Step | Hyp | Ref
| Expression |
1 | | nllytop 21496 |
. . . . . 6
⊢ (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Top) |
2 | 1 | adantl 467 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Top) |
3 | | nlly2i 21499 |
. . . . . . . . 9
⊢ ((𝑘 ∈ 𝑛-Locally 𝐴 ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴)) |
4 | 3 | 3adant1l 1185 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴)) |
5 | | simprl 754 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝑘) |
6 | | simprr2 1274 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ⊆ 𝑠) |
7 | | simplr 752 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ∈ 𝒫 𝑦) |
8 | 7 | elpwid 4310 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑦) |
9 | 6, 8 | sstrd 3762 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ⊆ 𝑦) |
10 | | selpw 4305 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝒫 𝑦 ↔ 𝑥 ⊆ 𝑦) |
11 | 9, 10 | sylibr 224 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ 𝒫 𝑦) |
12 | 5, 11 | elind 3949 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)) |
13 | | simprr1 1272 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑢 ∈ 𝑥) |
14 | | simpll1 1254 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴)) |
15 | 14 | simprd 483 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑘 ∈ 𝑛-Locally 𝐴) |
16 | 15, 1 | syl 17 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑘 ∈ Top) |
17 | | restabs 21189 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ Top ∧ 𝑥 ⊆ 𝑠 ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑘 ↾t 𝑠) ↾t 𝑥) = (𝑘 ↾t 𝑥)) |
18 | 16, 6, 7, 17 | syl3anc 1476 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ((𝑘 ↾t 𝑠) ↾t 𝑥) = (𝑘 ↾t 𝑥)) |
19 | | df-ss 3737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ⊆ 𝑠 ↔ (𝑥 ∩ 𝑠) = 𝑥) |
20 | 6, 19 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∩ 𝑠) = 𝑥) |
21 | | elrestr 16296 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ Top ∧ 𝑠 ∈ 𝒫 𝑦 ∧ 𝑥 ∈ 𝑘) → (𝑥 ∩ 𝑠) ∈ (𝑘 ↾t 𝑠)) |
22 | 16, 7, 5, 21 | syl3anc 1476 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∩ 𝑠) ∈ (𝑘 ↾t 𝑠)) |
23 | 20, 22 | eqeltrrd 2851 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝑥 ∈ (𝑘 ↾t 𝑠)) |
24 | | eleq2 2839 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑥 ∈ 𝑗 ↔ 𝑥 ∈ (𝑘 ↾t 𝑠))) |
25 | | oveq1 6802 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑘 ↾t 𝑠) → (𝑗 ↾t 𝑥) = ((𝑘 ↾t 𝑠) ↾t 𝑥)) |
26 | 25 | eleq1d 2835 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑗 ↾t 𝑥) ∈ 𝐴 ↔ ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴)) |
27 | 24, 26 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑘 ↾t 𝑠) → ((𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴) ↔ (𝑥 ∈ (𝑘 ↾t 𝑠) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴))) |
28 | 14 | simpld 482 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → 𝜑) |
29 | | restlly.1 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴) |
30 | 29 | expr 444 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴)) |
31 | 30 | ralrimiva 3115 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴)) |
32 | 28, 31 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ∀𝑗 ∈ 𝐴 (𝑥 ∈ 𝑗 → (𝑗 ↾t 𝑥) ∈ 𝐴)) |
33 | | simprr3 1276 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑘 ↾t 𝑠) ∈ 𝐴) |
34 | 27, 32, 33 | rspcdva 3466 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ↾t 𝑠) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴)) |
35 | 23, 34 | mpd 15 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → ((𝑘 ↾t 𝑠) ↾t 𝑥) ∈ 𝐴) |
36 | 18, 35 | eqeltrrd 2851 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑘 ↾t 𝑥) ∈ 𝐴) |
37 | 12, 13, 36 | jca32 505 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) ∧ (𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴))) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))) |
38 | 37 | ex 397 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → ((𝑥 ∈ 𝑘 ∧ (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴)) → (𝑥 ∈ (𝑘 ∩ 𝒫 𝑦) ∧ (𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)))) |
39 | 38 | reximdv2 3162 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) ∧ 𝑠 ∈ 𝒫 𝑦) → (∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))) |
40 | 39 | rexlimdva 3179 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → (∃𝑠 ∈ 𝒫 𝑦∃𝑥 ∈ 𝑘 (𝑢 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑠 ∧ (𝑘 ↾t 𝑠) ∈ 𝐴) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))) |
41 | 4, 40 | mpd 15 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ 𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)) |
42 | 41 | 3expb 1113 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) ∧ (𝑦 ∈ 𝑘 ∧ 𝑢 ∈ 𝑦)) → ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)) |
43 | 42 | ralrimivva 3120 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → ∀𝑦 ∈ 𝑘 ∀𝑢 ∈ 𝑦 ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴)) |
44 | | islly 21491 |
. . . . 5
⊢ (𝑘 ∈ Locally 𝐴 ↔ (𝑘 ∈ Top ∧ ∀𝑦 ∈ 𝑘 ∀𝑢 ∈ 𝑦 ∃𝑥 ∈ (𝑘 ∩ 𝒫 𝑦)(𝑢 ∈ 𝑥 ∧ (𝑘 ↾t 𝑥) ∈ 𝐴))) |
45 | 2, 43, 44 | sylanbrc 572 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑛-Locally 𝐴) → 𝑘 ∈ Locally 𝐴) |
46 | 45 | ex 397 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝑛-Locally 𝐴 → 𝑘 ∈ Locally 𝐴)) |
47 | 46 | ssrdv 3758 |
. 2
⊢ (𝜑 → 𝑛-Locally 𝐴 ⊆ Locally 𝐴) |
48 | | llyssnlly 21501 |
. . 3
⊢ Locally
𝐴 ⊆ 𝑛-Locally
𝐴 |
49 | 48 | a1i 11 |
. 2
⊢ (𝜑 → Locally 𝐴 ⊆ 𝑛-Locally 𝐴) |
50 | 47, 49 | eqssd 3769 |
1
⊢ (𝜑 → 𝑛-Locally 𝐴 = Locally 𝐴) |