Theorem txlly 23623

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 23459	. . 3 ⊢ (𝑅 ∈ Locally 𝐴 → 𝑅 ∈ Top)
2		llytop 23459	. . 3 ⊢ (𝑆 ∈ Locally 𝐴 → 𝑆 ∈ Top)
3		txtop 23556	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 603	. 2 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5		eltx 23555	. . . 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 773	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Locally 𝐴)
7		simprll 785	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢 ∈ 𝑅)
8		simprrl 787	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9		xp1st 7967	. . . . . . . . . 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 23461	. . . . . . . . 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 1380	. . . . . . . 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 775	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Locally 𝐴)
14		simprlr 786	. . . . . . . . 9 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣 ∈ 𝑆)
15		xp2nd 7968	. . . . . . . . . 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 23461	. . . . . . . . 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 1380	. . . . . . . 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 3213	. . . . . . . . 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 737	. . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . 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 23589	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
25	20, 21, 22, 23, 24	syl22anc 845	. . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . 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 5636	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ⊆ 𝑢 ∧ 𝑠 ⊆ 𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
29	26, 27, 28	syl2anc 591	. . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
31	29, 30	sylan9ssr 3931	. . . . . . . . . . . . . 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 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
33	32	elpw2 5265	. . . . . . . . . . . . . 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 4132	. . . . . . . . . . . 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 7974	. . . . . . . . . . . . . . 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 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)))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
39		simprl2 1227	. . . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . . . 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 5660	. . . . . . . . . . . . . 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 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)))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ (𝑟 × 𝑠))
43	38, 42	eqeltrd 2841	. . . . . . . . . . . 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 23618	. . . . . . . . . . . . . 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 845	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . . 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 7554	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ↾t 𝑟) ∈ 𝐴 ∧ (𝑆 ↾t 𝑠) ∈ 𝐴) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
50	46, 47, 49	syl2anc 591	. . . . . . . . . . . . . 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 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)))) → ((𝑅 ↾t 𝑟) ×t (𝑆 ↾t 𝑠)) ∈ 𝐴)
52	45, 51	eqeltrd 2841	. . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑟 × 𝑠)))
54		oveq2 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)))
55	54	eleq1d 2826	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))
56	53, 55	anbi12d 639	. . . . . . . . . . . . 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 3562	. . . . . . . . . . . 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 843	. . . . . . . . . . 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 458	. . . . . . . . . 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 3198	. . . . . . . . 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 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 706	. . . . . . 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 458	. . . . . 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 3198	. . . . 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 3155	. . . 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 3132	. 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 23455	. 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 590	1 ⊢ ((𝑅 ∈ Locally 𝐴 ∧ 𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ∩ cin 3884   ⊆ wss 3885  𝒫 cpw 4532  ⟨cop 4564   × cxp 5619  ‘cfv 6489  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934   ↾_t crest 17378  Topctop 22880  Locally clly 23451   ×_t ctx 23547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-rest 17380  df-topgen 17401  df-top 22881  df-bases 22933  df-lly 23453  df-tx 23549

This theorem is referenced by: (None)