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Theorem txnlly 22240
 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 22076 . . 3 (𝑅 ∈ 𝑛-Locally 𝐴𝑅 ∈ Top)
2 nllytop 22076 . . 3 (𝑆 ∈ 𝑛-Locally 𝐴𝑆 ∈ Top)
3 txtop 22172 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 598 . 2 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5 eltx 22171 . . . 4 ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6 simpll 766 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ 𝑛-Locally 𝐴)
7 simprll 778 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢𝑅)
8 simprrl 780 . . . . . . . . . 10 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9 xp1st 7707 . . . . . . . . . 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 22079 . . . . . . . . 9 ((𝑅 ∈ 𝑛-Locally 𝐴𝑢𝑅 ∧ (1st𝑦) ∈ 𝑢) → ∃𝑎 ∈ 𝒫 𝑢𝑟𝑅 ((1st𝑦) ∈ 𝑟𝑟𝑎 ∧ (𝑅t 𝑎) ∈ 𝐴))
126, 7, 10, 11syl3anc 1368 . . . . . . . 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 768 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ 𝑛-Locally 𝐴)
14 simprlr 779 . . . . . . . . 9 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣𝑆)
15 xp2nd 7708 . . . . . . . . . 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 22079 . . . . . . . . 9 ((𝑆 ∈ 𝑛-Locally 𝐴𝑣𝑆 ∧ (2nd𝑦) ∈ 𝑣) → ∃𝑏 ∈ 𝒫 𝑣𝑠𝑆 ((2nd𝑦) ∈ 𝑠𝑠𝑏 ∧ (𝑆t 𝑏) ∈ 𝐴))
1813, 14, 16, 17syl3anc 1368 . . . . . . . 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 3348 . . . . . . . . 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 3348 . . . . . . . . . . 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 729 . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Top)
2322ad2antrr 725 . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑟𝑅)
2726adantr 484 . . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . . 20 ((((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑎 ∈ 𝒫 𝑢𝑏 ∈ 𝒫 𝑣) ∧ (𝑟𝑅𝑠𝑆))) → 𝑠𝑆)
2928adantr 484 . . . . . . . . . . . . . . . . . . 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 22205 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
3123, 25, 27, 29, 30syl22anc 837 . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . 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 7714 . . . . . . . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . . . 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 5570 . . . . . . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . . . . . . 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 21722 . . . . . . . . . . . . . . . . . 18 (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆) ∧ 𝑦 ∈ (𝑟 × 𝑠)) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×t 𝑆))‘{𝑦}))
4021, 31, 38, 39syl3anc 1368 . . . . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑎𝑠𝑏) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
4441, 42, 43syl2anc 587 . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . 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 4522 . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . 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 4843 . . . . . . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . 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 4522 . . . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . . . . 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 4843 . . . . . . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . . 19 ((𝑎 𝑅𝑏 𝑆) → (𝑎 × 𝑏) ⊆ ( 𝑅 × 𝑆))
6051, 58, 59syl2anc 587 . . . . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . . . . 20 𝑅 = 𝑅
62 eqid 2822 . . . . . . . . . . . . . . . . . . . 20 𝑆 = 𝑆
6361, 62txuni 22195 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6423, 25, 63syl2anc 587 . . . . . . . . . . . . . . . . . 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 3982 . . . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . . 18 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6766ssnei2 21719 . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑢𝑏𝑣) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
7047, 54, 69syl2anc 587 . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
7271ad2antrr 725 . . . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . . . . . . 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 3472 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
7574elpw2 5224 . . . . . . . . . . . . . . . . 17 ((𝑎 × 𝑏) ∈ 𝒫 𝑥 ↔ (𝑎 × 𝑏) ⊆ 𝑥)
7673, 75sylibr 237 . . . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . . . . 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 22234 . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . 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 7327 . . . . . . . . . . . . . . . . 17 (((𝑅t 𝑎) ∈ 𝐴 ∧ (𝑆t 𝑏) ∈ 𝐴) → ((𝑅t 𝑎) ×t (𝑆t 𝑏)) ∈ 𝐴)
8480, 81, 83syl2anc 587 . . . . . . . . . . . . . . . 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 2914 . . . . . . . . . . . . . . 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 7148 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑎 × 𝑏) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)))
8786eleq1d 2898 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑎 × 𝑏) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑎 × 𝑏)) ∈ 𝐴))
8887rspcev 3598 . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 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 471 . . . . . . . . . . . 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 3280 . . . . . . . . . . 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 246 . . . . . . . . . 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 3280 . . . . . . . . 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 246 . . . . . . . 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 698 . . . . . . 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 460 . . . . . 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 3280 . . . . 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 3170 . . . 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 243 . . 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 3173 . 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 22072 . 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 586 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4511  {csn 4539  ⟨cop 4545  ∪ cuni 4813   × cxp 5530  ‘cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674   ↾t crest 16685  Topctop 21496  neicnei 21700  𝑛-Locally cnlly 22068   ×t ctx 22163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-rest 16687  df-topgen 16708  df-top 21497  df-topon 21514  df-bases 21549  df-nei 21701  df-nlly 22070  df-tx 22165 This theorem is referenced by:  xkohmeo  22418  cvmlift2lem13  32636
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