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Theorem txlly 21612
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 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
Assertion
Ref Expression
txlly ((𝑅 ∈ Locally 𝐴𝑆 ∈ 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.
StepHypRef Expression
1 llytop 21448 . . 3 (𝑅 ∈ Locally 𝐴𝑅 ∈ Top)
2 llytop 21448 . . 3 (𝑆 ∈ Locally 𝐴𝑆 ∈ Top)
3 txtop 21545 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 495 . 2 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5 eltx 21544 . . . 4 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6 simpll 807 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Locally 𝐴)
7 simprll 821 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢𝑅)
8 simprrl 823 . . . . . . . . . 10 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9 xp1st 7353 . . . . . . . . . 10 (𝑦 ∈ (𝑢 × 𝑣) → (1st𝑦) ∈ 𝑢)
108, 9syl 17 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (1st𝑦) ∈ 𝑢)
11 llyi 21450 . . . . . . . . 9 ((𝑅 ∈ Locally 𝐴𝑢𝑅 ∧ (1st𝑦) ∈ 𝑢) → ∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴))
126, 7, 10, 11syl3anc 1463 . . . . . . . 8 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴))
13 simplr 809 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Locally 𝐴)
14 simprlr 822 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣𝑆)
15 xp2nd 7354 . . . . . . . . . 10 (𝑦 ∈ (𝑢 × 𝑣) → (2nd𝑦) ∈ 𝑣)
168, 15syl 17 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (2nd𝑦) ∈ 𝑣)
17 llyi 21450 . . . . . . . . 9 ((𝑆 ∈ Locally 𝐴𝑣𝑆 ∧ (2nd𝑦) ∈ 𝑣) → ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))
1813, 14, 16, 17syl3anc 1463 . . . . . . . 8 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))
19 reeanv 3233 . . . . . . . . 9 (∃𝑟𝑅𝑠𝑆 ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)) ↔ (∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))
201ad3antrrr 768 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑅 ∈ Top)
212ad3antlr 769 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑆 ∈ Top)
22 simprll 821 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑟𝑅)
23 simprlr 822 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑠𝑆)
24 txopn 21578 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
2520, 21, 22, 23, 24syl22anc 1464 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
26 simprl1 1243 . . . . . . . . . . . . . . . 16 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → 𝑟𝑢)
27 simprr1 1249 . . . . . . . . . . . . . . . 16 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → 𝑠𝑣)
28 xpss12 5269 . . . . . . . . . . . . . . . 16 ((𝑟𝑢𝑠𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
2926, 27, 28syl2anc 696 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
30 simprrr 824 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
3129, 30sylan9ssr 3746 . . . . . . . . . . . . . 14 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ⊆ 𝑥)
32 vex 3331 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3332elpw2 4965 . . . . . . . . . . . . . 14 ((𝑟 × 𝑠) ∈ 𝒫 𝑥 ↔ (𝑟 × 𝑠) ⊆ 𝑥)
3431, 33sylibr 224 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ 𝒫 𝑥)
3525, 34elind 3929 . . . . . . . . . . . 12 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → (𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥))
36 1st2nd2 7360 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑢 × 𝑣) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
378, 36syl 17 . . . . . . . . . . . . . 14 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3837adantr 472 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
39 simprl2 1245 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (1st𝑦) ∈ 𝑟)
40 simprr2 1251 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (2nd𝑦) ∈ 𝑠)
41 opelxpi 5293 . . . . . . . . . . . . . . 15 (((1st𝑦) ∈ 𝑟 ∧ (2nd𝑦) ∈ 𝑠) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
4239, 40, 41syl2anc 696 . . . . . . . . . . . . . 14 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
4342adantl 473 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
4438, 43eqeltrd 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → 𝑦 ∈ (𝑟 × 𝑠))
45 txrest 21607 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅t 𝑟) ×t (𝑆t 𝑠)))
4620, 21, 22, 23, 45syl22anc 1464 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅t 𝑟) ×t (𝑆t 𝑠)))
47 simprl3 1247 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑅t 𝑟) ∈ 𝐴)
48 simprr3 1253 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑆t 𝑠) ∈ 𝐴)
49 txlly.1 . . . . . . . . . . . . . . . 16 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
5049caovcl 6981 . . . . . . . . . . . . . . 15 (((𝑅t 𝑟) ∈ 𝐴 ∧ (𝑆t 𝑠) ∈ 𝐴) → ((𝑅t 𝑟) ×t (𝑆t 𝑠)) ∈ 𝐴)
5147, 48, 50syl2anc 696 . . . . . . . . . . . . . 14 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → ((𝑅t 𝑟) ×t (𝑆t 𝑠)) ∈ 𝐴)
5251adantl 473 . . . . . . . . . . . . 13 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → ((𝑅t 𝑟) ×t (𝑆t 𝑠)) ∈ 𝐴)
5346, 52eqeltrd 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)
54 eleq2 2816 . . . . . . . . . . . . . 14 (𝑧 = (𝑟 × 𝑠) → (𝑦𝑧𝑦 ∈ (𝑟 × 𝑠)))
55 oveq2 6809 . . . . . . . . . . . . . . 15 (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)))
5655eleq1d 2812 . . . . . . . . . . . . . 14 (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))
5754, 56anbi12d 749 . . . . . . . . . . . . 13 (𝑧 = (𝑟 × 𝑠) → ((𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) ↔ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)))
5857rspcev 3437 . . . . . . . . . . . 12 (((𝑟 × 𝑠) ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
5935, 44, 53, 58syl12anc 1461 . . . . . . . . . . 11 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ ((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
6059expr 644 . . . . . . . . . 10 ((((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) ∧ (𝑟𝑅𝑠𝑆)) → (((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6160rexlimdvva 3164 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (∃𝑟𝑅𝑠𝑆 ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6219, 61syl5bir 233 . . . . . . . 8 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ((∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴)) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6312, 18, 62mp2and 717 . . . . . . 7 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
6463expr 644 . . . . . 6 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ (𝑢𝑅𝑣𝑆)) → ((𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6564rexlimdvva 3164 . . . . 5 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (∃𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∃𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6665ralimdv 3089 . . . 4 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥) → ∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
675, 66sylbid 230 . . 3 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6867ralrimiv 3091 . 2 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
69 islly 21444 . 2 ((𝑅 ×t 𝑆) ∈ Locally 𝐴 ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
704, 68, 69sylanbrc 701 1 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  wrex 3039  cin 3702  wss 3703  𝒫 cpw 4290  cop 4315   × cxp 5252  cfv 6037  (class class class)co 6801  1st c1st 7319  2nd c2nd 7320  t crest 16254  Topctop 20871  Locally clly 21440   ×t ctx 21536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-rest 16256  df-topgen 16277  df-top 20872  df-bases 20923  df-lly 21442  df-tx 21538
This theorem is referenced by: (None)
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