Theorem txlly 22987

Description: If the property 𝐴 is preserved under topological products, then so is the property of being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypothesis

Ref	Expression
txlly.1	⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)

Assertion

Ref	Expression
txlly	⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Distinct variable groups:   𝑗,𝑘,𝐴   𝑅,𝑗,𝑘   𝑆,𝑘

Allowed substitution hint:   𝑆(𝑗)

Proof of Theorem txlly

Dummy variables 𝑟 𝑠 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 22823	. . 3 ⊢ (𝑅 ∈ Locally 𝐴 → 𝑅 ∈ Top)
2		llytop 22823	. . 3 ⊢ (𝑆 ∈ Locally 𝐴 → 𝑆 ∈ Top)
3		txtop 22920	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5		eltx 22919	. . . 4 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6		simpll 765	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Locally 𝐴)
7		simprll 777	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢 ∈ 𝑅)
8		simprrl 779	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9		xp1st 7953	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → (1st ‘𝑦) ∈ 𝑢)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (1st ‘𝑦) ∈ 𝑢)
11		llyi 22825	. . . . . . . . 9 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑢 ∈ 𝑅 ∧ (1st ‘𝑦) ∈ 𝑢) → ∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴))
12	6, 7, 10, 11	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴))
13		simplr 767	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Locally 𝐴)
14		simprlr 778	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣 ∈ 𝑆)
15		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → (2nd ‘𝑦) ∈ 𝑣)
16	8, 15	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (2nd ‘𝑦) ∈ 𝑣)
17		llyi 22825	. . . . . . . . 9 ⊢ ((𝑆 ∈ Locally 𝐴 ∧ 𝑣 ∈ 𝑆 ∧ (2nd ‘𝑦) ∈ 𝑣) → ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))
18	13, 14, 16, 17	syl3anc 1371	. . . . . . . 8 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))
19		reeanv 3217	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) ↔ (∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))
20	1	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑅 ∈ Top)
21	2	ad3antlr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑆 ∈ Top)
22		simprll 777	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑟 ∈ 𝑅)
23		simprlr 778	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑠 ∈ 𝑆)
24		txopn 22953	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
25	20, 21, 22, 23, 24	syl22anc 837	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
26		simprl1 1218	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → 𝑟 ⊆ 𝑢)
27		simprr1 1221	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → 𝑠 ⊆ 𝑣)
28		xpss12 5648	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ⊆ 𝑢 ∧ 𝑠 ⊆ 𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
29	26, 27, 28	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
30		simprrr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
31	29, 30	sylan9ssr 3958	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ⊆ 𝑥)
32		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
33	32	elpw2 5302	. . . . . . . . . . . . . 14 ⊢ ((𝑟 × 𝑠) ∈ 𝒫 𝑥 ↔ (𝑟 × 𝑠) ⊆ 𝑥)
34	31, 33	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ 𝒫 𝑥)
35	25, 34	elind 4154	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥))
36		1st2nd2 7960	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
37	8, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
38	37	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
39		simprl2 1219	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (1st ‘𝑦) ∈ 𝑟)
40		simprr2 1222	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (2nd ‘𝑦) ∈ 𝑠)
41	39, 40	opelxpd 5671	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ (𝑟 × 𝑠))
42	41	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ (𝑟 × 𝑠))
43	38, 42	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → 𝑦 ∈ (𝑟 × 𝑠))
44		txrest 22982	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)))
45	20, 21, 22, 23, 44	syl22anc 837	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)))
46		simprl3 1220	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑅 ↾t 𝑟) ∈ 𝐴)
47		simprr3 1223	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → (𝑆 ↾t 𝑠) ∈ 𝐴)
48		txlly.1	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
49	48	caovcl 7548	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ↾t 𝑟) ∈ 𝐴 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
50	46, 47, 49	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴))) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
51	50	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
52	45, 51	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)
53		eleq2 2826	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑟 × 𝑠)))
54		oveq2 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)))
55	54	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))
56	53, 55	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) ↔ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)))
57	56	rspcev 3581	. . . . . . . . . . . 12 ⊢ (((𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
58	35, 43, 52, 57	syl12anc 835	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
59	58	expr 457	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
60	59	rexlimdvva 3205	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 ((𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
61	19, 60	biimtrrid 242	. . . . . . . 8 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑟 ∈ 𝑅 (𝑟 ⊆ 𝑢 ∧ (1st ‘𝑦) ∈ 𝑟 ∧ (𝑅 ↾t 𝑟) ∈ 𝐴) ∧ ∃𝑠 ∈ 𝑆 (𝑠 ⊆ 𝑣 ∧ (2nd ‘𝑦) ∈ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
62	12, 18, 61	mp2and 697	. . . . . . 7 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
63	62	expr 457	. . . . . 6 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ (𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆)) → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
64	63	rexlimdvva 3205	. . . . 5 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
65	64	ralimdv 3166	. . . 4 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
66	5, 65	sylbid 239	. . 3 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
67	66	ralrimiv 3142	. 2 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
68		islly 22819	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Locally 𝐴 ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦 ∈ 𝑥 ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
69	4, 67, 68	sylanbrc 583	1 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  ⟨cop 4592   × cxp 5631  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920   ↾_t crest 17302  Topctop 22242  Locally clly 22815   ×_t ctx 22911

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-rest 17304  df-topgen 17325  df-top 22243  df-bases 22296  df-lly 22817  df-tx 22913

This theorem is referenced by: (None)