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Theorem ovnsubadd 39262
Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ovnsubadd.1 (𝜑𝑋 ∈ Fin)
ovnsubadd.2 (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
Assertion
Ref Expression
ovnsubadd (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
Distinct variable groups:   𝐴,𝑛   𝑛,𝑋   𝜑,𝑛

Proof of Theorem ovnsubadd
Dummy variables 𝑎 𝑒 𝑖 𝑗 𝑘 𝑙 𝑦 𝑧 𝑏 𝑑 𝑓 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6084 . . . . . 6 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
21fveq1d 6086 . . . . 5 (𝑋 = ∅ → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)))
32adantl 480 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)))
4 ovnsubadd.2 . . . . . . . . . . . 12 (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
54adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
6 simpr 475 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
75, 6ffvelrnd 6249 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
8 elpwi 4112 . . . . . . . . . 10 ((𝐴𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
97, 8syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109ralrimiva 2944 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
11 iunss 4487 . . . . . . . 8 ( 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
1210, 11sylibr 222 . . . . . . 7 (𝜑 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
1312adantr 479 . . . . . 6 ((𝜑𝑋 = ∅) → 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
14 oveq2 6531 . . . . . . 7 (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅))
1514adantl 480 . . . . . 6 ((𝜑𝑋 = ∅) → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅))
1613, 15sseqtrd 3599 . . . . 5 ((𝜑𝑋 = ∅) → 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 ∅))
1716ovn0val 39240 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = 0)
183, 17eqtrd 2639 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = 0)
19 nnex 10869 . . . . . 6 ℕ ∈ V
2019a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
21 ovnsubadd.1 . . . . . . . 8 (𝜑𝑋 ∈ Fin)
2221adantr 479 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
2322, 9ovncl 39257 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴𝑛)) ∈ (0[,]+∞))
24 eqid 2605 . . . . . 6 (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))
2523, 24fmptd 6273 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛))):ℕ⟶(0[,]+∞))
2620, 25sge0ge0 39077 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2726adantr 479 . . 3 ((𝜑𝑋 = ∅) → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2818, 27eqbrtrd 4595 . 2 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2921, 12ovnxrcl 39259 . . . 4 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ∈ ℝ*)
3029adantr 479 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ∈ ℝ*)
3120, 25sge0xrcl 39078 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) ∈ ℝ*)
3231adantr 479 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) ∈ ℝ*)
3321ad2antrr 757 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin)
34 neqne 2785 . . . . 5 𝑋 = ∅ → 𝑋 ≠ ∅)
3534ad2antlr 758 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅)
364ad2antrr 757 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
37 simpr 475 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
38 eqid 2605 . . . 4 (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}) = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
39 sseq1 3584 . . . . . 6 (𝑏 = 𝑎 → (𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
4039rabbidv 3159 . . . . 5 (𝑏 = 𝑎 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
4140cbvmptv 4668 . . . 4 (𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)}) = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
42 eqid 2605 . . . 4 ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
43 fveq2 6084 . . . . . . . . . . . . . . . . . . . . . 22 (𝑜 = 𝑗 → (𝑙𝑜) = (𝑙𝑗))
4443coeq2d 5190 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 = 𝑗 → ([,) ∘ (𝑙𝑜)) = ([,) ∘ (𝑙𝑗)))
4544fveq1d 6086 . . . . . . . . . . . . . . . . . . . 20 (𝑜 = 𝑗 → (([,) ∘ (𝑙𝑜))‘𝑑) = (([,) ∘ (𝑙𝑗))‘𝑑))
4645ixpeq2dv 7783 . . . . . . . . . . . . . . . . . . 19 (𝑜 = 𝑗X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = X𝑑𝑋 (([,) ∘ (𝑙𝑗))‘𝑑))
47 fveq2 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑘 → (([,) ∘ (𝑙𝑗))‘𝑑) = (([,) ∘ (𝑙𝑗))‘𝑘))
4847cbvixpv 7785 . . . . . . . . . . . . . . . . . . 19 X𝑑𝑋 (([,) ∘ (𝑙𝑗))‘𝑑) = X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)
4946, 48syl6eq 2655 . . . . . . . . . . . . . . . . . 18 (𝑜 = 𝑗X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘))
5049cbviunv 4485 . . . . . . . . . . . . . . . . 17 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)
5150sseq2i 3588 . . . . . . . . . . . . . . . 16 (𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) ↔ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘))
5251a1i 11 . . . . . . . . . . . . . . 15 (𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → (𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) ↔ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
5352rabbiia 3156 . . . . . . . . . . . . . 14 {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)} = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)}
5453mpteq2i 4659 . . . . . . . . . . . . 13 (𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)}) = (𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
5554fveq1i 6085 . . . . . . . . . . . 12 ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑑)
56 fveq2 6084 . . . . . . . . . . . 12 (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎))
5755, 56syl5eq 2651 . . . . . . . . . . 11 (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎))
5857eleq2d 2668 . . . . . . . . . 10 (𝑑 = 𝑎 → (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ↔ 𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎)))
