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Theorem txlly 22239
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 22075 . . 3 (𝑅 ∈ Locally 𝐴𝑅 ∈ Top)
2 llytop 22075 . . 3 (𝑆 ∈ Locally 𝐴𝑆 ∈ Top)
3 txtop 22172 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 598 . 2 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Top)
5 eltx 22171 . . . 4 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦𝑥𝑢𝑅𝑣𝑆 (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥)))
6 simpll 766 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑅 ∈ Locally 𝐴)
7 simprll 778 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑢𝑅)
8 simprrl 780 . . . . . . . . . 10 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑦 ∈ (𝑢 × 𝑣))
9 xp1st 7707 . . . . . . . . . 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 22077 . . . . . . . . 9 ((𝑅 ∈ Locally 𝐴𝑢𝑅 ∧ (1st𝑦) ∈ 𝑢) → ∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴))
126, 7, 10, 11syl3anc 1368 . . . . . . . 8 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑟𝑅 (𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴))
13 simplr 768 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑆 ∈ Locally 𝐴)
14 simprlr 779 . . . . . . . . 9 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → 𝑣𝑆)
15 xp2nd 7708 . . . . . . . . . 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 22077 . . . . . . . . 9 ((𝑆 ∈ Locally 𝐴𝑣𝑆 ∧ (2nd𝑦) ∈ 𝑣) → ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))
1813, 14, 16, 17syl3anc 1368 . . . . . . . 8 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → ∃𝑠𝑆 (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))
19 reeanv 3348 . . . . . . . . 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 729 . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 22205 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → (𝑟 × 𝑠) ∈ (𝑅 ×t 𝑆))
2520, 21, 22, 23, 24syl22anc 837 . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . . . 16 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → 𝑟𝑢)
27 simprr1 1218 . . . . . . . . . . . . . . . 16 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → 𝑠𝑣)
28 xpss12 5547 . . . . . . . . . . . . . . . 16 ((𝑟𝑢𝑠𝑣) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
2926, 27, 28syl2anc 587 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑟 × 𝑠) ⊆ (𝑢 × 𝑣))
30 simprrr 781 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) ∧ ((𝑢𝑅𝑣𝑆) ∧ (𝑦 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑥))) → (𝑢 × 𝑣) ⊆ 𝑥)
3129, 30sylan9ssr 3956 . . . . . . . . . . . . . 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 3472 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3332elpw2 5224 . . . . . . . . . . . . . 14 ((𝑟 × 𝑠) ∈ 𝒫 𝑥 ↔ (𝑟 × 𝑠) ⊆ 𝑥)
3431, 33sylibr 237 . . . . . . . . . . . . 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 4145 . . . . . . . . . . . 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 7714 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (1st𝑦) ∈ 𝑟)
40 simprr2 1219 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (2nd𝑦) ∈ 𝑠)
4139, 40opelxpd 5570 . . . . . . . . . . . . . 14 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝑟 × 𝑠))
4241adantl 485 . . . . . . . . . . . . 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))
4338, 42eqeltrd 2914 . . . . . . . . . . . 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))
44 txrest 22234 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑟𝑅𝑠𝑆)) → ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) = ((𝑅t 𝑟) ×t (𝑆t 𝑠)))
4520, 21, 22, 23, 44syl22anc 837 . . . . . . . . . . . . 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)))
46 simprl3 1217 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑅t 𝑟) ∈ 𝐴)
47 simprr3 1220 . . . . . . . . . . . . . . 15 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → (𝑆t 𝑠) ∈ 𝐴)
48 txlly.1 . . . . . . . . . . . . . . . 16 ((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)
4948caovcl 7327 . . . . . . . . . . . . . . 15 (((𝑅t 𝑟) ∈ 𝐴 ∧ (𝑆t 𝑠) ∈ 𝐴) → ((𝑅t 𝑟) ×t (𝑆t 𝑠)) ∈ 𝐴)
5046, 47, 49syl2anc 587 . . . . . . . . . . . . . 14 (((𝑟𝑅𝑠𝑆) ∧ ((𝑟𝑢 ∧ (1st𝑦) ∈ 𝑟 ∧ (𝑅t 𝑟) ∈ 𝐴) ∧ (𝑠𝑣 ∧ (2nd𝑦) ∈ 𝑠 ∧ (𝑆t 𝑠) ∈ 𝐴))) → ((𝑅t 𝑟) ×t (𝑆t 𝑠)) ∈ 𝐴)
5150adantl 485 . . . . . . . . . . . . 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)
5245, 51eqeltrd 2914 . . . . . . . . . . . 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)
53 eleq2 2902 . . . . . . . . . . . . . 14 (𝑧 = (𝑟 × 𝑠) → (𝑦𝑧𝑦 ∈ (𝑟 × 𝑠)))
54 oveq2 7148 . . . . . . . . . . . . . . 15 (𝑧 = (𝑟 × 𝑠) → ((𝑅 ×t 𝑆) ↾t 𝑧) = ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)))
5554eleq1d 2898 . . . . . . . . . . . . . 14 (𝑧 = (𝑟 × 𝑠) → (((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴 ↔ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴))
5653, 55anbi12d 633 . . . . . . . . . . . . 13 (𝑧 = (𝑟 × 𝑠) → ((𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴) ↔ (𝑦 ∈ (𝑟 × 𝑠) ∧ ((𝑅 ×t 𝑆) ↾t (𝑟 × 𝑠)) ∈ 𝐴)))
5756rspcev 3598 . . . . . . . . . . . 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))
5835, 43, 52, 57syl12anc 835 . . . . . . . . . . 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))
5958expr 460 . . . . . . . . . 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)))
6059rexlimdvva 3280 . . . . . . . . 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)))
6119, 60syl5bir 246 . . . . . . . 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)))
6212, 18, 61mp2and 698 . . . . . . 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))
6362expr 460 . . . . . 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)))
6463rexlimdvva 3280 . . . . 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)))
6564ralimdv 3170 . . . 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)))
665, 65sylbid 243 . . 3 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑥 ∈ (𝑅 ×t 𝑆) → ∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴)))
6766ralrimiv 3173 . 2 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → ∀𝑥 ∈ (𝑅 ×t 𝑆)∀𝑦𝑥𝑧 ∈ ((𝑅 ×t 𝑆) ∩ 𝒫 𝑥)(𝑦𝑧 ∧ ((𝑅 ×t 𝑆) ↾t 𝑧) ∈ 𝐴))
68 islly 22071 . 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)))
694, 67, 68sylanbrc 586 1 ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cin 3907  wss 3908  𝒫 cpw 4511  cop 4545   × cxp 5530  cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674  t crest 16685  Topctop 21496  Locally clly 22067   ×t ctx 22163
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-rest 16687  df-topgen 16708  df-top 21497  df-bases 21549  df-lly 22069  df-tx 22165
This theorem is referenced by: (None)
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