MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldllycmp Structured version   Visualization version   GIF version

Theorem cldllycmp 21208
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 21199.) (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 21186 . . 3 (𝐽 ∈ 𝑛-Locally Comp → 𝐽 ∈ Top)
2 resttop 20874 . . 3 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 488 . 2 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐽t 𝐴) ∈ Top)
4 elrest 16009 . . . 4 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝐴)))
5 simpll 789 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝐽 ∈ 𝑛-Locally Comp)
6 simprl 793 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑢𝐽)
7 inss1 3811 . . . . . . . . . . 11 (𝑢𝐴) ⊆ 𝑢
8 simprr 795 . . . . . . . . . . 11 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦 ∈ (𝑢𝐴))
97, 8sseldi 3581 . . . . . . . . . 10 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → 𝑦𝑢)
10 nlly2i 21189 . . . . . . . . . 10 ((𝐽 ∈ 𝑛-Locally Comp ∧ 𝑢𝐽𝑦𝑢) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
115, 6, 9, 10syl3anc 1323 . . . . . . . . 9 (((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) → ∃𝑠 ∈ 𝒫 𝑢𝑤𝐽 (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))
123ad2antrr 761 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝐴) ∈ Top)
131ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐽 ∈ Top)
14 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 ∈ (Clsd‘𝐽))
15 simprlr 802 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝐽)
16 elrestr 16010 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 ∈ (Clsd‘𝐽) ∧ 𝑤𝐽) → (𝑤𝐴) ∈ (𝐽t 𝐴))
1713, 14, 15, 16syl3anc 1323 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ (𝐽t 𝐴))
18 simprr1 1107 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝑤)
19 inss2 3812 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ⊆ 𝐴
20 simplrr 800 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢𝐴))
2119, 20sseldi 3581 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦𝐴)
2218, 21elind 3776 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑤𝐴))
23 opnneip 20833 . . . . . . . . . . . . . . 15 (((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ (𝐽t 𝐴) ∧ 𝑦 ∈ (𝑤𝐴)) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
2412, 17, 22, 23syl3anc 1323 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
25 simprr2 1108 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑤𝑠)
26 ssrin 3816 . . . . . . . . . . . . . . 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 3812 . . . . . . . . . . . . . . 15 (𝑠𝐴) ⊆ 𝐴
29 eqid 2621 . . . . . . . . . . . . . . . . . 18 𝐽 = 𝐽
3029cldss 20743 . . . . . . . . . . . . . . . . 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 20876 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → 𝐴 = (𝐽t 𝐴))
3313, 31, 32syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝐴 = (𝐽t 𝐴))
3428, 33syl5sseq 3632 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ⊆ (𝐽t 𝐴))
35 eqid 2621 . . . . . . . . . . . . . . 15 (𝐽t 𝐴) = (𝐽t 𝐴)
3635ssnei2 20830 . . . . . . . . . . . . . 14 ((((𝐽t 𝐴) ∈ Top ∧ (𝑤𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦})) ∧ ((𝑤𝐴) ⊆ (𝑠𝐴) ∧ (𝑠𝐴) ⊆ (𝐽t 𝐴))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
3712, 24, 27, 34, 36syl22anc 1324 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ ((nei‘(𝐽t 𝐴))‘{𝑦}))
38 simprll 801 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 𝑢)
3938elpwid 4141 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠𝑢)
40 ssrin 3816 . . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
4342inex1 4759 . . . . . . . . . . . . . . 15 (𝑠𝐴) ∈ V
4443elpw 4136 . . . . . . . . . . . . . 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 3776 . . . . . . . . . . . 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 20879 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝐴𝐴 ∈ (Clsd‘𝐽)) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
4913, 47, 14, 48syl3anc 1323 . . . . . . . . . . . . . 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 3811 . . . . . . . . . . . . . . . 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 20879 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (𝑠𝐴) ⊆ 𝑠𝑠 ∈ 𝒫 𝑢) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) = (𝐽t (𝑠𝐴)))
5313, 51, 38, 52syl3anc 1323 . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝐽t 𝑠) ∈ Comp)
56 incom 3783 . . . . . . . . . . . . . . 15 (𝑠𝐴) = (𝐴𝑠)
57 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝐴𝑠) = (𝐴𝑠)
58 ineq1 3785 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝐴 → (𝑣𝑠) = (𝐴𝑠))
5958eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝐴 → ((𝐴𝑠) = (𝑣𝑠) ↔ (𝐴𝑠) = (𝐴𝑠)))
6059rspcev 3295 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ (Clsd‘𝐽) ∧ (𝐴𝑠) = (𝐴𝑠)) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
6114, 57, 60sylancl 693 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠))
62 simplrl 799 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑢𝐽)
63 elssuni 4433 . . . . . . . . . . . . . . . . . . 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 3593 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → 𝑠 𝐽)
6629restcld 20886 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑠 𝐽) → ((𝐴𝑠) ∈ (Clsd‘(𝐽t 𝑠)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)(𝐴𝑠) = (𝑣𝑠)))
6713, 65, 66syl2anc 692 . . . . . . . . . . . . . . . 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 2702 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠)))
70 cmpcld 21115 . . . . . . . . . . . . . 14 (((𝐽t 𝑠) ∈ Comp ∧ (𝑠𝐴) ∈ (Clsd‘(𝐽t 𝑠))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
7155, 69, 70syl2anc 692 . . . . . . . . . . . . 13 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝑠) ↾t (𝑠𝐴)) ∈ Comp)
7254, 71eqeltrd 2698 . . . . . . . . . . . 12 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ (𝑢𝐽𝑦 ∈ (𝑢𝐴))) ∧ ((𝑠 ∈ 𝒫 𝑢𝑤𝐽) ∧ (𝑦𝑤𝑤𝑠 ∧ (𝐽t 𝑠) ∈ Comp))) → ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp)
73 oveq2 6612 . . . . . . . . . . . . . 14 (𝑣 = (𝑠𝐴) → ((𝐽t 𝐴) ↾t 𝑣) = ((𝐽t 𝐴) ↾t (𝑠𝐴)))
7473eleq1d 2683 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐴) → (((𝐽t 𝐴) ↾t 𝑣) ∈ Comp ↔ ((𝐽t 𝐴) ↾t (𝑠𝐴)) ∈ Comp))
7574rspcev 3295 . . . . . . . . . . . 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 692 . . . . . . . . . . 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 3031 . . . . . . . . 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 679 . . . . . . 7 ((((𝐽 ∈ 𝑛-Locally Comp ∧ 𝐴 ∈ (Clsd‘𝐽)) ∧ 𝑢𝐽) ∧ 𝑦 ∈ (𝑢𝐴)) → ∃𝑣 ∈ (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴))((𝐽t 𝐴) ↾t 𝑣) ∈ Comp)
8180ralrimiva 2960 . . . . . 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 4133 . . . . . . . . 9 (𝑥 = (𝑢𝐴) → 𝒫 𝑥 = 𝒫 (𝑢𝐴))
8382ineq2d 3792 . . . . . . . 8 (𝑥 = (𝑢𝐴) → (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘(𝐽t 𝐴))‘{𝑦}) ∩ 𝒫 (𝑢𝐴)))
8483rexeqdv 3134 . . . . . . 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 3135 . . . . . 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 3024 . . . 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 2959 . 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 21182 . 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 697 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  Clsdccld 20730  neicnei 20811  Compccmp 21099  𝑛-Locally cnlly 21178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-nei 20812  df-cmp 21100  df-nlly 21180
This theorem is referenced by:  rellycmp  22664
  Copyright terms: Public domain W3C validator