Step | Hyp | Ref
| Expression |
1 | | fvex 6201 |
. . . 4
⊢ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∈ V |
2 | | nnenom 12779 |
. . . 4
⊢ ℕ
≈ ω |
3 | 1, 2 | axcc3 9260 |
. . 3
⊢
∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
4 | | simprl 794 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → 𝑔 Fn ℕ) |
5 | | nfv 1843 |
. . . . . . . . 9
⊢
Ⅎ𝑛𝜑 |
6 | | nfra1 2941 |
. . . . . . . . 9
⊢
Ⅎ𝑛∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
7 | 5, 6 | nfan 1828 |
. . . . . . . 8
⊢
Ⅎ𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
8 | | rspa 2930 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
9 | 8 | adantll 750 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
10 | | ovnsubaddlem2.x |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ∈ Fin) |
11 | 10 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
12 | | ovnsubaddlem2.n0 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 ≠ ∅) |
13 | 12 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅) |
14 | | ovnsubaddlem2.a |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐴:ℕ⟶𝒫 (ℝ
↑𝑚 𝑋)) |
15 | 14 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:ℕ⟶𝒫 (ℝ
↑𝑚 𝑋)) |
16 | | simpr 477 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
17 | 15, 16 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
18 | | elpwi 4168 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴‘𝑛) ∈ 𝒫 (ℝ
↑𝑚 𝑋) → (𝐴‘𝑛) ⊆ (ℝ ↑𝑚
𝑋)) |
19 | 17, 18 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (ℝ ↑𝑚
𝑋)) |
20 | | ovnsubaddlem2.e |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
21 | 20 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈
ℝ+) |
22 | | nnnn0 11299 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
23 | | 2nn 11185 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℕ |
24 | 23 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ0
→ 2 ∈ ℕ) |
25 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℕ0) |
26 | | nnexpcl 12873 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ0) → (2↑𝑛) ∈ ℕ) |
27 | 24, 25, 26 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ0
→ (2↑𝑛) ∈
ℕ) |
28 | | nnrp 11842 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2↑𝑛) ∈
ℕ → (2↑𝑛)
∈ ℝ+) |
29 | 27, 28 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ0
→ (2↑𝑛) ∈
ℝ+) |
30 | 22, 29 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ →
(2↑𝑛) ∈
ℝ+) |
31 | 30 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈
ℝ+) |
32 | 21, 31 | rpdivcld 11889 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 / (2↑𝑛)) ∈
ℝ+) |
33 | | ovnsubaddlem2.c |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ
↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)
∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
34 | | ovnsubaddlem2.l |
. . . . . . . . . . . . . . . 16
⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ)
↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) |
35 | | ovnsubaddlem2.d |
. . . . . . . . . . . . . . . 16
⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ
↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)})) |
36 | 11, 13, 19, 32, 33, 34, 35 | ovncvrrp 40778 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
37 | | n0 3931 |
. . . . . . . . . . . . . . 15
⊢ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
38 | 36, 37 | sylibr 224 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅) |
39 | 38 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅) |
40 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
41 | 39, 40 | mpd 15 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
42 | 41 | ex 450 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
43 | 42 | adantlr 751 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → ((((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
44 | 9, 43 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
45 | 44 | ex 450 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑛 ∈ ℕ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
46 | 7, 45 | ralrimi 2957 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
47 | 46 | adantrl 752 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
48 | 4, 47 | jca 554 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
49 | 48 | ex 450 |
. . . 4
⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) |
50 | 49 | eximdv 1846 |
. . 3
⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ≠ ∅ → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))))) |
51 | 3, 50 | mpi 20 |
. 2
⊢ (𝜑 → ∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
52 | | simpl 473 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝜑) |
53 | | simprl 794 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → 𝑔 Fn ℕ) |
54 | | simprr 796 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
55 | | nnf1oxpnn 39384 |
. . . . . 6
⊢
∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ) |
56 | | simpl1 1064 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝜑) |
57 | | simpl2 1065 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑔 Fn ℕ) |
58 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑞 = 𝑛 → (𝑔‘𝑞) = (𝑔‘𝑛)) |
59 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑞 = 𝑛 → (𝐴‘𝑞) = (𝐴‘𝑛)) |
60 | 59 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 = 𝑛 → (𝐷‘(𝐴‘𝑞)) = (𝐷‘(𝐴‘𝑛))) |
61 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑞 = 𝑛 → (2↑𝑞) = (2↑𝑛)) |
