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Theorem ovncvrrp 43685
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 2739 . . . 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 43683 . . 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 1137 . . . . . 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 1139 . . . . . 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 3282 . . . . . . . . . 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 219 . . . . . . . . 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 499 . . . . . . . 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 484 . . . . . . 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 1131 . . . . . 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 1921 . . . . . . . 8 𝑖(𝜑𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15 nfe1 2155 . . . . . . . 8 𝑖𝑖(𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
16 simp1l 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)))))) → 𝜑)
17 simp2 1138 . . . . . . . . . . . 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 1202 . . . . . . . . . . . 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 6686 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑖 → (𝑙𝑗) = (𝑖𝑗))
2120coeq2d 5715 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑖 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝑖𝑗)))
2221fveq1d 6689 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑖 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
2322ixpeq2dv 8536 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑖X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
2423iuneq2d 4920 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑖 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘))
2524sseq2d 3919 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑖 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)))
2625elrab 3593 . . . . . . . . . . . . . . 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 237 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 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 3912 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
3130rabbidv 3382 . . . . . . . . . . . . . . . 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 7218 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
3332, 3ssexd 5202 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
34 elpwg 4501 . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
37 ovex 7216 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
3837rabex 5210 . . . . . . . . . . . . . . . . 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 6811 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
4140eqcomd 2745 . . . . . . . . . . . . . 14 (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = (𝐶𝐴))
42413ad2ant1 1134 . . . . . . . . . . . . 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 2836 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘)) → 𝑖 ∈ (𝐶𝐴))
4416, 17, 18, 43syl3anc 1372 . . . . . . . . . . 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 5711 . . . . . . . . . . . . . . . . . . . . . . 23 ( = (𝑖𝑗) → ([,) ∘ ) = ([,) ∘ (𝑖𝑗)))
4746fveq1d 6689 . . . . . . . . . . . . . . . . . . . . . 22 ( = (𝑖𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝑖𝑗))‘𝑘))
4847fveq2d 6691 . . . . . . . . . . . . . . . . . . . . 21 ( = (𝑖𝑗) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
4948prodeq2ad 42716 . . . . . . . . . . . . . . . . . . . 20 ( = (𝑖𝑗) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
50 elmapi 8472 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
5150adantr 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)
5351, 52ffvelrnd 6875 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
54 prodex 15366 . . . . . . . . . . . . . . . . . . . . 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 6811 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝑖𝑗)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))
5756mpteq2dva 5135 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))
5857fveq2d 6691 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
5958adantr 484 . . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . . . . 17 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = 𝑧)
6261adantl 485 . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = 𝑧)
64633adant1 1131 . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . 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 5062 . . . . . . . . . . . . 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 1177 . . . . . . . . . . . 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 1181 . . . . . . . . . . 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 515 . . . . . . . . . 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 2183 . . . . . . . . 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 1120 . . . . . . . 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 3228 . . . . . . 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 410 . . . . . 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 837 . . . . 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 1120 . . . 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 3194 . . 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 3282 . . . . . . . 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 227 . . . . . . 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 219 . . . . . 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 485 . . . . 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 2900 . . . . . . . . . 10 𝑏(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
84 nfcv 2900 . . . . . . . . . . 11 𝑎+
85 nfv 1921 . . . . . . . . . . . 12 𝑎^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)
86 nfmpt1 5138 . . . . . . . . . . . . . 14 𝑎(𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
8729, 86nfcxfr 2898 . . . . . . . . . . . . 13 𝑎𝐶
