Step | Hyp | Ref
| Expression |
1 | | llytop 21085 |
. . 3
⊢ (𝑅 ∈ Locally 𝐴 → 𝑅 ∈ Top) |
2 | | llytop 21085 |
. . 3
⊢ (𝑆 ∈ Locally 𝐴 → 𝑆 ∈ Top) |
3 | | txtop 21182 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
4 | 1, 2, 3 | syl2an 493 |
. 2
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top) |
5 | | eltx 21181 |
. . . 4
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) |
6 | | simpll 786 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Locally 𝐴) |
7 | | simprll 798 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢 ∈ 𝑅) |
8 | | simprrl 800 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣)) |
9 | | xp1st 7089 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝑢 × 𝑣) → (1st ‘𝑦) ∈ 𝑢) |
10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (1st ‘𝑦) ∈ 𝑢) |
11 | | llyi 21087 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑢 ∈ 𝑅 ∧ (1st ‘𝑦) ∈ 𝑢) → ∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴)) |
12 | 6, 7, 10, 11 | syl3anc 1318 |
. . . . . . . 8
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴)) |
13 | | simplr 788 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Locally 𝐴) |
14 | | simprlr 799 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣 ∈ 𝑆) |
15 | | xp2nd 7090 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝑢 × 𝑣) → (2nd ‘𝑦) ∈ 𝑣) |
16 | 8, 15 | syl 17 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (2nd ‘𝑦) ∈ 𝑣) |
17 | | llyi 21087 |
. . . . . . . . 9
⊢ ((𝑆 ∈ Locally 𝐴 ∧ 𝑣 ∈ 𝑆 ∧ (2nd ‘𝑦) ∈ 𝑣) → ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) |
18 | 13, 14, 16, 17 | syl3anc 1318 |
. . . . . . . 8
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) |
19 | | reeanv 3086 |
. . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑠 ∈ 𝑆 ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) ↔ (∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) |
20 | 1 | ad3antrrr 762 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑅 ∈ Top) |
21 | 2 | ad3antlr 763 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑆 ∈ Top) |
22 | | simprll 798 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑟 ∈ 𝑅) |
23 | | simprlr 799 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑠 ∈ 𝑆) |
24 | | txopn 21215 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆)) |
25 | 20, 21, 22, 23, 24 | syl22anc 1319 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆)) |
26 | | simprl1 1099 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → 𝑟 ⊆ 𝑢) |
27 | | simprr1 1102 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑣) |
28 | | xpss12 5148 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ⊆ 𝑢 ∧ 𝑠 ⊆ 𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣)) |
29 | 26, 27, 28 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣)) |
30 | | simprrr 801 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥) |
31 | 29, 30 | sylan9ssr 3582 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ⊆ 𝑥) |
32 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
33 | 32 | elpw2 4755 |
. . . . . . . . . . . . . 14
⊢ ((𝑟 × 𝑠) ∈ 𝒫 𝑥 ↔ (𝑟 × 𝑠) ⊆ 𝑥) |
34 | 31, 33 | sylibr 223 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ 𝒫 𝑥) |
35 | 25, 34 | elind 3760 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)) |
36 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
37 | 8, 36 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
38 | 37 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
39 | | simprl2 1100 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (1st ‘𝑦) ∈ 𝑟) |
40 | | simprr2 1103 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (2nd ‘𝑦) ∈ 𝑠) |
41 | | opelxpi 5072 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑦) ∈ 𝑟 ∧ (2nd ‘𝑦) ∈ 𝑠) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∈ (𝑟 × 𝑠)) |
42 | 39, 40, 41 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∈
(𝑟 × 𝑠)) |
43 | 42 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∈
(𝑟 × 𝑠)) |
44 | 38, 43 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑦 ∈ (𝑟 × 𝑠)) |
45 | | txrest 21244 |
. . . . . . . . . . . . . 14
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠))) |
46 | 20, 21, 22, 23, 45 | syl22anc 1319 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠))) |
47 | | simprl3 1101 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑅 ↾t 𝑟) ∈ 𝐴) |
48 | | simprr3 1104 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑆 ↾t 𝑠) ∈ 𝐴) |
49 | | txlly.1 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴) |
50 | 49 | caovcl 6726 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ↾t 𝑟) ∈ 𝐴 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴) |
51 | 47, 48, 50 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴) |
52 | 51 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴) |
53 | 46, 52 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴) |
54 | | eleq2 2677 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑟 × 𝑠) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑟 × 𝑠))) |
55 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠))) |
56 | 55 | eleq1d 2672 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)) |
57 | 54, 56 | anbi12d 743 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) ↔ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))) |
58 | 57 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ (((𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
59 | 35, 44, 53, 58 | syl12anc 1316 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
60 | 59 | expr 641 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
61 | 60 | rexlimdvva 3020 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
62 | 19, 61 | syl5bir 232 |
. . . . . . . 8
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
63 | 12, 18, 62 | mp2and 711 |
. . . . . . 7
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
64 | 63 | expr 641 |
. . . . . 6
⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ (𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
65 | 64 | rexlimdvva 3020 |
. . . . 5
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
66 | 65 | ralimdv 2946 |
. . . 4
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
67 | 5, 66 | sylbid 229 |
. . 3
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
68 | 67 | ralrimiv 2948 |
. 2
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
69 | | islly 21081 |
. 2
⊢ ((𝑅 ×t 𝑆) ∈ Locally 𝐴 ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))) |
70 | 4, 68, 69 | sylanbrc 695 |
1
⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴) |