| Step | Hyp | Ref
| Expression |
| 1 | | fveq2 6906 |
. . . . . 6
⊢ (𝑋 = ∅ →
(voln*‘𝑋) =
(voln*‘∅)) |
| 2 | 1 | fveq1d 6908 |
. . . . 5
⊢ (𝑋 = ∅ →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛))) |
| 3 | 2 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = ((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛))) |
| 4 | | ovnsubadd.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ
↑m 𝑋)) |
| 5 | 4 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ
↑m 𝑋)) |
| 6 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
| 7 | 5, 6 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 8 | | elpwi 4607 |
. . . . . . . . . 10
⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ
↑m 𝑋)
→ (𝐴‘𝑛) ⊆ (ℝ
↑m 𝑋)) |
| 9 | 7, 8 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 10 | 9 | ralrimiva 3146 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 11 | | iunss 5045 |
. . . . . . . 8
⊢ (∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 12 | 10, 11 | sylibr 234 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 13 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 14 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑋 = ∅ → (ℝ
↑m 𝑋) =
(ℝ ↑m ∅)) |
| 15 | 14 | adantl 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ
↑m 𝑋) =
(ℝ ↑m ∅)) |
| 16 | 13, 15 | sseqtrd 4020 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑛 ∈ ℕ (𝐴‘𝑛) ⊆ (ℝ ↑m
∅)) |
| 17 | 16 | ovn0val 46565 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) →
((voln*‘∅)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0) |
| 18 | 3, 17 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) = 0) |
| 19 | | nnex 12272 |
. . . . . 6
⊢ ℕ
∈ V |
| 20 | 19 | a1i 11 |
. . . . 5
⊢ (𝜑 → ℕ ∈
V) |
| 21 | | ovnsubadd.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 22 | 21 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
| 23 | 22, 9 | ovncl 46582 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln*‘𝑋)‘(𝐴‘𝑛)) ∈ (0[,]+∞)) |
| 24 | | eqid 2737 |
. . . . . 6
⊢ (𝑛 ∈ ℕ ↦
((voln*‘𝑋)‘(𝐴‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))) |
| 25 | 23, 24 | fmptd 7134 |
. . . . 5
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))):ℕ⟶(0[,]+∞)) |
| 26 | 20, 25 | sge0ge0 46399 |
. . . 4
⊢ (𝜑 → 0 ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) |
| 27 | 26 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) |
| 28 | 18, 27 | eqbrtrd 5165 |
. 2
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) |
| 29 | 21, 12 | ovnxrcl 46584 |
. . . 4
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈
ℝ*) |
| 30 | 29 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ∈
ℝ*) |
| 31 | 20, 25 | sge0xrcl 46400 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈
ℝ*) |
| 32 | 31 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) ∈
ℝ*) |
| 33 | 21 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ∈ Fin) |
| 34 | | neqne 2948 |
. . . . 5
⊢ (¬
𝑋 = ∅ → 𝑋 ≠ ∅) |
| 35 | 34 | ad2antlr 727 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑋 ≠ ∅) |
| 36 | 4 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝐴:ℕ⟶𝒫
(ℝ ↑m 𝑋)) |
| 37 | | simpr 484 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈
ℝ+) |
| 38 | | eqid 2737 |
. . . 4
⊢ (𝑎 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ {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)))))}) = (𝑎 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {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)))))}) |
| 39 | | sseq1 4009 |
. . . . . 6
⊢ (𝑏 = 𝑎 → (𝑏 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝑎 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))) |
| 40 | 39 | rabbidv 3444 |
. . . . 5
⊢ (𝑏 = 𝑎 → {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝑏 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝑎 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 41 | 40 | cbvmptv 5255 |
. . . 4
⊢ (𝑏 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ {𝑙 ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) = (𝑎 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 42 | | eqid 2737 |
. . . 4
⊢ (ℎ ∈ ((ℝ ×
ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘))) |
| 43 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑜 = 𝑗 → (𝑙‘𝑜) = (𝑙‘𝑗)) |
| 44 | 43 | coeq2d 5873 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑜 = 𝑗 → ([,) ∘ (𝑙‘𝑜)) = ([,) ∘ (𝑙‘𝑗))) |
| 45 | 44 | fveq1d 6908 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑜 = 𝑗 → (([,) ∘ (𝑙‘𝑜))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑑)) |
| 46 | 45 | ixpeq2dv 8953 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑)) |
| 47 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 = 𝑘 → (([,) ∘ (𝑙‘𝑗))‘𝑑) = (([,) ∘ (𝑙‘𝑗))‘𝑘)) |
| 48 | 47 | cbvixpv 8955 |
. . . . . . . . . . . . . . . . . 18
⊢ X𝑑 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) |
| 49 | 46, 48 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑜 = 𝑗 → X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)) |
| 50 | 49 | cbviunv 5040 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) = ∪ 𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) |
| 51 | 50 | sseq2i 4013 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑) ↔ 𝑏 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)) |
| 52 | 51 | rabbii 3442 |
. . . . . . . . . . . . . 14
⊢ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)} = {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝑏 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} |
| 53 | 52 | mpteq2i 5247 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ {𝑙 ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)}) = (𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 54 | 53 | fveq1i 6907 |