62 | 61 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 = 𝑛 → (𝐸 / (2↑𝑞)) = (𝐸 / (2↑𝑛))) |
63 | 60, 62 | fveq12d 6197 |
. . . . . . . . . . . . . 14
⊢ (𝑞 = 𝑛 → ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) = ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
64 | 58, 63 | eleq12d 2695 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑛 → ((𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) |
65 | 64 | cbvralv 3171 |
. . . . . . . . . . . 12
⊢
(∀𝑞 ∈
ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) ↔ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
66 | 65 | biimpri 218 |
. . . . . . . . . . 11
⊢
(∀𝑛 ∈
ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) |
67 | 66 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) |
68 | 67 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) →
∀𝑞 ∈ ℕ
(𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) |
69 | | simpr 477 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) |
70 | 10 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin) |
71 | 70 | 3ad2antl1 1223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ∈ Fin) |
72 | 12 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅) |
73 | 72 | 3ad2antl1 1223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑋 ≠ ∅) |
74 | 14 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫
(ℝ ↑𝑚 𝑋)) |
75 | 74 | 3ad2antl1 1223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐴:ℕ⟶𝒫
(ℝ ↑𝑚 𝑋)) |
76 | 20 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈
ℝ+) |
77 | 76 | 3ad2antl1 1223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝐸 ∈
ℝ+) |
78 | | ovnsubaddlem2.z |
. . . . . . . . . 10
⊢ 𝑍 = (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)))))}) |
79 | | coeq2 5280 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = 𝑖 → ([,) ∘ ℎ) = ([,) ∘ 𝑖)) |
80 | 79 | fveq1d 6193 |
. . . . . . . . . . . . . 14
⊢ (ℎ = 𝑖 → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ 𝑖)‘𝑘)) |
81 | 80 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑖 → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(([,) ∘ 𝑖)‘𝑘))) |
82 | 81 | prodeq2ad 39824 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) |
83 | 82 | cbvmptv 4750 |
. . . . . . . . . . 11
⊢ (ℎ ∈ ((ℝ ×
ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))) = (𝑖 ∈ ((ℝ × ℝ)
↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) |
84 | 34, 83 | eqtri 2644 |
. . . . . . . . . 10
⊢ 𝐿 = (𝑖 ∈ ((ℝ × ℝ)
↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))) |
85 | 65 | biimpi 206 |
. . . . . . . . . . . . 13
⊢
(∀𝑞 ∈
ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
86 | 85 | 3ad2ant3 1084 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) → ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
87 | 86 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) →
∀𝑛 ∈ ℕ
(𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
88 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℕ) |
89 | | rspa 2930 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
90 | 87, 88, 89 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) |
91 | | simpr 477 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) → 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) |
92 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑞 = 𝑚 → (𝑓‘𝑞) = (𝑓‘𝑚)) |
93 | 92 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑚 → (1st ‘(𝑓‘𝑞)) = (1st ‘(𝑓‘𝑚))) |
94 | 93 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑚 → (𝑔‘(1st ‘(𝑓‘𝑞))) = (𝑔‘(1st ‘(𝑓‘𝑚)))) |
95 | 92 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑚 → (2nd ‘(𝑓‘𝑞)) = (2nd ‘(𝑓‘𝑚))) |
96 | 94, 95 | fveq12d 6197 |
. . . . . . . . . . 11
⊢ (𝑞 = 𝑚 → ((𝑔‘(1st ‘(𝑓‘𝑞)))‘(2nd ‘(𝑓‘𝑞))) = ((𝑔‘(1st ‘(𝑓‘𝑚)))‘(2nd ‘(𝑓‘𝑚)))) |
97 | 96 | cbvmptv 4750 |
. . . . . . . . . 10
⊢ (𝑞 ∈ ℕ ↦ ((𝑔‘(1st
‘(𝑓‘𝑞)))‘(2nd
‘(𝑓‘𝑞)))) = (𝑚 ∈ ℕ ↦ ((𝑔‘(1st ‘(𝑓‘𝑚)))‘(2nd ‘(𝑓‘𝑚)))) |
98 | 71, 73, 75, 77, 78, 33, 84, 35, 90, 91, 97 | ovnsubaddlem1 40784 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑞 ∈ ℕ (𝑔‘𝑞) ∈ ((𝐷‘(𝐴‘𝑞))‘(𝐸 / (2↑𝑞)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) |
99 | 56, 57, 68, 69, 98 | syl31anc 1329 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) ∧ 𝑓:ℕ–1-1-onto→(ℕ × ℕ)) →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) |
100 | 99 | ex 450 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (𝑓:ℕ–1-1-onto→(ℕ × ℕ) →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))) |
101 | 100 | exlimdv 1861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → (∃𝑓 𝑓:ℕ–1-1-onto→(ℕ × ℕ) →
((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))) |
102 | 55, 101 | mpi 20 |
. . . . 5
⊢ ((𝜑 ∧ 𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) |
103 | 52, 53, 54, 102 | syl3anc 1326 |
. . . 4
⊢ ((𝜑 ∧ (𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛))))) → ((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) |
104 | 103 | ex 450 |
. . 3
⊢ (𝜑 → ((𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))) |
105 | 104 | exlimdv 1861 |
. 2
⊢ (𝜑 → (∃𝑔(𝑔 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∈ ((𝐷‘(𝐴‘𝑛))‘(𝐸 / (2↑𝑛)))) → ((voln*‘𝑋)‘∪
𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸))) |
106 | 51, 105 | mpd 15 |
1
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤
((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴‘𝑛)))) +𝑒 𝐸)) |