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Theorem cldllycmp 21346
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 21337.) (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 21324 . . 3 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2 resttop 21012 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 487 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
4 elrest 16135 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝐴)))
5 simpll 805 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6 simprl 809 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑢𝐽)
7 inss1 3866 . . . . . . . . . . 11 (𝑢𝐴) ⊆ 𝑢
8 simprr 811 . . . . . . . . . . 11 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦 ∈ (𝑢𝐴))
97, 8sseldi 3634 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦𝑢)
10 nlly2i 21327 . . . . . . . . . 10 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑢𝐽𝑦𝑢) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
115, 6, 9, 10syl3anc 1366 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
123ad2antrr 762 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝐴) ∈ Top)
131ad3antrrr 766 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐽 ∈ Top)
14 simpllr 815 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 ∈ (Clsd‘𝐽))
15 simprlr 820 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝐽)
16 elrestr 16136 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤𝐽) → (𝑤𝐴) ∈ (𝐽t 𝐴))
1713, 14, 15, 16syl3anc 1366 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ (𝐽t 𝐴))
18 simprr1 1129 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝑤)
19 inss2 3867 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ⊆ 𝐴
20 simplrr 818 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢𝐴))
2119, 20sseldi 3634 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝐴)
2218, 21elind 3831 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑤𝐴))
23 opnneip 20971 . . . . . . . . . . . . . . 15 (((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ (𝐽t 𝐴) ∧ 𝑦 ∈ (𝑤𝐴)) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
2412, 17, 22, 23syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
25 simprr2 1130 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝑠)
26 ssrin 3871 . . . . . . . . . . . . . . 15 (𝑤𝑠 → (𝑤𝐴) ⊆ (𝑠𝐴))
2725, 26syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ⊆ (𝑠𝐴))
28 inss2 3867 . . . . . . . . . . . . . . 15 (𝑠𝐴) ⊆ 𝐴
29 eqid 2651 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
3029cldss 20881 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ (Clsd‘𝐽) → 𝐴 𝐽)
3114, 30syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 𝐽)
3229restuni 21014 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → 𝐴 = (𝐽t 𝐴))
3313, 31, 32syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 = (𝐽t 𝐴))
3428, 33syl5sseq 3686 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ (𝐽t 𝐴))
35 eqid 2651 . . . . . . . . . . . . . . 15 (𝐽t 𝐴) = (𝐽t 𝐴)
3635ssnei2 20968 . . . . . . . . . . . . . 14 ((((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦})) ∧ ((𝑤𝐴) ⊆ (𝑠𝐴) ∧ (𝑠𝐴) ⊆ (𝐽t 𝐴))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
3712, 24, 27, 34, 36syl22anc 1367 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
38 simprll 819 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 𝑢)
3938elpwid 4203 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠𝑢)
40 ssrin 3871 . . . . . . . . . . . . . . 15 (𝑠𝑢 → (𝑠𝐴) ⊆ (𝑢𝐴))
4139, 40syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ (𝑢𝐴))
42 vex 3234 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
4342inex1 4832 . . . . . . . . . . . . . . 15 (𝑠𝐴) ∈ V
4443elpw 4197 . . . . . . . . . . . . . 14 ((𝑠𝐴) ∈ 𝒫 (𝑢𝐴) ↔ (𝑠𝐴) ⊆ (𝑢𝐴))
4541, 44sylibr 224 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ 𝒫 (𝑢𝐴))
4637, 45elind 3831 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)))
4728a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ 𝐴)
48 restabs 21017 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝐴𝐴 ∈ (Clsd‘𝐽)) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
4913, 47, 14, 48syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
50 inss1 3866 . . . . . . . . . . . . . . . 16 (𝑠𝐴) ⊆ 𝑠
5150a1i 11 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ 𝑠)
52 restabs 21017 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝑠𝑠 ∈ 𝒫 𝑢) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
5313, 51, 38, 52syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
5449, 53eqtr4d 2688 . . . . . . . . . . . . 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)))
55 simprr3 1131 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝑠) ∈ Comp)
56 incom 3838 . . . . . . . . . . . . . . 15 (𝑠𝐴) = (𝐴𝑠)
57 eqid 2651 . . . . . . . . . . . . . . . . 17 (𝐴𝑠) = (𝐴𝑠)
58 ineq1 3840 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (𝑣𝑠) = (𝐴𝑠))
5958eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ((𝐴𝑠) = (𝑣𝑠) ↔ (𝐴𝑠) = (𝐴𝑠)))
6059rspcev 3340 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴𝑠) = (𝐴𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
6114, 57, 60sylancl 695 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
62 simplrl 817 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑢𝐽)
63 elssuni 4499 . . . . . . . . . . . . . . . . . . 19 (𝑢𝐽𝑢 𝐽)
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑢 𝐽)
6539, 64sstrd 3646 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 𝐽)
6629restcld 21024 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠)))
6713, 65, 66syl2anc 694 . . . . . . . . . . . . . . . 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)))
6861, 67mpbird 247 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)))
6956, 68syl5eqel 2734 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠)))
70 cmpcld 21253 . . . . . . . . . . . . . 14 (((𝐽t 𝑠) ∈ Comp ∧ (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
7155, 69, 70syl2anc 694 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
7254, 71eqeltrd 2730 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp)
73 oveq2 6698 . . . . . . . . . . . . . 14 (𝑣 = (𝑠𝐴) → ((𝐽t 𝐴) ↾t 𝑣) = ((𝐽t 𝐴) ↾t (𝑠𝐴)))
7473eleq1d 2715 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐴) → (((𝐽t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp))
7574rspcev 3340 . . . . . . . . . . . 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)
7646, 72, 75syl2anc 694 . . . . . . . . . . 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)
7776expr 642 . . . . . . . . . 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))
7877rexlimdvva 3067 . . . . . . . . 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))
7911, 78mpd 15 . . . . . . . 8 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
8079anassrs 681 . . . . . . 7 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) ∧ 𝑦 ∈ (𝑢𝐴)) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
8180ralrimiva 2995 . . . . . 6 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) → ∀𝑦 ∈ (𝑢𝐴)∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
82 pweq 4194 . . . . . . . . 9 (𝑥 = (𝑢𝐴) → 𝒫 𝑥 = 𝒫 (𝑢𝐴))
8382ineq2d 3847 . . . . . . . 8 (𝑥 = (𝑢𝐴) → (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)))
8483rexeqdv 3175 . . . . . . 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))
8584raleqbi1dv 3176 . . . . . 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))
8681, 85syl5ibrcom 237 . . . . 5 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) → (𝑥 = (𝑢𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8786rexlimdva 3060 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (∃𝑢𝐽 𝑥 = (𝑢𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
884, 87sylbid 230 . . 3 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽t 𝐴) → ∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
8988ralrimiv 2994 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ (𝐽t 𝐴)∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
90 isnlly 21320 . 2 ((𝐽t 𝐴) ∈ 𝑛-Locally Comp ↔ ((𝐽t 𝐴) ∈ Top ∧ ∀𝑥 ∈ (𝐽t 𝐴)∀𝑦𝑥𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥)((𝐽t 𝐴) ↾t 𝑣) ∈ Comp))
913, 89, 90sylanbrc 699 1 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468  cfv 5926  (class class class)co 6690  t crest 16128  Topctop 20746  Clsdccld 20868  neicnei 20949  Compccmp 21237  𝑛-Locally cnlly 21316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-nei 20950  df-cmp 21238  df-nlly 21318
This theorem is referenced by:  rellycmp  22803
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