Step | Hyp | Ref
| Expression |
1 | | nnre 11723 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℝ) |
2 | 1 | adantl 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ) |
3 | | stoweidlem60.17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
4 | 3 | rpred 12514 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ∈ ℝ) |
5 | 4 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ) |
6 | 3 | rpne0d 12519 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸 ≠ 0) |
7 | 6 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ≠ 0) |
8 | 2, 5, 7 | redivcld 11546 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ) |
9 | | 1red 10720 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈
ℝ) |
10 | 8, 9 | readdcld 10748 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ) |
11 | 10 | adantr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ) |
12 | | arch 11973 |
. . . . . . . . 9
⊢ (((𝑚 / 𝐸) + 1) ∈ ℝ → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛) |
13 | 11, 12 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛) |
14 | | stoweidlem60.2 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡𝜑 |
15 | | nfv 1921 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡 𝑚 ∈ ℕ |
16 | 14, 15 | nfan 1906 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ) |
17 | | nfra1 3131 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚 |
18 | 16, 17 | nfan 1906 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) |
19 | | nfv 1921 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡 𝑛 ∈ ℕ |
20 | 18, 19 | nfan 1906 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡(((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) |
21 | | nfv 1921 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡((𝑚 / 𝐸) + 1) < 𝑛 |
22 | 20, 21 | nfan 1906 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) |
23 | | simp-5l 785 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝜑) |
24 | | stoweidlem60.3 |
. . . . . . . . . . . . . . . 16
⊢ 𝐾 = (topGen‘ran
(,)) |
25 | | stoweidlem60.4 |
. . . . . . . . . . . . . . . 16
⊢ 𝑇 = ∪
𝐽 |
26 | | stoweidlem60.5 |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (𝐽 Cn 𝐾) |
27 | | stoweidlem60.15 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹 ∈ 𝐶) |
28 | 24, 25, 26, 27 | fcnre 42106 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
29 | 28 | ffvelrnda 6861 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
30 | 23, 29 | sylancom 591 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
31 | | simp-5r 786 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ) |
32 | 31 | nnred 11731 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℝ) |
33 | | simpllr 776 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℕ) |
34 | 33 | nnred 11731 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℝ) |
35 | | 1red 10720 |
. . . . . . . . . . . . . . 15
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
36 | 34, 35 | resubcld 11146 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝑛 − 1) ∈ ℝ) |
37 | 23, 4 | syl 17 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ) |
38 | 36, 37 | remulcld 10749 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ) |
39 | | simpllr 776 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) |
40 | 39 | r19.21bi 3121 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < 𝑚) |
41 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛) |
42 | | simpr 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛) |
43 | | simpl1 1192 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑) |
44 | | simpl2 1193 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ) |
45 | 43, 44, 8 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ) |
46 | | 1red 10720 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ) |
47 | | simpl3 1194 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ) |
48 | 47 | nnred 11731 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ) |
49 | 45, 46, 48 | ltaddsubd 11318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1))) |
50 | 42, 49 | mpbid 235 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1)) |
51 | 1 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ) |
52 | 51 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ) |
53 | 48, 46 | resubcld 11146 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ) |
54 | 4 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ) |
55 | 54 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ) |
56 | 3 | rpgt0d 12517 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 0 < 𝐸) |
57 | 43, 56 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸) |
58 | | ltdivmul2 11595 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧
(𝐸 ∈ ℝ ∧ 0
< 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸))) |
59 | 52, 53, 55, 57, 58 | syl112anc 1375 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸))) |
60 | 50, 59 | mpbid 235 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸)) |
61 | 23, 31, 33, 41, 60 | syl31anc 1374 |
. . . . . . . . . . . . 13
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸)) |
62 | 30, 32, 38, 40, 61 | lttrd 10879 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
63 | 62 | ex 416 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑡 ∈ 𝑇 → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
64 | 22, 63 | ralrimi 3128 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑚 ∈ ℕ) ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
65 | 64 | ex 416 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) → (((𝑚 / 𝐸) + 1) < 𝑛 → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
66 | 65 | reximdva 3184 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → (∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
67 | 13, 66 | mpd 15 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
68 | | stoweidlem60.1 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐹 |
69 | | stoweidlem60.8 |
. . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ Comp) |
70 | | stoweidlem60.9 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ≠ ∅) |
71 | 68, 14, 24, 69, 25, 70, 26, 27 | rfcnnnub 42117 |
. . . . . . 7
⊢ (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) |
72 | 67, 71 | r19.29a 3199 |
. . . . . 6
⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
73 | | df-rex 3059 |
. . . . . 6
⊢
(∃𝑛 ∈
ℕ ∀𝑡 ∈
𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
74 | 72, 73 | sylib 221 |
. . . . 5
⊢ (𝜑 → ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
75 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) |
76 | 14, 19 | nfan 1906 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ) |
