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Theorem txnlly 21949
Description: If the property 𝐴 is preserved under topological products, then so is the property of being n-locally 𝐴. (Contributed by Mario Carneiro, 13-Apr-2015.)
Hypothesis
Ref Expression
txlly.1 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
Assertion
Ref Expression
txnlly ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴)
Distinct variable groups:   𝑗,𝑘,𝐴   𝑅,𝑗,𝑘   𝑆,𝑘
Allowed substitution hint:   𝑆(𝑗)

Proof of Theorem txnlly
Dummy variables 𝑎 𝑏 𝑟 𝑠 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 21785 . . 3 (𝑅 ∈ 𝑛-Locally 𝐴𝑅 ∈ Top)
2 nllytop 21785 . . 3 (𝑆 ∈ 𝑛-Locally 𝐴𝑆 ∈ Top)
3 txtop 21881 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 586 . 2 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5 eltx 21880 . . . 4 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6 simpll 754 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ 𝑛-Locally 𝐴)
7 simprll 766 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢𝑅)
8 simprrl 768 . . . . . . . . . 10 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9 xp1st 7533 . . . . . . . . . 10 (𝑦 ∈ (𝑢 × 𝑣) → (1st𝑦) ∈ 𝑢)
108, 9syl 17 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (1st𝑦) ∈ 𝑢)
11 nlly2i 21788 . . . . . . . . 9 ((𝑅 ∈ 𝑛-Locally 𝐴𝑢𝑅 ∧ (1st𝑦) ∈ 𝑢) → ∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴))
126, 7, 10, 11syl3anc 1351 . . . . . . . 8 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴))
13 simplr 756 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ 𝑛-Locally 𝐴)
14 simprlr 767 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣𝑆)
15 xp2nd 7534 . . . . . . . . . 10 (𝑦 ∈ (𝑢 × 𝑣) → (2nd𝑦) ∈ 𝑣)
168, 15syl 17 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (2nd𝑦) ∈ 𝑣)
17 nlly2i 21788 . . . . . . . . 9 ((𝑆 ∈ 𝑛-Locally 𝐴𝑣𝑆 ∧ (2nd𝑦) ∈ 𝑣) → ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))
1813, 14, 16, 17syl3anc 1351 . . . . . . . 8 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))
19 reeanv 3309 . . . . . . . . 9 (∃𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣(∃𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) ↔ (∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)))
20 reeanv 3309 . . . . . . . . . . 11 (∃𝑟𝑅𝑠𝑆 (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) ↔ (∃𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)))
214ad3antrrr 717 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑅 ×t 𝑆) ∈ Top)
221ad2antrr 713 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Top)
2322ad2antrr 713 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑅 ∈ Top)
2413, 2syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Top)
2524ad2antrr 713 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑆 ∈ Top)
26 simprrl 768 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑟𝑅)
2726adantr 473 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑟𝑅)
28 simprrr 769 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑠𝑆)
2928adantr 473 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑠𝑆)
30 txopn 21914 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
3123, 25, 27, 29, 30syl22anc 826 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
328ad2antrr 713 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑢 × 𝑣))
33 1st2nd2 7540 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3432, 33syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
35 simprl1 1198 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (1st𝑦) ∈ 𝑟)
36 simprr1 1201 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (2nd𝑦) ∈ 𝑠)
3735, 36opelxpd 5445 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
3834, 37eqeltrd 2867 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑟 × 𝑠))
39 opnneip 21431 . . . . . . . . . . . . . . . . . 18 (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆) ∧ 𝑦 ∈ (𝑟 × 𝑠)) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦}))
4021, 31, 38, 39syl3anc 1351 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦}))
41 simprl2 1199 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑟𝑎)
42 simprr2 1202 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑠𝑏)
43 xpss12 5422 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑎𝑠𝑏) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
4441, 42, 43syl2anc 576 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
45 simprll 766 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑎 ∈ 𝒫 𝑢)
4645adantr 473 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑎 ∈ 𝒫 𝑢)
4746elpwid 4434 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑎𝑢)
487ad2antrr 713 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑢𝑅)
49 elssuni 4741 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑅𝑢 𝑅)
5048, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑢 𝑅)
5147, 50sstrd 3869 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑎 𝑅)
52 simprlr 767 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑏 ∈ 𝒫 𝑣)
5352adantr 473 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑏 ∈ 𝒫 𝑣)
5453elpwid 4434 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑏𝑣)
5514ad2antrr 713 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑣𝑆)
56 elssuni 4741 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑆𝑣 𝑆)
5755, 56syl 17 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑣 𝑆)
5854, 57sstrd 3869 . . . . . . . . . . . . . . . . . . 19 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → 𝑏 𝑆)