59 fveq2 6084 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑘 → (([,) ∘ )‘𝑑) = (([,) ∘ )‘𝑘))
6059fveq2d 6088 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑘 → (vol‘(([,) ∘ )‘𝑑)) = (vol‘(([,) ∘ )‘𝑘)))
6160cbvprodv 14427 . . . . . . . . . . . . . . . . 17 𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)) = ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))
6261mpteq2i 4659 . . . . . . . . . . . . . . . 16 ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
6362a1i 11 . . . . . . . . . . . . . . 15 (𝑜 = 𝑗 → ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))))
64 fveq2 6084 . . . . . . . . . . . . . . 15 (𝑜 = 𝑗 → (𝑚𝑜) = (𝑚𝑗))
6563, 64fveq12d 6090 . . . . . . . . . . . . . 14 (𝑜 = 𝑗 → (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)) = (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))
6665cbvmptv 4668 . . . . . . . . . . . . 13 (𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜))) = (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))
6766fveq2i 6087 . . . . . . . . . . . 12 ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗))))
6867a1i 11 . . . . . . . . . . 11 (𝑑 = 𝑎 → (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))))
69 fveq2 6084 . . . . . . . . . . . 12 (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎))
7069oveq1d 6538 . . . . . . . . . . 11 (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))
7168, 70breq12d 4586 . . . . . . . . . 10 (𝑑 = 𝑎 → ((Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)))
7258, 71anbi12d 742 . . . . . . . . 9 (𝑑 = 𝑎 → ((𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∧ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)) ↔ (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))))
7372rabbidva2 3157 . . . . . . . 8 (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)})
74 fveq1 6083 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → (𝑚𝑗) = (𝑖𝑗))
7574fveq2d 6088 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)) = (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))
7675mpteq2dv 4663 . . . . . . . . . . 11 (𝑚 = 𝑖 → (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗))) = (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗))))
7776fveq2d 6088 . . . . . . . . . 10 (𝑚 = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))))
7877breq1d 4583 . . . . . . . . 9 (𝑚 = 𝑖 → ((Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)))
7978cbvrabv 3167 . . . . . . . 8 {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}
8073, 79syl6eq 2655 . . . . . . 7 (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)})
8180mpteq2dv 4663 . . . . . 6 (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑓 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}))
82 oveq2 6531 . . . . . . . . 9 (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))
8382breq2d 4585 . . . . . . . 8 (𝑓 = 𝑒 → ((Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)))
8483rabbidv 3159 . . . . . . 7 (𝑓 = 𝑒 → {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
8584cbvmptv 4668 . . . . . 6 (𝑓 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})
8681, 85syl6eq 2655 . . . . 5 (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
8786cbvmptv 4668 . . . 4 (𝑑 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑)})‘𝑑) ∣ (Σ^‘(𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)})) = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))
8833, 35, 36, 37, 38, 41, 42, 87ovnsubaddlem2 39261 . . 3 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝑦))
8930, 32, 88xrlexaddrp 38309 . 2 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
9028, 89pm2.61dan 827 1 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  wss 3535  c0 3869  𝒫 cpw 4103   ciun 4445   class class class wbr 4573  cmpt 4633   × cxp 5022  ccom 5028  wf 5782  cfv 5786  (class class class)co 6523  𝑚 cmap 7717  Xcixp 7767  Fincfn 7814  cr 9787  0cc0 9788  +∞cpnf 9923  *cxr 9925  cle 9927  cn 10863  +crp 11660   +𝑒 cxad 11772  [,)cico 12000  [,]cicc 12001  cprod 14416  volcvol 22952  Σ^csumge0 39055  voln*covoln 39226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cc 9113  ax-ac2 9141  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866  ax-addf 9867  ax-mulf 9868
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-disj 4544  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-of 6768  df-om 6931  df-1st 7032  df-2nd 7033  df-tpos 7212  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-map 7719  df-pm 7720  df-ixp 7768  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-fi 8173  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-acn 8624  df-ac 8795  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-q 11617  df-rp 11661  df-xneg 11774  df-xadd 11775  df-xmul 11776  df-ioo 12002  df-ico 12004  df-icc 12005  df-fz 12149  df-fzo 12286  df-fl 12406  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-clim 14009  df-rlim 14010  df-sum 14207  df-prod 14417  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-starv 15725  df-tset 15729  df-ple 15730  df-ds 15733  df-unif 15734  df-rest 15848  df-0g 15867  df-topgen 15869  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-grp 17190  df-minusg 17191  df-subg 17356  df-cmn 17960  df-abl 17961  df-mgp 18255  df-ur 18267  df-ring 18314  df-cring 18315  df-oppr 18388  df-dvdsr 18406  df-unit 18407  df-invr 18437  df-dvr 18448  df-drng 18514  df-psmet 19501  df-xmet 19502  df-met 19503  df-bl 19504  df-mopn 19505  df-cnfld 19510  df-top 20459  df-bases 20460  df-topon 20461  df-cmp 20938  df-ovol 22953  df-vol 22954  df-sumge0 39056  df-ovoln 39227
This theorem is referenced by:  ovnome  39263  ovnsubadd2lem  39335
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