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Theorem txnlly 21345
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 21181 . . 3 (𝑅 ∈ 𝑛-Locally 𝐴𝑅 ∈ Top)
2 nllytop 21181 . . 3 (𝑆 ∈ 𝑛-Locally 𝐴𝑆 ∈ Top)
3 txtop 21277 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 494 . 2 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5 eltx 21276 . . . 4 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6 simpll 789 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ 𝑛-Locally 𝐴)
7 simprll 801 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢𝑅)
8 simprrl 803 . . . . . . . . . 10 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9 xp1st 7146 . . . . . . . . . 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 21184 . . . . . . . . 9 ((𝑅 ∈ 𝑛-Locally 𝐴𝑢𝑅 ∧ (1st𝑦) ∈ 𝑢) → ∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴))
126, 7, 10, 11syl3anc 1323 . . . . . . . 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 791 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ 𝑛-Locally 𝐴)
14 simprlr 802 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣𝑆)
15 xp2nd 7147 . . . . . . . . . 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 21184 . . . . . . . . 9 ((𝑆 ∈ 𝑛-Locally 𝐴𝑣𝑆 ∧ (2nd𝑦) ∈ 𝑣) → ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))
1813, 14, 16, 17syl3anc 1323 . . . . . . . 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 3102 . . . . . . . . 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 3102 . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Top)
2322ad2antrr 761 . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . 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 803 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑟𝑅)
2726adantr 481 . . . . . . . . . . . . . . . . . . 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 804 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑠𝑆)
2928adantr 481 . . . . . . . . . . . . . . . . . . 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 21310 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
3123, 25, 27, 29, 30syl22anc 1324 . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . . . . . 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 1104 . . . . . . . . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . . . . . . . . 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)
37 opelxpi 5113 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑦) ∈ 𝑟 ∧ (2nd𝑦) ∈ 𝑠) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
3835, 36, 37syl2anc 692 . . . . . . . . . . . . . . . . . . 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))
3934, 38eqeltrd 2704 . . . . . . . . . . . . . . . . . 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))
40 opnneip 20828 . . . . . . . . . . . . . . . . . 18 (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆) ∧ 𝑦 ∈ (𝑟 × 𝑠)) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦}))
4121, 31, 39, 40syl3anc 1323 . . . . . . . . . . . . . . . . 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}))
42 simprl2 1105 . . . . . . . . . . . . . . . . . 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)
43 simprr2 1108 . . . . . . . . . . . . . . . . . 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)
44 xpss12 5191 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑎𝑠𝑏) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
4542, 43, 44syl2anc 692 . . . . . . . . . . . . . . . . 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))
46 simprll 801 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑎 ∈ 𝒫 𝑢)
4746adantr 481 . . . . . . . . . . . . . . . . . . . . 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)
4847elpwid 4146 . . . . . . . . . . . . . . . . . . . 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)
497ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 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)
50 elssuni 4438 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝑅𝑢 𝑅)
5149, 50syl 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)
5248, 51sstrd 3598 . . . . . . . . . . . . . . . . . . 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)
53 simprlr 802 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑏 ∈ 𝒫 𝑣)
5453adantr 481 . . . . . . . . . . . . . . . . . . . . 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)
5554elpwid 4146 . . . . . . . . . . . . . . . . . . . 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)
5614ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 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)
57 elssuni 4438 . . . . . . . . . . . . . . . . . . . . 21 (𝑣𝑆𝑣 𝑆)
5856, 57syl 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)
5955, 58sstrd 3598 . . . . . . . . . . . . . . . . . . 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)
60 xpss12 5191 . . . . . . . . . . . . . . . . . . 19 ((𝑎 𝑅𝑏 𝑆) → (𝑎 × 𝑏) ⊆ ( 𝑅 × 𝑆))