59 xpss12 5422 . . . . . . . . . . . . . . . . . . 19 ((𝑎 𝑅𝑏 𝑆) → (𝑎 × 𝑏) ⊆ ( 𝑅 × 𝑆))
6051, 58, 59syl2anc 576 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ ( 𝑅 × 𝑆))
61 eqid 2779 . . . . . . . . . . . . . . . . . . . 20 𝑅 = 𝑅
62 eqid 2779 . . . . . . . . . . . . . . . . . . . 20 𝑆 = 𝑆
6361, 62txuni 21904 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6423, 25, 63syl2anc 576 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6560, 64sseqtrd 3898 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (𝑅 ×t 𝑆))
66 eqid 2779 . . . . . . . . . . . . . . . . . 18 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6766ssnei2 21428 . . . . . . . . . . . . . . . . 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}))
6821, 40, 44, 65, 67syl22anc 826 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦}))
69 xpss12 5422 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑢𝑏𝑣) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
7047, 54, 69syl2anc 576 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
71 simprrr 769 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
7271ad2antrr 713 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑢 × 𝑣) ⊆ 𝑥)
7370, 72sstrd 3869 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ 𝑥)
74 vex 3419 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
7574elpw2 5104 . . . . . . . . . . . . . . . . 17 ((𝑎 × 𝑏) ∈ 𝒫 𝑥 ↔ (𝑎 × 𝑏) ⊆ 𝑥)
7673, 75sylibr 226 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ 𝒫 𝑥)
7768, 76elind 4060 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥))
78 txrest 21943 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣)) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) = ((𝑅t 𝑎) ×t (𝑆t 𝑏)))
7923, 25, 46, 53, 78syl22anc 826 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) = ((𝑅t 𝑎) ×t (𝑆t 𝑏)))
80 simprl3 1200 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑅t 𝑎) ∈ 𝐴)
81 simprr3 1203 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → (𝑆t 𝑏) ∈ 𝐴)
82 txlly.1 . . . . . . . . . . . . . . . . . 18 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
8382caovcl 7158 . . . . . . . . . . . . . . . . 17 (((𝑅t 𝑎) ∈ 𝐴 ∧ (𝑆t 𝑏) ∈ 𝐴) → ((𝑅t 𝑎) ×t (𝑆t 𝑏)) ∈ 𝐴)
8480, 81, 83syl2anc 576 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ((𝑅t 𝑎) ×t (𝑆t 𝑏)) ∈ 𝐴)
8579, 84eqeltrd 2867 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴)
86 oveq2 6984 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎 × 𝑏) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)))
8786eleq1d 2851 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎 × 𝑏) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴))
8887rspcev 3536 . . . . . . . . . . . . . . 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)
8977, 85, 88syl2anc 576 . . . . . . . . . . . . . 14 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) ∧ (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)
9089ex 405 . . . . . . . . . . . . 13 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → ((((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9190anassrs 460 . . . . . . . . . . . 12 (((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣)) ∧ (𝑟𝑅𝑠𝑆)) → ((((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9291rexlimdvva 3240 . . . . . . . . . . 11 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣)) → (∃𝑟𝑅𝑠𝑆 (((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9320, 92syl5bir 235 . . . . . . . . . 10 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣)) → ((∃𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9493rexlimdvva 3240 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (∃𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣(∃𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9519, 94syl5bir 235 . . . . . . . 8 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴)) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9612, 18, 95mp2and 686 . . . . . . 7 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)
9796expr 449 . . . . . 6 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ (𝑢𝑅𝑣𝑆)) → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9897rexlimdvva 3240 . . . . 5 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (∃𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
9998ralimdv 3129 . . . 4 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∀𝑦𝑥𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
1005, 99sylbid 232 . . 3 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦𝑥𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
101100ralrimiv 3132 . 2 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦𝑥𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)
102 isnlly 21781 . 2 ((𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴 ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦𝑥𝑧 ∈ (((nei‘(𝑅 ×t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥)((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
1034, 101, 102sylanbrc 575 1 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3089  wrex 3090  cin 3829  wss 3830  𝒫 cpw 4422  {csn 4441  cop 4447   cuni 4712   × cxp 5405  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  t crest 16550  Topctop 21205  neicnei 21409  𝑛-Locally cnlly 21777   ×t ctx 21872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-rest 16552  df-topgen 16573  df-top 21206  df-topon 21223  df-bases 21258  df-nei 21410  df-nlly 21779  df-tx 21874
This theorem is referenced by:  xkohmeo  22127  cvmlift2lem13  32144
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