| Step | Hyp | Ref
| Expression |
| 1 | | nllytop 23481 |
. . 3
⊢ (𝑅 ∈ 𝑛-Locally 𝐴 → 𝑅 ∈ Top) |
| 2 | | nllytop 23481 |
. . 3
⊢ (𝑆 ∈ 𝑛-Locally 𝐴 → 𝑆 ∈ Top) |
| 3 | | txtop 23577 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
| 4 | 1, 2, 3 | syl2an 596 |
. 2
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top) |
| 5 | | eltx 23576 |
. . . 4
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) |
| 6 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ 𝑛-Locally 𝐴) |
| 7 | | simprll 779 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢 ∈ 𝑅) |
| 8 | | simprrl 781 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣)) |
| 9 | | xp1st 8046 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝑢 × 𝑣) → (1st ‘𝑦) ∈ 𝑢) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (1st ‘𝑦) ∈ 𝑢) |
| 11 | | nlly2i 23484 |
. . . . . . . . 9
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑢 ∈ 𝑅 ∧ (1st ‘𝑦) ∈ 𝑢) → ∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴)) |
| 12 | 6, 7, 10, 11 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴)) |
| 13 | | simplr 769 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ 𝑛-Locally 𝐴) |
| 14 | | simprlr 780 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣 ∈ 𝑆) |
| 15 | | xp2nd 8047 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝑢 × 𝑣) → (2nd ‘𝑦) ∈ 𝑣) |
| 16 | 8, 15 | syl 17 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (2nd ‘𝑦) ∈ 𝑣) |
| 17 | | nlly2i 23484 |
. . . . . . . . 9
⊢ ((𝑆 ∈ 𝑛-Locally 𝐴 ∧ 𝑣 ∈ 𝑆 ∧ (2nd ‘𝑦) ∈ 𝑣) → ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) |
| 18 | 13, 14, 16, 17 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) |
| 19 | | reeanv 3229 |
. . . . . . . . 9
⊢
(∃𝑎 ∈
𝒫 𝑢∃𝑏 ∈ 𝒫 𝑣(∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) ↔ (∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) |
| 20 | | reeanv 3229 |
. . . . . . . . . . 11
⊢
(∃𝑟 ∈
𝑅 ∃𝑠 ∈ 𝑆 (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) ↔ (∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) |
| 21 | 4 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑅 ×t 𝑆) ∈ Top) |
| 22 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Top) |
| 23 | 22 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑅 ∈ Top) |
| 24 | 13, 2 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Top) |
| 25 | 24 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑆 ∈ Top) |
| 26 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑟 ∈ 𝑅) |
| 27 | 26 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑟 ∈ 𝑅) |
| 28 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑠 ∈ 𝑆) |
| 29 | 28 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑠 ∈ 𝑆) |
| 30 | | txopn 23610 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆)) |
| 31 | 23, 25, 27, 29, 30 | syl22anc 839 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆)) |
| 32 | 8 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑢 × 𝑣)) |
| 33 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
| 34 | 32, 33 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
| 35 | | simprl1 1219 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (1st ‘𝑦) ∈ 𝑟) |
| 36 | | simprr1 1222 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (2nd ‘𝑦) ∈ 𝑠) |
| 37 | 35, 36 | opelxpd 5724 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∈
(𝑟 × 𝑠)) |
| 38 | 34, 37 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑟 × 𝑠)) |
| 39 | | opnneip 23127 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆) ∧ 𝑦 ∈ (𝑟 × 𝑠)) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦})) |
| 40 | 21, 31, 38, 39 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦})) |
| 41 | | simprl2 1220 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑟 ⊆ 𝑎) |
| 42 | | simprr2 1223 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑠 ⊆ 𝑏) |
| 43 | | xpss12 5700 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑟 ⊆ 𝑎 ∧ 𝑠 ⊆ 𝑏) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏)) |
| 44 | 41, 42, 43 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏)) |
| 45 | | simprll 779 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑎 ∈ 𝒫 𝑢) |
| 46 | 45 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑎 ∈ 𝒫 𝑢) |
| 47 | 46 | elpwid 4609 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑎 ⊆ 𝑢) |
| 48 | 7 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑢 ∈ 𝑅) |
| 49 | | elssuni 4937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 ∈ 𝑅 → 𝑢 ⊆ ∪ 𝑅) |
| 50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑢 ⊆ ∪ 𝑅) |
| 51 | 47, 50 | sstrd 3994 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑎 ⊆ ∪ 𝑅) |
| 52 | | simprlr 780 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑏 ∈ 𝒫 𝑣) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑏 ∈ 𝒫 𝑣) |
| 54 | 53 | elpwid 4609 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑏 ⊆ 𝑣) |
| 55 | 14 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑣 ∈ 𝑆) |
| 56 | | elssuni 4937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ 𝑆 → 𝑣 ⊆ ∪ 𝑆) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑣 ⊆ ∪ 𝑆) |
| 58 | 54, 57 | sstrd 3994 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → 𝑏 ⊆ ∪ 𝑆) |
