syl3anc 1368 . . . . . . . . . . 11 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → 𝑖 ∈ (𝐶𝐴))
45 ovncvrrp.l . . . . . . . . . . . . . . . . . . . 20 𝐿 = ( ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
46 coeq2 5852 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑖𝑗) → ([,) ∘ ) = ([,) ∘ (𝑖𝑗)))
4746fveq1d 6887 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑖𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
4847fveq2d 6889 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑖𝑗) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
4948prodeq2ad 44880 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑖𝑗) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
50 elmapi 8845 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
5150adantr 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)
5351, 52ffvelcdmd 7081 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
54 prodex 15857 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) ∈ V
5554a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) ∈ V)
5645, 49, 53, 55fvmptd3 7015 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝑖𝑗)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
5756mpteq2dva 5241 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
5857fveq2d 6889 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
5958adantr 480 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
60 id 22 . . . . . . . . . . . . . . . . . 18 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
6160eqcomd 2732 . . . . . . . . . . . . . . . . 17 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = 𝑧)
6261adantl 481 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = 𝑧)
6359, 62eqtrd 2766 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = 𝑧)
64633adant1 1127 . . . . . . . . . . . . . 14 ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = 𝑧)
65 simp1 1133 . . . . . . . . . . . . . 14 ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
6664, 65eqbrtrd 5163 . . . . . . . . . . . . 13 ((𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
67663adant1l 1173 . . . . . . . . . . . 12 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
68673adant3l 1177 . . . . . . . . . . 11 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
6944, 68jca 511 . . . . . . . . . 10 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
706919.8ad 2167 . . . . . . . . 9 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
71703exp 1116 . . . . . . . 8 ((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))))
7214, 15, 71rexlimd 3257 . . . . . . 7 ((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))))
7372imp 406 . . . . . 6 (((𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
747, 8, 13, 73syl21anc 835 . . . . 5 ((𝜑𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ∧ 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
75743exp 1116 . . . 4 (𝜑 → (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} → (𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))))
7675rexlimdv 3147 . . 3 (𝜑 → (∃𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))))
776, 76mpd 15 . 2 (𝜑 → ∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
78 rabid 3446 . . . . . . . 8 (𝑖 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ↔ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
7978bicomi 223 . . . . . . 7 ((𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) ↔ 𝑖 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
8079biimpi 215 . . . . . 6 ((𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
8180adantl 481 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
82 ovncvrrp.d . . . . . . . . 9 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
83 nfcv 2897 . . . . . . . . . 10 𝑏(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
84 nfcv 2897 . . . . . . . . . . 11 𝑎+
85 nfv 1909 . . . . . . . . . . . 12 𝑎^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)
86 nfmpt1 5249 . . . . . . . . . . . . . 14 𝑎(𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
8729, 86nfcxfr 2895 . . . . . . . . . . . . 13 𝑎𝐶
88 nfcv 2897 . . . . . . . . . . . . 13 𝑎𝑏
8987, 88nffv 6895 . . . . . . . . . . . 12 𝑎(𝐶𝑏)
9085, 89nfrabw 3462 . . . . . . . . . . 11 𝑎{𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}
9184, 90nfmpt 5248 . . . . . . . . . 10 𝑎(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
92 fveq2 6885 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝐶𝑎) = (𝐶𝑏))
9392eleq2d 2813 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝑏)))
94 fveq2 6885 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝑏))
9594oveq1d 7420 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))
9695breq2d 5153 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)))
9793, 96anbi12d 630 . . . . . . . . . . . 12 (𝑎 = 𝑏 → ((𝑖 ∈ (𝐶𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))))
9897rabbidva2 3428 . . . . . . . . . . 11 (𝑎 = 𝑏 → {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)} = {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
9998mpteq2dv 5243 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
10083, 91, 99cbvmpt 5252 . . . . . . . . 9 (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
10182, 100eqtri 2754 . . . . . . . 8 𝐷 = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
102 fveq2 6885 . . . . . . . . . . . 12 (𝑏 = 𝐴 → (𝐶𝑏) = (𝐶𝐴))
103102eleq2d 2813 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑖 ∈ (𝐶𝑏) ↔ 𝑖 ∈ (𝐶𝐴)))
104 fveq2 6885 . . . . . . . . . . . . 13 (𝑏 = 𝐴 → ((voln*‘𝑋)‘𝑏) = ((voln*‘𝑋)‘𝐴))
105104oveq1d 7420 . . . . . . . . . . . 12 (𝑏 = 𝐴 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))
106105breq2d 5153 . . . . . . . . . . 11 (𝑏 = 𝐴 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)))
107103, 106anbi12d 630 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝑖 ∈ (𝐶𝑏) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)) ↔ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))))
108107rabbidva2 3428 . . . . . . . . 9 (𝑏 = 𝐴 → {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)})
109108mpteq2dv 5243 . . . . . . . 8 (𝑏 = 𝐴 → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}))
11036adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
111 rpex 44628 . . . . . . . . . 10 + ∈ V
112111mptex 7220 . . . . . . . . 9 (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}) ∈ V
113112a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}) ∈ V)
114101, 109, 110, 113fvmptd3 7015 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → (𝐷𝐴) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)}))
115 oveq2 7413 . . . . . . . . . 10 (𝑒 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
116115breq2d 5153 . . . . . . . . 9 (𝑒 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
117116rabbidv 3434 . . . . . . . 8 (𝑒 = 𝐸 → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
118117adantl 481 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) ∧ 𝑒 = 𝐸) → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
1194adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐸 ∈ ℝ+)
120 fvex 6898 . . . . . . . . 9 (𝐶𝐴) ∈ V
121120rabex 5325 . . . . . . . 8 {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
122121a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V)
123114, 118, 119, 122fvmptd 6999 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → ((𝐷𝐴)‘𝐸) = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
124123eqcomd 2732 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} = ((𝐷𝐴)‘𝐸))
12581, 124eleqtrd 2829 . . . 4 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ ((𝐷𝐴)‘𝐸))
126125ex 412 . . 3 (𝜑 → ((𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ ((𝐷𝐴)‘𝐸)))
127126eximdv 1912 . 2 (𝜑 → (∃𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → ∃𝑖 𝑖 ∈ ((𝐷𝐴)‘𝐸)))
12877, 127mpd 15 1 (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷𝐴)‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2934  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  c0 4317  𝒫 cpw 4597   ciun 4990   class class class wbr 5141  cmpt 5224   × cxp 5667  ccom 5673  wf 6533  cfv 6537  (class class class)co 7405  m cmap 8822  Xcixp 8893  Fincfn 8941  cr 11111  *cxr 11251  cle 11253  cn 12216  +crp 12980   +𝑒 cxad 13096  [,)cico 13332  cprod 15855  volcvol 25347  Σ^csumge0 45650  voln*covoln 45824
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-fl 13763  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-rlim 15439  df-sum 15639  df-prod 15856  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-rest 17377  df-0g 17396  df-topgen 17398  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-subg 19050  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-cring 20141  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-dvr 20303  df-drng 20589  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-cnfld 21241  df-top 22751  df-topon 22768  df-bases 22804  df-cmp 23246  df-ovol 25348  df-vol 25349  df-sumge0 45651  df-ovoln 45825
This theorem is referenced by:  ovnsubaddlem2  45859  hspmbllem3  45916
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