6152, 59, 60syl2anc 692 . . . . . . . . . . . . . . . . . 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))
62 eqid 2626 . . . . . . . . . . . . . . . . . . . 20 𝑅 = 𝑅
63 eqid 2626 . . . . . . . . . . . . . . . . . . . 20 𝑆 = 𝑆
6462, 63txuni 21300 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6523, 25, 64syl2anc 692 . . . . . . . . . . . . . . . . . 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))
6661, 65sseqtrd 3625 . . . . . . . . . . . . . . . . 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))
67 eqid 2626 . . . . . . . . . . . . . . . . . 18 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6867ssnei2 20825 . . . . . . . . . . . . . . . . 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}))
6921, 41, 45, 66, 68syl22anc 1324 . . . . . . . . . . . . . . . 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}))
70 xpss12 5191 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑢𝑏𝑣) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
7148, 55, 70syl2anc 692 . . . . . . . . . . . . . . . . . 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))
72 simprrr 804 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
7372ad2antrr 761 . . . . . . . . . . . . . . . . . 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)
7471, 73sstrd 3598 . . . . . . . . . . . . . . . . 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)
75 vex 3194 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
7675elpw2 4793 . . . . . . . . . . . . . . . . 17 ((𝑎 × 𝑏) ∈ 𝒫 𝑥 ↔ (𝑎 × 𝑏) ⊆ 𝑥)
7774, 76sylibr 224 . . . . . . . . . . . . . . . 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)
7869, 77elind 3781 . . . . . . . . . . . . . . 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))
79 txrest 21339 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣)) → ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) = ((𝑅t 𝑎) ×t (𝑆t 𝑏)))
8023, 25, 47, 54, 79syl22anc 1324 . . . . . . . . . . . . . . . 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)))
81 simprl3 1106 . . . . . . . . . . . . . . . . 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)
82 simprr3 1109 . . . . . . . . . . . . . . . . 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)
83 txlly.1 . . . . . . . . . . . . . . . . . 18 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
8483caovcl 6782 . . . . . . . . . . . . . . . . 17 (((𝑅t 𝑎) ∈ 𝐴 ∧ (𝑆t 𝑏) ∈ 𝐴) → ((𝑅t 𝑎) ×t (𝑆t 𝑏)) ∈ 𝐴)
8581, 82, 84syl2anc 692 . . . . . . . . . . . . . . . 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)
8680, 85eqeltrd 2704 . . . . . . . . . . . . . . 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)
87 oveq2 6613 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎 × 𝑏) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)))
8887eleq1d 2688 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎 × 𝑏) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴))
8988rspcev 3300 . . . . . . . . . . . . . . 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)
9078, 86, 89syl2anc 692 . . . . . . . . . . . . . 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)
9190ex 450 . . . . . . . . . . . . 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))
9291anassrs 679 . . . . . . . . . . . 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))
9392rexlimdvva 3036 . . . . . . . . . . 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))
9420, 93syl5bir 233 . . . . . . . . . 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))
9594rexlimdvva 3036 . . . . . . . . 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))
9619, 95syl5bir 233 . . . . . . . 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))
9712, 18, 96mp2and 714 . . . . . . 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)
9897expr 642 . . . . . 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))
9998rexlimdvva 3036 . . . . 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))
10099ralimdv 2962 . . . 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))
1015, 100sylbid 230 . . 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))
102101ralrimiv 2964 . 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)
103 isnlly 21177 . 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))
1044, 102, 103sylanbrc 697 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 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cin 3559  wss 3560  𝒫 cpw 4135  {csn 4153  cop 4159   cuni 4407   × cxp 5077  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  t crest 15997  Topctop 20612  neicnei 20806  𝑛-Locally cnlly 21173   ×t ctx 21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-rest 15999  df-topgen 16020  df-top 20616  df-bases 20617  df-topon 20618  df-nei 20807  df-nlly 21175  df-tx 21270
This theorem is referenced by:  xkohmeo  21523  cvmlift2lem13  30997
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