| 59 | | xpss12 5700 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ⊆ ∪ 𝑅
∧ 𝑏 ⊆ ∪ 𝑆)
→ (𝑎 × 𝑏) ⊆ (∪ 𝑅
× ∪ 𝑆)) |
| 60 | 51, 58, 59 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (∪ 𝑅 × ∪ 𝑆)) |
| 61 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑅 =
∪ 𝑅 |
| 62 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑆 =
∪ 𝑆 |
| 63 | 61, 62 | txuni 23600 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅
× ∪ 𝑆) = ∪ (𝑅 ×t 𝑆)) |
| 64 | 23, 25, 63 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (∪
𝑅 × ∪ 𝑆) =
∪ (𝑅 ×t 𝑆)) |
| 65 | 60, 64 | sseqtrd 4020 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ ∪ (𝑅 ×t 𝑆)) |
| 66 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ (𝑅
×t 𝑆) =
∪ (𝑅 ×t 𝑆) |
| 67 | 66 | ssnei2 23124 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ×t 𝑆) ∈ Top ∧ (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦})) ∧ ((𝑟 × 𝑠) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ ∪ (𝑅 ×t 𝑆))) → (𝑎 × 𝑏) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦})) |
| 68 | 21, 40, 44, 65, 67 | syl22anc 839 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦})) |
| 69 | | xpss12 5700 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ⊆ 𝑢 ∧ 𝑏 ⊆ 𝑣) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣)) |
| 70 | 47, 54, 69 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣)) |
| 71 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥) |
| 72 | 71 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑢 × 𝑣) ⊆ 𝑥) |
| 73 | 70, 72 | sstrd 3994 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ 𝑥) |
| 74 | | vex 3484 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑥 ∈ V |
| 75 | 74 | elpw2 5334 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑎 × 𝑏) ∈ 𝒫 𝑥 ↔ (𝑎 × 𝑏) ⊆ 𝑥) |
| 76 | 73, 75 | sylibr 234 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ 𝒫 𝑥) |
| 77 | 68, 76 | elind 4200 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)) |
| 78 | | txrest 23639 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣)) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) = ((𝑅 ↾t 𝑎) ×t (𝑆 ↾t 𝑏))) |
| 79 | 23, 25, 46, 53, 78 | syl22anc 839 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) = ((𝑅 ↾t 𝑎) ×t (𝑆 ↾t 𝑏))) |
| 80 | | simprl3 1221 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑅 ↾t 𝑎) ∈ 𝐴) |
| 81 | | simprr3 1224 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → (𝑆 ↾t 𝑏) ∈ 𝐴) |
| 82 | | txlly.1 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴) |
| 83 | 82 | caovcl 7627 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ↾t 𝑎) ∈ 𝐴 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴) → ((𝑅 ↾t 𝑎) ×t (𝑆 ↾t 𝑏)) ∈ 𝐴) |
| 84 | 80, 81, 83 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → ((𝑅 ↾t 𝑎) ×t (𝑆 ↾t 𝑏)) ∈ 𝐴) |
| 85 | 79, 84 | eqeltrd 2841 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴) |
| 86 | | oveq2 7439 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = (𝑎 × 𝑏) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏))) |
| 87 | 86 | eleq1d 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑎 × 𝑏) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴)) |
| 88 | 87 | rspcev 3622 |
. . . . . . . . . . . . . . 15
⊢ (((𝑎 × 𝑏) ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥) ∧ ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) |
| 89 | 77, 85, 88 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) ∧ (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴))) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) |
| 90 | 89 | ex 412 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 91 | 90 | anassrs 467 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally 𝐴 ∧
𝑆 ∈ 𝑛-Locally
𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 92 | 91 | rexlimdvva 3213 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣)) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 93 | 20, 92 | biimtrrid 243 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢 ∧ 𝑏 ∈ 𝒫 𝑣)) → ((∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 94 | 93 | rexlimdvva 3213 |
. . . . . . . . 9
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (∃𝑎 ∈ 𝒫 𝑢∃𝑏 ∈ 𝒫 𝑣(∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 95 | 19, 94 | biimtrrid 243 |
. . . . . . . 8
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 96 | 12, 18, 95 | mp2and 699 |
. . . . . . 7
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) |
| 97 | 96 | expr 456 |
. . . . . 6
⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ (𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 98 | 97 | rexlimdvva 3213 |
. . . . 5
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 99 | 98 | ralimdv 3169 |
. . . 4
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 100 | 5, 99 | sylbid 240 |
. . 3
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 101 | 100 | ralrimiv 3145 |
. 2
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) |
| 102 | | isnlly 23477 |
. 2
⊢ ((𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴 ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)) |
| 103 | 4, 101, 102 | sylanbrc 583 |
1
⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴) |