Step | Hyp | Ref
| Expression |
1 | | breq2 5083 |
. . . . 5
⊢ (𝑗 = 𝑖 → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < 𝑖)) |
2 | 1 | cbvrabv 3425 |
. . . 4
⊢ {𝑗 ∈ ℕ ∣ (1 /
𝐷) < 𝑗} = {𝑖 ∈ ℕ ∣ (1 / 𝐷) < 𝑖} |
3 | | stoweidlem49.4 |
. . . 4
⊢ (𝜑 → 𝐷 ∈
ℝ+) |
4 | | stoweidlem49.5 |
. . . 4
⊢ (𝜑 → 𝐷 < 1) |
5 | 2, 3, 4 | stoweidlem14 43537 |
. . 3
⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) |
6 | | eqid 2740 |
. . . . . 6
⊢ (𝑖 ∈ ℕ0
↦ ((1 / (𝑘 ·
𝐷))↑𝑖)) = (𝑖 ∈ ℕ0 ↦ ((1 /
(𝑘 · 𝐷))↑𝑖)) |
7 | | eqid 2740 |
. . . . . 6
⊢ (𝑖 ∈ ℕ0
↦ (((𝑘 · 𝐷) / 2)↑𝑖)) = (𝑖 ∈ ℕ0 ↦ (((𝑘 · 𝐷) / 2)↑𝑖)) |
8 | | nnre 11991 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
9 | 8 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ) |
10 | 3 | rpred 12783 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ ℝ) |
11 | 10 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ ℝ) |
12 | 9, 11 | remulcld 11016 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝐷) ∈ ℝ) |
13 | 12 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → (𝑘 · 𝐷) ∈ ℝ) |
14 | | simprl 768 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → 1 < (𝑘 · 𝐷)) |
15 | 12 | rehalfcld 12231 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 · 𝐷) / 2) ∈ ℝ) |
16 | | nngt0 12015 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → 0 <
𝑘) |
17 | 16 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝑘) |
18 | 3 | rpgt0d 12786 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < 𝐷) |
19 | 18 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐷) |
20 | 9, 11, 17, 19 | mulgt0d 11141 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (𝑘 · 𝐷)) |
21 | | 2re 12058 |
. . . . . . . . . . 11
⊢ 2 ∈
ℝ |
22 | | 2pos 12087 |
. . . . . . . . . . 11
⊢ 0 <
2 |
23 | 21, 22 | pm3.2i 471 |
. . . . . . . . . 10
⊢ (2 ∈
ℝ ∧ 0 < 2) |
24 | 23 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2 ∈ ℝ
∧ 0 < 2)) |
25 | | divgt0 11854 |
. . . . . . . . 9
⊢ ((((𝑘 · 𝐷) ∈ ℝ ∧ 0 < (𝑘 · 𝐷)) ∧ (2 ∈ ℝ ∧ 0 < 2))
→ 0 < ((𝑘 ·
𝐷) / 2)) |
26 | 12, 20, 24, 25 | syl21anc 835 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < ((𝑘 · 𝐷) / 2)) |
27 | 15, 26 | elrpd 12780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 · 𝐷) / 2) ∈
ℝ+) |
28 | 27 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → ((𝑘 · 𝐷) / 2) ∈
ℝ+) |
29 | | simprr 770 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → ((𝑘 · 𝐷) / 2) < 1) |
30 | | stoweidlem49.14 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
31 | 30 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → 𝐸 ∈
ℝ+) |
32 | 6, 7, 13, 14, 28, 29, 31 | stoweidlem7 43530 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) → ∃𝑛 ∈ ℕ ((1 −
𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) |
33 | 32 | ex 413 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1) → ∃𝑛 ∈ ℕ ((1 −
𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸))) |
34 | 33 | reximdva 3205 |
. . 3
⊢ (𝜑 → (∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1) → ∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ((1 −
𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸))) |
35 | 5, 34 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) |
36 | | stoweidlem49.1 |
. . . . 5
⊢
Ⅎ𝑡𝑃 |
37 | | stoweidlem49.2 |
. . . . . . 7
⊢
Ⅎ𝑡𝜑 |
38 | | nfv 1921 |
. . . . . . 7
⊢
Ⅎ𝑡(𝑘 ∈ ℕ ∧ 𝑛 ∈
ℕ) |
39 | 37, 38 | nfan 1906 |
. . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) |
40 | | nfv 1921 |
. . . . . 6
⊢
Ⅎ𝑡((1 −
𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸) |
41 | 39, 40 | nfan 1906 |
. . . . 5
⊢
Ⅎ𝑡((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) |
42 | | stoweidlem49.3 |
. . . . 5
⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} |
43 | | eqid 2740 |
. . . . 5
⊢ (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑛))↑(𝑘↑𝑛))) = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑛))↑(𝑘↑𝑛))) |
44 | | simplrr 775 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝑛 ∈ ℕ) |
45 | | simplrl 774 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝑘 ∈ ℕ) |
46 | 3 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝐷 ∈
ℝ+) |
47 | 4 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝐷 < 1) |
48 | | stoweidlem49.6 |
. . . . . 6
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
49 | 48 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝑃 ∈ 𝐴) |
50 | | stoweidlem49.7 |
. . . . . 6
⊢ (𝜑 → 𝑃:𝑇⟶ℝ) |
51 | 50 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝑃:𝑇⟶ℝ) |
52 | | stoweidlem49.8 |
. . . . . 6
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) |
53 | 52 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) |
54 | | stoweidlem49.9 |
. . . . . 6
⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) |
55 | 54 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) |
56 | | stoweidlem49.10 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
57 | 56 | ad4ant14 749 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
58 | | simp1ll 1235 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → 𝜑) |
59 | | stoweidlem49.11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
60 | 58, 59 | syld3an1 1409 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
61 | | stoweidlem49.12 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
62 | 58, 61 | syld3an1 1409 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
63 | | stoweidlem49.13 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
64 | 63 | ad4ant14 749 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
65 | 30 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → 𝐸 ∈
ℝ+) |
66 | | simprl 768 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → (1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛))) |
67 | | simprr 770 |
. . . . 5
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸) |
68 | 36, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 55, 57, 60, 62, 64, 65, 66, 67 | stoweidlem45 43568 |
. . . 4
⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸)) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) |
69 | 68 | ex 413 |
. . 3
⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))) |
70 | 69 | rexlimdvva 3225 |
. 2
⊢ (𝜑 → (∃𝑘 ∈ ℕ ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (((𝑘 · 𝐷) / 2)↑𝑛)) ∧ (1 / ((𝑘 · 𝐷)↑𝑛)) < 𝐸) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))) |
71 | 35, 70 | mpd 15 |
1
⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) |