. . . . . . . . . . . 12
⊢ ((𝑏 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ {𝑙 ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑑) |
| 55 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎)) |
| 56 | 54, 55 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) = ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎)) |
| 57 | 56 | eleq2d 2827 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑎 → (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ↔ 𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎))) |
| 58 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑑 = 𝑘 → (vol‘(([,) ∘ ℎ)‘𝑑)) = (vol‘(([,) ∘ ℎ)‘𝑘))) |
| 59 | 58 | cbvprodv 15950 |
. . . . . . . . . . . . . . . . 17
⊢
∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) |
| 60 | 59 | mpteq2i 5247 |
. . . . . . . . . . . . . . . 16
⊢ (ℎ ∈ ((ℝ ×
ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑))) = (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘))) |
| 61 | 60 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑜 = 𝑗 → (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑))) = (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))) |
| 62 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑜 = 𝑗 → (𝑚‘𝑜) = (𝑚‘𝑗)) |
| 63 | 61, 62 | fveq12d 6913 |
. . . . . . . . . . . . . 14
⊢ (𝑜 = 𝑗 → ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)) = ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗))) |
| 64 | 63 | cbvmptv 5255 |
. . . . . . . . . . . . 13
⊢ (𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ ×
ℝ) ↑m 𝑋) ↦ ∏𝑑 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑑)))‘(𝑚‘𝑜))) = (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗))) |
| 65 | 64 | fveq2i 6909 |
. . . . . . . . . . . 12
⊢
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) |
| 66 | 65 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 →
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗))))) |
| 67 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → ((voln*‘𝑋)‘𝑑) = ((voln*‘𝑋)‘𝑎)) |
| 68 | 67 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)) |
| 69 | 66, 68 | breq12d 5156 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑎 →
((Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))) |
| 70 | 57, 69 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑑 = 𝑎 → ((𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∧
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)) ↔ (𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)))) |
| 71 | 70 | rabbidva2 3438 |
. . . . . . . 8
⊢ (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}) |
| 72 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑖 → (𝑚‘𝑗) = (𝑖‘𝑗)) |
| 73 | 72 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑖 → ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)) = ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗))) |
| 74 | 73 | mpteq2dv 5244 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑖 → (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗))) = (𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) |
| 75 | 74 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑖 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗))))) |
| 76 | 75 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑚 = 𝑖 →
((Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓))) |
| 77 | 76 | cbvrabv 3447 |
. . . . . . . 8
⊢ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑚‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} |
| 78 | 71, 77 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝑑 = 𝑎 → {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}) |
| 79 | 78 | mpteq2dv 5244 |
. . . . . 6
⊢ (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑓 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)})) |
| 80 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑓 = 𝑒 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) = (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)) |
| 81 | 80 | breq2d 5155 |
. . . . . . . 8
⊢ (𝑓 = 𝑒 →
((Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒))) |
| 82 | 81 | rabbidv 3444 |
. . . . . . 7
⊢ (𝑓 = 𝑒 → {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)} = {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}) |
| 83 | 82 | cbvmptv 5255 |
. . . . . 6
⊢ (𝑓 ∈ ℝ+
↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ {𝑙 ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}) |
| 84 | 79, 83 | eqtrdi 2793 |
. . . . 5
⊢ (𝑑 = 𝑎 → (𝑓 ∈ ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((voln*‘𝑋)‘𝑑) +𝑒 𝑓)}) = (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) |
| 85 | 84 | cbvmptv 5255 |
. . . 4
⊢ (𝑑 ∈ 𝒫 (ℝ
↑m 𝑋)
↦ (𝑓 ∈
ℝ+ ↦ {𝑚 ∈ ((𝑏 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑏 ⊆ ∪ 𝑜 ∈ ℕ X𝑑 ∈ 𝑋 (([,) ∘ (𝑙‘𝑜))‘𝑑)})‘𝑑) ∣
(Σ^‘(𝑜 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑑 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑑)))‘(𝑚‘𝑜)))) ≤ (((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 X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ ((ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) |
| 86 | 33, 35, 36, 37, 38, 41, 42, 85 | ovnsubaddlem2 46586 |
. . 3
⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑦 ∈ ℝ+) →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝑦)) |
| 87 | 30, 32, 86 | xrlexaddrp 45363 |
. 2
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) |
| 88 | 28, 87 | pm2.61dan 813 |
1
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛))))) |