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Theorem txnlly 22696

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	⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×_t 𝑘) ∈ 𝐴)

**Assertion**
Ref	Expression
txnlly	⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 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.

Step	Hyp	Ref	Expression
1		nllytop 22532	. . 3 ⊢ (𝑅 ∈ 𝑛-Locally 𝐴 → 𝑅 ∈ Top)
2		nllytop 22532	. . 3 ⊢ (𝑆 ∈ 𝑛-Locally 𝐴 → 𝑆 ∈ Top)
3		txtop 22628	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×_t 𝑆) ∈ Top)
5		eltx 22627	. . . 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 763	. . . . . . . . 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 775	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢 ∈ 𝑅)
8		simprrl 777	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9		xp1st 7836	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → (1^st ‘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))) → (1^st ‘𝑦) ∈ 𝑢)
11		nlly2i 22535	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑢 ∈ 𝑅 ∧ (1^st ‘𝑦) ∈ 𝑢) → ∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴))
12	6, 7, 10, 11	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴))
13		simplr 765	. . . . . . . . 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 776	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣 ∈ 𝑆)
15		xp2nd 7837	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → (2^nd ‘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))) → (2^nd ‘𝑦) ∈ 𝑣)
17		nlly2i 22535	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑛-Locally 𝐴 ∧ 𝑣 ∈ 𝑆 ∧ (2^nd ‘𝑦) ∈ 𝑣) → ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))
18	13, 14, 16, 17	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))
19		reeanv 3292	. . . . . . . . 9 ⊢ (∃𝑎 ∈ 𝒫 𝑢∃𝑏 ∈ 𝒫 𝑣(∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴)) ↔ (∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴)))
20		reeanv 3292	. . . . . . . . . . 11 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴)) ↔ (∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴)))
21	4	ad3antrrr 726	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑅 ×_t 𝑆) ∈ Top)
22	1	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Top)
23	22	ad2antrr 722	. . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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 722	. . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑆 ∈ Top)
26		simprrl 777	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑟 ∈ 𝑅)
28		simprrr 778	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑠 ∈ 𝑆)
30		txopn 22661	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×_t 𝑆))
31	23, 25, 27, 29, 30	syl22anc 835	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ (𝑅 ×_t 𝑆))
32	8	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑢 × 𝑣))
33		1st2nd2 7843	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
35		simprl1 1216	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (1^st ‘𝑦) ∈ 𝑟)
36		simprr1 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (2^nd ‘𝑦) ∈ 𝑠)
37	35, 36	opelxpd 5618	. . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∈ (𝑟 × 𝑠))
38	34, 37	eqeltrd 2839	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑦 ∈ (𝑟 × 𝑠))
39		opnneip 22178	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ∈ ((nei‘(𝑅 ×_t 𝑆))‘{𝑦}))
41		simprl2 1217	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑟 ⊆ 𝑎)
42		simprr2 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑠 ⊆ 𝑏)
43		xpss12 5595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑟 ⊆ 𝑎 ∧ 𝑠 ⊆ 𝑏) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
44	41, 42, 43	syl2anc 583	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑎 × 𝑏))
45		simprll 775	. . . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑎 ∈ 𝒫 𝑢)
47	46	elpwid 4541	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑎 ⊆ 𝑢)
48	7	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑢 ∈ 𝑅)
49		elssuni 4868	. . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑢 ⊆ ∪ 𝑅)
51	47, 50	sstrd 3927	. . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑎 ⊆ ∪ 𝑅)
52		simprlr 776	. . . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑏 ∈ 𝒫 𝑣)
54	53	elpwid 4541	. . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑏 ⊆ 𝑣)
55	14	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑣 ∈ 𝑆)
56		elssuni 4868	. . . . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑣 ⊆ ∪ 𝑆)
58	54, 57	sstrd 3927	. . . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → 𝑏 ⊆ ∪ 𝑆)
59		xpss12 5595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ⊆ ∪ 𝑅 ∧ 𝑏 ⊆ ∪ 𝑆) → (𝑎 × 𝑏) ⊆ (∪ 𝑅 × ∪ 𝑆))