77 | | stoweidlem60.6 |
. . . . . . . . . . 11
⊢ 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) |
78 | | stoweidlem60.7 |
. . . . . . . . . . 11
⊢ 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) |
79 | | eqid 2738 |
. . . . . . . . . . 11
⊢ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} |
80 | | eqid 2738 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷‘𝑗)(𝑦‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < (𝑦‘𝑡))}) = (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷‘𝑗)(𝑦‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < (𝑦‘𝑡))}) |
81 | 69 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Comp) |
82 | | stoweidlem60.10 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
83 | 82 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ 𝐶) |
84 | | stoweidlem60.11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
85 | 84 | 3adant1r 1178 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
86 | | stoweidlem60.12 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
87 | 86 | 3adant1r 1178 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
88 | | stoweidlem60.13 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) |
89 | 88 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) |
90 | | stoweidlem60.14 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
91 | 90 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) |
92 | 27 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ 𝐶) |
93 | 3 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈
ℝ+) |
94 | | stoweidlem60.18 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸 < (1 / 3)) |
95 | 94 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 < (1 / 3)) |
96 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
97 | 68, 76, 24, 25, 26, 77, 78, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 96 | stoweidlem59 43142 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
98 | 97 | adantrr 717 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
99 | | 19.42v 1961 |
. . . . . . . . 9
⊢
(∃𝑥((𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
100 | 75, 98, 99 | sylanbrc 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
101 | | 3anass 1096 |
. . . . . . . . 9
⊢ (((𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
102 | 101 | exbii 1854 |
. . . . . . . 8
⊢
(∃𝑥((𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) ↔ ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
103 | 100, 102 | sylibr 237 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
104 | 103 | ex 416 |
. . . . . 6
⊢ (𝜑 → ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
105 | 104 | eximdv 1924 |
. . . . 5
⊢ (𝜑 → (∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))) |
106 | 74, 105 | mpd 15 |
. . . 4
⊢ (𝜑 → ∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
107 | | simpl 486 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝜑) |
108 | | simpr1l 1231 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑛 ∈ ℕ) |
109 | | simpr2 1196 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑥:(0...𝑛)⟶𝐴) |
110 | | nfv 1921 |
. . . . . . . . . 10
⊢
Ⅎ𝑡 𝑥:(0...𝑛)⟶𝐴 |
111 | 14, 19, 110 | nf3an 1908 |
. . . . . . . . 9
⊢
Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) |
112 | | simp2 1138 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ) |
113 | | simp3 1139 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴) |
114 | | simp1 1137 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑) |
115 | 114, 84 | syl3an1 1164 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
116 | 114, 86 | syl3an1 1164 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
117 | 88 | 3ad2antl1 1186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) |
118 | 3 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈
ℝ+) |
119 | 118 | rpred 12514 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ) |
120 | 82 | sselda 3877 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶) |
121 | 24, 25, 26, 120 | fcnre 42106 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
122 | 121 | 3ad2antl1 1186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
123 | 111, 112,
113, 115, 116, 117, 119, 122 | stoweidlem17 43100 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴) |
124 | 107, 108,
109, 123 | syl3anc 1372 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴) |
125 | | nfv 1921 |
. . . . . . . . 9
⊢
Ⅎ𝑗𝜑 |
126 | | nfv 1921 |
. . . . . . . . . 10
⊢
Ⅎ𝑗(𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
127 | | nfv 1921 |
. . . . . . . . . 10
⊢
Ⅎ𝑗 𝑥:(0...𝑛)⟶𝐴 |
128 | | nfra1 3131 |
. . . . . . . . . 10
⊢
Ⅎ𝑗∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
129 | 126, 127,
128 | nf3an 1908 |
. . . . . . . . 9
⊢
Ⅎ𝑗((𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
130 | 125, 129 | nfan 1906 |
. . . . . . . 8
⊢
Ⅎ𝑗(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
131 | | nfra1 3131 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸) |
132 | 19, 131 | nfan 1906 |
. . . . . . . . . 10
⊢
Ⅎ𝑡(𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
133 | | nfcv 2899 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(0...𝑛) |
134 | | nfra1 3131 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) |
135 | | nfra1 3131 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) |
136 | | nfra1 3131 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡) |
137 | 134, 135,
136 | nf3an 1908 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
138 | 133, 137 | nfralw 3138 |
. . . . . . . . . 10
⊢
Ⅎ𝑡∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
139 | 132, 110,
138 | nf3an 1908 |
. . . . . . . . 9
⊢
Ⅎ𝑡((𝑛 ∈ ℕ ∧
∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
140 | 14, 139 | nfan 1906 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) |
141 | | eqid 2738 |
. . . . . . . 8
⊢ (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) |
142 | 69 | uniexd 7486 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝐽
∈ V) |
143 | 25, 142 | eqeltrid 2837 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ V) |
144 | 143 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑇 ∈ V) |
145 | 28 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐹:𝑇⟶ℝ) |
146 | | stoweidlem60.16 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡)) |
147 | 146 | r19.21bi 3121 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡)) |
148 | 147 | adantlr 715 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡)) |
149 | | simpr1r 1232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
