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Theorem cldllycmp 23419
Description: A closed subspace of a locally compact space is also locally compact. (The analogous result for open subspaces follows from the more general nllyrest 23410.) (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
cldllycmp ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)

Proof of Theorem cldllycmp
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nllytop 23397 . . 3 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2 resttop 23084 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 578 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
4 elrest 17416 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝐴)))
5 simpll 765 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6 simprl 769 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑢𝐽)
7 simprr 771 . . . . . . . . . . 11 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦 ∈ (𝑢𝐴))
87elin1d 4200 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦𝑢)
9 nlly2i 23400 . . . . . . . . . 10 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑢𝐽𝑦𝑢) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
105, 6, 8, 9syl3anc 1368 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
113ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝐴) ∈ Top)
121ad3antrrr 728 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐽 ∈ Top)
13 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 ∈ (Clsd‘𝐽))
14 simprlr 778 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝐽)
15 elrestr 17417 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤𝐽) → (𝑤𝐴) ∈ (𝐽t 𝐴))
1612, 13, 14, 15syl3anc 1368 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ (𝐽t 𝐴))
17 simprr1 1218 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝑤)
18 simplrr 776 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢𝐴))
1918elin2d 4201 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝐴)
2017, 19elind 4196 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑤𝐴))
21 opnneip 23043 . . . . . . . . . . . . . . 15 (((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ (𝐽t 𝐴) ∧ 𝑦 ∈ (𝑤𝐴)) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
2211, 16, 20, 21syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
23 simprr2 1219 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝑠)
2423ssrind 4238 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ⊆ (𝑠𝐴))
25 inss2 4232 . . . . . . . . . . . . . . 15 (𝑠𝐴) ⊆ 𝐴
26 eqid 2728 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
2726cldss 22953 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
2813, 27syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 𝐽)
2926restuni 23086 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → 𝐴 = (𝐽t 𝐴))
3012, 28, 29syl2anc 582 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 = (𝐽t 𝐴))
3125, 30sseqtrid 4034 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ (𝐽t 𝐴))
32 eqid 2728 . . . . . . . . . . . . . . 15 (𝐽t 𝐴) = (𝐽t 𝐴)
3332ssnei2 23040 . . . . . . . . . . . . . 14 ((((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦})) ∧ ((𝑤𝐴) ⊆ (𝑠𝐴) ∧ (𝑠𝐴) ⊆ (𝐽t 𝐴))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
3411, 22, 24, 31, 33syl22anc 837 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
35 simprll 777 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 𝑢)
3635elpwid 4615 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠𝑢)
3736ssrind 4238 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ (𝑢𝐴))
38 vex 3477 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
3938inex1 5321 . . . . . . . . . . . . . . 15 (𝑠𝐴) ∈ V
4039elpw 4610 . . . . . . . . . . . . . 14 ((𝑠𝐴) ∈ 𝒫 (𝑢𝐴) ↔ (𝑠𝐴) ⊆ (𝑢𝐴))
4137, 40sylibr 233 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ 𝒫 (𝑢𝐴))
4234, 41elind 4196 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)))
4325a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ 𝐴)
44 restabs 23089 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝐴𝐴 ∈ (Clsd‘𝐽)) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
4512, 43, 13, 44syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
46 inss1 4231 . . . . . . . . . . . . . . . 16 (𝑠𝐴) ⊆ 𝑠
4746a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ 𝑠)
48 restabs 23089 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝑠𝑠 ∈ 𝒫 𝑢) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
4912, 47, 35, 48syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
5045, 49eqtr4d 2771 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = ((𝐽t 𝑠) ↾t (𝑠𝐴)))
51 simprr3 1220 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝑠) ∈ Comp)
52 incom 4203 . . . . . . . . . . . . . . 15 (𝑠𝐴) = (𝐴𝑠)
53 eqid 2728 . . . . . . . . . . . . . . . . 17 (𝐴𝑠) = (𝐴𝑠)
54 ineq1 4207 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → (𝑣𝑠) = (𝐴𝑠))
5554rspceeqv 3633 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴𝑠) = (𝐴𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
5613, 53, 55sylancl 584 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
57 simplrl 775 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑢𝐽)
58 elssuni 4944 . . . . . . . . . . . . . . . . . . 19 (𝑢𝐽𝑢 𝐽)
5957, 58syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑢 𝐽)
6036, 59sstrd 3992 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 𝐽)
6126restcld 23096 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠)))
6212, 60, 61syl2anc 582 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠)))
6356, 62mpbird 256 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)))
6452, 63eqeltrid 2833 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠)))
65 cmpcld 23326 . . . . . . . . . . . . . 14 (((𝐽t 𝑠) ∈ Comp ∧ (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
6651, 64, 65syl2anc 582 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
6750, 66eqeltrd 2829 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp)
68 oveq2 7434 . . . . . . . . . . . . . 14 (𝑣 = (𝑠𝐴) → ((𝐽t 𝐴) ↾t 𝑣) = ((𝐽t 𝐴) ↾t (𝑠𝐴)))
6968eleq1d 2814 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐴) → (((𝐽t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp))
7069rspcev 3611 . . . . . . . . . . . 12 (((𝑠𝐴) ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)) ∧ ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
7142, 67, 70syl2anc 582 . . . . . . . . . . 11 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
7271expr 455 . . . . . . . . . 10 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ (𝑠 ∈ 𝒫 𝑢𝑤𝐽)) → ((𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
7372rexlimdvva 3209 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → (∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
7410, 73mpd 15 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
7574anassrs 466 . . . . . . 7 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) ∧ 𝑦 ∈ (𝑢𝐴)) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
7675ralrimiva 3143 . . . . . 6 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) → ∀𝑦 ∈ (𝑢𝐴)∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
77 pweq 4620 . . . . . . . . 9 (𝑥 = (𝑢𝐴) → 𝒫 𝑥 = 𝒫 (𝑢𝐴))
7877ineq2d 4214 . . . . . . . 8 (𝑥 = (𝑢𝐴) → (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)))
7978rexeqdv 3324 . . . . . . 7 (𝑥 = (𝑢𝐴) → (∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp ↔ ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8079raleqbi1dv 3331 . . . . . 6 (𝑥 = (𝑢𝐴) → (∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp ↔ ∀𝑦 ∈ (𝑢𝐴)∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8176, 80syl5ibrcom 246 . . . . 5 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) → (𝑥 = (𝑢𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8281rexlimdva 3152 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (∃𝑢𝐽 𝑥 = (𝑢𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
834, 82sylbid 239 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽t 𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8483ralrimiv 3142 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ (𝐽t 𝐴)∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
85 isnlly 23393 . 2 ((𝐽t 𝐴) ∈ 𝑛-Locally Comp ↔ ((𝐽t 𝐴) ∈ Top ∧ ∀𝑥 ∈ (𝐽t 𝐴)∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
863, 84, 85sylanbrc 581 1 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cin 3948  wss 3949  𝒫 cpw 4606  {csn 4632   cuni 4912  cfv 6553  (class class class)co 7426  t crest 17409  Topctop 22815  Clsdccld 22940  neicnei 23021  Compccmp 23310  𝑛-Locally cnlly 23389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-fin 8974  df-fi 9442  df-rest 17411  df-topgen 17432  df-top 22816  df-topon 22833  df-bases 22869  df-cld 22943  df-nei 23022  df-cmp 23311  df-nlly 23391
This theorem is referenced by:  rellycmp  24903
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