Theorem txlly 22238

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 22074	. . 3 ⊢ (𝑅 ∈ Locally 𝐴 → 𝑅 ∈ Top)
2		llytop 22074	. . 3 ⊢ (𝑆 ∈ Locally 𝐴 → 𝑆 ∈ Top)
3		txtop 22171	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5		eltx 22170	. . . 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 7715	. . . . . . . . . 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 22076	. . . . . . . . 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 1367	. . . . . . . 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 7716	. . . . . . . . . 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 22076	. . . . . . . . 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 1367	. . . . . . . 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 3367	. . . . . . . . 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 22204	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
25	20, 21, 22, 23, 24	syl22anc 836	. . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . . 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 5564	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ⊆ 𝑢 ∧ 𝑠 ⊆ 𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
29	26, 27, 28	syl2anc 586	. . . . . . . . . . . . . . 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 3980	. . . . . . . . . . . . . 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 3497	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
33	32	elpw2 5240	. . . . . . . . . . . . . 14 ⊢ ((𝑟 × 𝑠) ∈ 𝒫 𝑥 ↔ (𝑟 × 𝑠) ⊆ 𝑥)
34	31, 33	sylibr 236	. . . . . . . . . . . . 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 4170	. . . . . . . . . . . 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 7722	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . 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 5587	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2913	. . . . . . . . . . . 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 22233	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . 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 7336	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ↾t 𝑟) ∈ 𝐴 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
50	46, 47, 49	syl2anc 586	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2913	. . . . . . . . . . . 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 2901	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑟 × 𝑠)))
54		oveq2 7158	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)))
55	54	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))
56	53, 55	anbi12d 632	. . . . . . . . . . . . 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 3622	. . . . . . . . . . . 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 834	. . . . . . . . . . 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 459	. . . . . . . . . 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 3294	. . . . . . . . 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	syl5bir 245	. . . . . . . 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 459	. . . . . 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 3294	. . . . 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 3178	. . . 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 242	. . 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 3181	. 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 22070	. 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 585	1 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  ⟨cop 4566   × cxp 5547  ‘cfv 6349  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682   ↾_t crest 16688  Topctop 21495  Locally clly 22066   ×_t ctx 22162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-rest 16690  df-topgen 16711  df-top 21496  df-bases 21548  df-lly 22068  df-tx 22164

This theorem is referenced by: (None)