150 | 149 | r19.21bi 3121 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) |
151 | 3 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐸 ∈
ℝ+) |
152 | 94 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐸 < (1 / 3)) |
153 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝜑) |
154 | | simplr2 1217 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑥:(0...𝑛)⟶𝐴) |
155 | | simpr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ (0...𝑛)) |
156 | | simp1 1137 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → 𝜑) |
157 | | ffvelrn 6859 |
. . . . . . . . . . 11
⊢ ((𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴) |
158 | 157 | 3adant1 1131 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴) |
159 | 82 | sselda 3877 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗) ∈ 𝐶) |
160 | 24, 25, 26, 159 | fcnre 42106 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗):𝑇⟶ℝ) |
161 | 156, 158,
160 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ) |
162 | 153, 154,
155, 161 | syl3anc 1372 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ) |
163 | | simp1r3 1272 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
164 | | r19.26-3 3086 |
. . . . . . . . . . 11
⊢
(∀𝑗 ∈
(0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) ↔ (∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
165 | 164 | simp1bi 1146 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1)) |
166 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((0 ≤
((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
167 | 166 | 2ralimi 3076 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
168 | 163, 165,
167 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
169 | | simp2 1138 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑗 ∈ (0...𝑛)) |
170 | | simp3 1139 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
171 | | rspa 3119 |
. . . . . . . . . 10
⊢
((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
172 | 171 | r19.21bi 3121 |
. . . . . . . . 9
⊢
(((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
173 | 168, 169,
170, 172 | syl21anc 837 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑥‘𝑗)‘𝑡)) |
174 | | simpr 488 |
. . . . . . . . . . 11
⊢ ((0 ≤
((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ((𝑥‘𝑗)‘𝑡) ≤ 1) |
175 | 174 | 2ralimi 3076 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1) |
176 | 163, 165,
175 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1) |
177 | | rspa 3119 |
. . . . . . . . . 10
⊢
((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1) |
178 | 177 | r19.21bi 3121 |
. . . . . . . . 9
⊢
(((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ 𝑇) → ((𝑥‘𝑗)‘𝑡) ≤ 1) |
179 | 176, 169,
170, 178 | syl21anc 837 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑥‘𝑗)‘𝑡) ≤ 1) |
180 | | simp1r3 1272 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
181 | 164 | simp2bi 1147 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)) |
182 | 180, 181 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)) |
183 | | simp2 1138 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → 𝑗 ∈ (0...𝑛)) |
184 | | simp3 1139 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → 𝑡 ∈ (𝐷‘𝑗)) |
185 | | rspa 3119 |
. . . . . . . . . 10
⊢
((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)) |
186 | 185 | r19.21bi 3121 |
. . . . . . . . 9
⊢
(((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)) |
187 | 182, 183,
184, 186 | syl21anc 837 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)) |
188 | | simp1r3 1272 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) |
189 | 164 | simp3bi 1148 |
. . . . . . . . . 10
⊢
(∀𝑗 ∈
(0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
190 | 188, 189 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
191 | | simp2 1138 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → 𝑗 ∈ (0...𝑛)) |
192 | | simp3 1139 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → 𝑡 ∈ (𝐵‘𝑗)) |
193 | | rspa 3119 |
. . . . . . . . . 10
⊢
((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
194 | 193 | r19.21bi 3121 |
. . . . . . . . 9
⊢
(((∀𝑗 ∈
(0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐵‘𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
195 | 190, 191,
192, 194 | syl21anc 837 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) |
196 | 68, 130, 140, 77, 78, 141, 108, 144, 145, 148, 150, 151, 152, 162, 173, 179, 187, 195 | stoweidlem34 43117 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))) |
197 | | nfmpt1 5128 |
. . . . . . . . . 10
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) |
198 | 197 | nfeq2 2916 |
. . . . . . . . 9
⊢
Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) |
199 | | fveq1 6673 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)) |
200 | 199 | breq1d 5040 |
. . . . . . . . . . . 12
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ↔ ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸))) |
201 | 199 | breq2d 5042 |
. . . . . . . . . . . 12
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡) ↔ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))) |
202 | 200, 201 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)) ↔ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))) |
203 | 202 | anbi2d 632 |
. . . . . . . . . 10
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))))) |
204 | 203 | rexbidv 3207 |
. . . . . . . . 9
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))))) |
205 | 198, 204 | ralbid 3145 |
. . . . . . . 8
⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))))) |
206 | 205 | rspcev 3526 |
. . . . . . 7
⊢ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |
207 | 124, 196,
206 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |
208 | 207 | ex 416 |
. . . . 5
⊢ (𝜑 → (((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
209 | 208 | 2eximdv 1926 |
. . . 4
⊢ (𝜑 → (∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) → ∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
210 | 106, 209 | mpd 15 |
. . 3
⊢ (𝜑 → ∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |
211 | | idd 24 |
. . . 4
⊢ (𝜑 → (∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
212 | 211 | exlimdv 1940 |
. . 3
⊢ (𝜑 → (∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
213 | 210, 212 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |
214 | | idd 24 |
. . 3
⊢ (𝜑 → (∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
215 | 214 | exlimdv 1940 |
. 2
⊢ (𝜑 → (∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))) |
216 | 213, 215 | mpd 15 |
1
⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) |