Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnsubadd Structured version   Visualization version   GIF version

 Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
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

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 6103 . . . . . 6 (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
21fveq1d 6105 . . . . 5 (𝑋 = ∅ → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)))
32adantl 481 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)))
4 ovnsubadd.2 . . . . . . . . . . . 12 (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
54adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
6 simpr 476 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
75, 6ffvelrnd 6268 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋))
8 elpwi 4117 . . . . . . . . . 10 ((𝐴𝑛) ∈ 𝒫 (ℝ ↑𝑚 𝑋) → (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
97, 8syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
109ralrimiva 2949 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
11 iunss 4497 . . . . . . . 8 ( 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
1210, 11sylibr 223 . . . . . . 7 (𝜑 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
1312adantr 480 . . . . . 6 ((𝜑𝑋 = ∅) → 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 𝑋))
14 oveq2 6557 . . . . . . 7 (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅))
1514adantl 481 . . . . . 6 ((𝜑𝑋 = ∅) → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅))
1613, 15sseqtrd 3604 . . . . 5 ((𝜑𝑋 = ∅) → 𝑛 ∈ ℕ (𝐴𝑛) ⊆ (ℝ ↑𝑚 ∅))
1716ovn0val 39440 . . . 4 ((𝜑𝑋 = ∅) → ((voln*‘∅)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = 0)
183, 17eqtrd 2644 . . 3 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) = 0)
19 nnex 10903 . . . . . 6 ℕ ∈ V
2019a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
21 ovnsubadd.1 . . . . . . . 8 (𝜑𝑋 ∈ Fin)
2221adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
2322, 9ovncl 39457 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴𝑛)) ∈ (0[,]+∞))
24 eqid 2610 . . . . . 6 (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))
2523, 24fmptd 6292 . . . . 5 (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛))):ℕ⟶(0[,]+∞))
2620, 25sge0ge0 39277 . . . 4 (𝜑 → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2726adantr 480 . . 3 ((𝜑𝑋 = ∅) → 0 ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2818, 27eqbrtrd 4605 . 2 ((𝜑𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
2921, 12ovnxrcl 39459 . . . 4 (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ∈ ℝ*)
3029adantr 480 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ∈ ℝ*)
3120, 25sge0xrcl 39278 . . . 4 (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) ∈ ℝ*)
3231adantr 480 . . 3 ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) ∈ ℝ*)
3321ad2antrr 758 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin)
34 neqne 2790 . . . . 5 𝑋 = ∅ → 𝑋 ≠ ∅)
3534ad2antlr 759 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅)
364ad2antrr 758 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))
37 simpr 476 . . . 4 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
38 eqid 2610 . . . 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 3589 . . . . . 6 (𝑏 = 𝑎 → (𝑏 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
4039rabbidv 3164 . . . . 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 4678 . . . 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 2610 . . . 4 ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))) = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
43 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑜 = 𝑗 → (𝑙𝑜) = (𝑙𝑗))
4443coeq2d 5206 . . . . . . . . . . . . . . . . . . . . 21 (𝑜 = 𝑗 → ([,) ∘ (𝑙𝑜)) = ([,) ∘ (𝑙𝑗)))
4544fveq1d 6105 . . . . . . . . . . . . . . . . . . . 20 (𝑜 = 𝑗 → (([,) ∘ (𝑙𝑜))‘𝑑) = (([,) ∘ (𝑙𝑗))‘𝑑))
4645ixpeq2dv 7810 . . . . . . . . . . . . . . . . . . 19 (𝑜 = 𝑗X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = X𝑑𝑋 (([,) ∘ (𝑙𝑗))‘𝑑))
47 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = 𝑘 → (([,) ∘ (𝑙𝑗))‘𝑑) = (([,) ∘ (𝑙𝑗))‘𝑘))
4847cbvixpv 7812 . . . . . . . . . . . . . . . . . . 19 X𝑑𝑋 (([,) ∘ (𝑙𝑗))‘𝑑) = X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)
4946, 48syl6eq 2660 . . . . . . . . . . . . . . . . . 18 (𝑜 = 𝑗X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘))
5049cbviunv 4495 . . . . . . . . . . . . . . . . 17 𝑜 ∈ ℕ X𝑑𝑋 (([,) ∘ (𝑙𝑜))‘𝑑) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)
5150sseq2i 3593 . . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . 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 4669 . . . . . . . . . . . . 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 6104 . . . . . . . . . . . 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 6103 . . . . . . . . . . . 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 2656 . . . . . . . . . . 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 2673 . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑘 → (([,) ∘ )‘𝑑) = (([,) ∘ )‘𝑘))
6059fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑘 → (vol‘(([,) ∘ )‘𝑑)) = (vol‘(([,) ∘ )‘𝑘)))
6160cbvprodv 14485 . . . . . . . . . . . . . . . . 17 𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)) = ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))
6261mpteq2i 4669 . . . . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . 15 (𝑜 = 𝑗 → (𝑚𝑜) = (𝑚𝑗))
6563, 64fveq12d 6109 . . . . . . . . . . . . . 14 (𝑜 = 𝑗 → (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜)) = (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))
6665cbvmptv 4678 . . . . . . . . . . . . 13 (𝑜 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑑𝑋 (vol‘(([,) ∘ )‘𝑑)))‘(𝑚𝑜))) = (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)))
6766fveq2i 6106 . . . . . . . . . . . 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 6103 . . . . . . . . . . . 12 (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎))
7069oveq1d 6564 . . . . . . . . . . 11 (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))
7168, 70breq12d 4596 . . . . . . . . . 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 743 . . . . . . . . 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 3162 . . . . . . . 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 6102 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → (𝑚𝑗) = (𝑖𝑗))
7574fveq2d 6107 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗)) = (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗)))
7675mpteq2dv 4673 . . . . . . . . . . 11 (𝑚 = 𝑖 → (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑚𝑗))) = (𝑗 ∈ ℕ ↦ (( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))‘(𝑖𝑗))))
7776fveq2d 6107 . . . . . . . . . 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 4593 . . . . . . . . 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 3172 . . . . . . . 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 2660 . . . . . . 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 4673 . . . . . 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 6557 . . . . . . . . 9 (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))
8382breq2d 4595 . . . . . . . 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 3164 . . . . . . 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 4678 . . . . . 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 2660 . . . . 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 4678 . . . 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 39461 . . 3 (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝑦))
8930, 32, 88xrlexaddrp 38509 . 2 ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
9028, 89pm2.61dan 828 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ*cxr 9952   ≤ cle 9954  ℕcn 10897  ℝ+crp 11708   +𝑒 cxad 11820  [,)cico 12048  [,]cicc 12049  ∏cprod 14474  volcvol 23039  Σ^csumge0 39255  voln*covoln 39426 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-sumge0 39256  df-ovoln 39427 This theorem is referenced by:  ovnome  39463  ovnsubadd2lem  39535
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