88 nfcv 2900 . . . . . . . . . . . . 13 𝑎𝑏
8987, 88nffv 6697 . . . . . . . . . . . 12 𝑎(𝐶𝑏)
9085, 89nfrabw 3289 . . . . . . . . . . 11 𝑎{𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}
9184, 90nfmpt 5137 . . . . . . . . . 10 𝑎(𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)})
92 fveq2 6687 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝐶𝑎) = (𝐶𝑏))
9392eleq2d 2819 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝑏)))
94 fveq2 6687 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝑏))
9594oveq1d 7198 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑒) = (((voln*‘𝑋)‘𝑏) +𝑒 𝑒))
9695breq2d 5052 . . . . . . . . . . . . 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 634 . . . . . . . . . . . 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 3378 . . . . . . . . . . 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 5136 . . . . . . . . . 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 5141 . . . . . . . . 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 2762 . . . . . . . 8 𝐷 = (𝑏 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑏) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑏) +𝑒 𝑒)}))
102 fveq2 6687 . . . . . . . . . . . 12 (𝑏 = 𝐴 → (𝐶𝑏) = (𝐶𝐴))
103102eleq2d 2819 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑖 ∈ (𝐶𝑏) ↔ 𝑖 ∈ (𝐶𝐴)))
104 fveq2 6687 . . . . . . . . . . . . 13 (𝑏 = 𝐴 → ((voln*‘𝑋)‘𝑏) = ((voln*‘𝑋)‘𝐴))
105104oveq1d 7198 . . . . . . . . . . . 12 (𝑏 = 𝐴 → (((voln*‘𝑋)‘𝑏) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑒))
106105breq2d 5052 . . . . . . . . . . 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 634 . . . . . . . . . 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 3378 . . . . . . . . 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 5136 . . . . . . . 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 484 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
111 rpex 42464 . . . . . . . . . 10 + ∈ V
112111mptex 7009 . . . . . . . . 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 6811 . . . . . . 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 7191 . . . . . . . . . 10 (𝑒 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
116115breq2d 5052 . . . . . . . . 9 (𝑒 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑒) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
117116rabbidv 3382 . . . . . . . 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 485 . . . . . . 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 484 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝐸 ∈ ℝ+)
120 fvex 6700 . . . . . . . . 9 (𝐶𝐴) ∈ V
121120rabex 5210 . . . . . . . 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 6795 . . . . . 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 2745 . . . . 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 2836 . . . 4 ((𝜑 ∧ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) → 𝑖 ∈ ((𝐷𝐴)‘𝐸))
126125ex 416 . . 3 (𝜑 → ((𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) → 𝑖 ∈ ((𝐷𝐴)‘𝐸)))
127126eximdv 1924 . 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 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wne 2935  wrex 3055  {crab 3058  Vcvv 3400  wss 3853  c0 4221  𝒫 cpw 4498   ciun 4891   class class class wbr 5040  cmpt 5120   × cxp 5533  ccom 5539  wf 6346  cfv 6350  (class class class)co 7183  m cmap 8450  Xcixp 8520  Fincfn 8568  cr 10627  *cxr 10765  cle 10767  cn 11729  +crp 12485   +𝑒 cxad 12601  [,)cico 12836  cprod 15364  volcvol 24228  Σ^csumge0 43483  voln*covoln 43657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-inf2 9190  ax-cnex 10684  ax-resscn 10685  ax-1cn 10686  ax-icn 10687  ax-addcl 10688  ax-addrcl 10689  ax-mulcl 10690  ax-mulrcl 10691  ax-mulcom 10692  ax-addass 10693  ax-mulass 10694  ax-distr 10695  ax-i2m1 10696  ax-1ne0 10697  ax-1rid 10698  ax-rnegex 10699  ax-rrecex 10700  ax-cnre 10701  ax-pre-lttri 10702  ax-pre-lttrn 10703  ax-pre-ltadd 10704  ax-pre-mulgt0 10705  ax-pre-sup 10706  ax-addf 10707  ax-mulf 10708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-se 5494  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7140  df-ov 7186  df-oprab 7187  df-mpo 7188  df-of 7438  df-om 7613  df-1st 7727  df-2nd 7728  df-tpos 7934  df-wrecs 7989  df-recs 8050  df-rdg 8088  df-1o 8144  df-2o 8145  df-er 8333  df-map 8452  df-pm 8453  df-ixp 8521  df-en 8569  df-dom 8570  df-sdom 8571  df-fin 8572  df-fi 8961  df-sup 8992  df-inf 8993  df-oi 9060  df-dju 9416  df-card 9454  df-pnf 10768  df-mnf 10769  df-xr 10770  df-ltxr 10771  df-le 10772  df-sub 10963  df-neg 10964  df-div 11389  df-nn 11730  df-2 11792  df-3 11793  df-4 11794  df-5 11795  df-6 11796  df-7 11797  df-8 11798  df-9 11799  df-n0 11990  df-z 12076  df-dec 12193  df-uz 12338  df-q 12444  df-rp 12486  df-xneg 12603  df-xadd 12604  df-xmul 12605  df-ioo 12838  df-ico 12840  df-icc 12841  df-fz 12995  df-fzo 13138  df-fl 13266  df-seq 13474  df-exp 13535  df-hash 13796  df-cj 14561  df-re 14562  df-im 14563  df-sqrt 14697  df-abs 14698  df-clim 14948  df-rlim 14949  df-sum 15149  df-prod 15365  df-struct 16601  df-ndx 16602  df-slot 16603  df-base 16605  df-sets 16606  df-ress 16607  df-plusg 16694  df-mulr 16695  df-starv 16696  df-tset 16700  df-ple 16701  df-ds 16703  df-unif 16704  df-rest 16812  df-0g 16831  df-topgen 16833  df-mgm 17981  df-sgrp 18030  df-mnd 18041  df-grp 18235  df-minusg 18236  df-subg 18407  df-cmn 19039  df-abl 19040  df-mgp 19372  df-ur 19384  df-ring 19431  df-cring 19432  df-oppr 19508  df-dvdsr 19526  df-unit 19527  df-invr 19557  df-dvr 19568  df-drng 19636  df-psmet 20222  df-xmet 20223  df-met 20224  df-bl 20225  df-mopn 20226  df-cnfld 20231  df-top 21658  df-topon 21675  df-bases 21710  df-cmp 22151  df-ovol 24229  df-vol 24230  df-sumge0 43484  df-ovoln 43658
This theorem is referenced by:  ovnsubaddlem2  43692  hspmbllem3  43749
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