60	51, 58, 59	syl2anc 583	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (∪ 𝑅 × ∪ 𝑆))
61		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑅 = ∪ 𝑅
62		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑆 = ∪ 𝑆
63	61, 62	txuni 22651	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
64	23, 25, 63	syl2anc 583	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
65	60, 64	sseqtrd 3957	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ ∪ (𝑅 ×_t 𝑆))
66		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
67	66	ssnei2 22175	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ ((nei‘(𝑅 ×_t 𝑆))‘{𝑦}))
69		xpss12 5595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ⊆ 𝑢 ∧ 𝑏 ⊆ 𝑣) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
70	47, 54, 69	syl2anc 583	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ (𝑢 × 𝑣))
71		simprrr 778	. . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑢 × 𝑣) ⊆ 𝑥)
73	70, 72	sstrd 3927	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ⊆ 𝑥)
74		vex 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
75	74	elpw2 5264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 × 𝑏) ∈ 𝒫 𝑥 ↔ (𝑎 × 𝑏) ⊆ 𝑥)
76	73, 75	sylibr 233	. . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ 𝒫 𝑥)
77	68, 76	elind 4124	. . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑎 × 𝑏) ∈ (((nei‘(𝑅 ×_t 𝑆))‘{𝑦}) ∩ 𝒫 𝑥))
78		txrest 22690	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → ((𝑅 ×_t 𝑆) ↾_t (𝑎 × 𝑏)) = ((𝑅 ↾_t 𝑎) ×_t (𝑆 ↾_t 𝑏)))
80		simprl3 1218	. . . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → (𝑅 ↾_t 𝑎) ∈ 𝐴)
81		simprr3 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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 7444	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ↾_t 𝑎) ∈ 𝐴 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴) → ((𝑅 ↾_t 𝑎) ×_t (𝑆 ↾_t 𝑏)) ∈ 𝐴)
84	80, 81, 83	syl2anc 583	. . . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → ((𝑅 ↾_t 𝑎) ×_t (𝑆 ↾_t 𝑏)) ∈ 𝐴)
85	79, 84	eqeltrd 2839	. . . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘𝑦) ∈ 𝑠 ∧ 𝑠 ⊆ 𝑏 ∧ (𝑆 ↾_t 𝑏) ∈ 𝐴))) → ((𝑅 ×_t 𝑆) ↾_t (𝑎 × 𝑏)) ∈ 𝐴)
86		oveq2 7263	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑎 × 𝑏) → ((𝑅 ×_t 𝑆) ↾_t 𝑧) = ((𝑅 ×_t 𝑆) ↾_t (𝑎 × 𝑏)))
87	86	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑎 × 𝑏) → (((𝑅 ×_t 𝑆) ↾_t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×_t 𝑆) ↾_t (𝑎 × 𝑏)) ∈ 𝐴))
88	87	rspcev 3552	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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))) ∧ (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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))) → ((((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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)) → ((((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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 3222	. . . . . . . . . . 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 (((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ((2^nd ‘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	syl5bir 242	. . . . . . . . . 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 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2^nd ‘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 3222	. . . . . . . . 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 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 ((2^nd ‘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	syl5bir 242	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑎 ∈ 𝒫 𝑢∃𝑟 ∈ 𝑅 ((1^st ‘𝑦) ∈ 𝑟 ∧ 𝑟 ⊆ 𝑎 ∧ (𝑅 ↾_t 𝑎) ∈ 𝐴) ∧ ∃𝑏 ∈ 𝒫 𝑣∃𝑠 ∈ 𝑆 ((2^nd ‘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 695	. . . . . . 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 3222	. . . . 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 3103	. . . 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 239	. . 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 3106	. 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 22528	. 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 582	1 ⊢ ((𝑅 ∈ 𝑛-Locally 𝐴 ∧ 𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×_t 𝑆) ∈ 𝑛-Locally 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4530 {csn 4558 ⟨cop 4564 ∪ cuni 4836 × cxp 5578 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 ↾_t crest 17048 Topctop 21950 neicnei 22156 𝑛-Locally cnlly 22524 ×_t ctx 22619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-rest 17050 df-topgen 17071 df-top 21951 df-topon 21968 df-bases 22004 df-nei 22157 df-nlly 22526 df-tx 22621

This theorem is referenced by: xkohmeo 22874 cvmlift2lem13 33177

W3C validator