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

Theorem nllyidm 22203
 Description: Idempotence of the "n-locally" predicate, i.e. being "n-locally 𝐴 " is a local property. (Use loclly 22201 to show 𝑛-Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴.) (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
nllyidm Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴

Proof of Theorem nllyidm
Dummy variables 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 22186 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ Top)
2 llyi 22188 . . . . . . 7 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))
3 simprr3 1220 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴)
4 simprl 770 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑗)
5 ssidd 3917 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑢)
6 simpl1 1188 . . . . . . . . . . . 12 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Locally 𝑛-Locally 𝐴)
76, 1syl 17 . . . . . . . . . . 11 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑗 ∈ Top)
8 restopn2 21891 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
97, 4, 8syl2anc 587 . . . . . . . . . 10 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
104, 5, 9mpbir2and 712 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
11 simprr2 1219 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑦𝑢)
12 nlly2i 22190 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 𝑛-Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
133, 10, 11, 12syl3anc 1368 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
14 restopn2 21891 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
157, 4, 14syl2anc 587 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
1615adantr 484 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗t 𝑢) ↔ (𝑧𝑗𝑧𝑢)))
177adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
18 simpr2l 1229 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑗)
19 simpr31 1260 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑧)
20 opnneip 21833 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑧𝑗𝑦𝑧) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
2117, 18, 19, 20syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧 ∈ ((nei‘𝑗)‘{𝑦}))
22 simpr32 1261 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑧𝑣)
23 simpr1 1191 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑢)
2423elpwid 4508 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
254adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
26 elssuni 4833 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑗𝑢 𝑗)
2725, 26syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢 𝑗)
2824, 27sstrd 3904 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 𝑗)
29 eqid 2758 . . . . . . . . . . . . . . . . . 18 𝑗 = 𝑗
3029ssnei2 21830 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑧 ∈ ((nei‘𝑗)‘{𝑦})) ∧ (𝑧𝑣𝑣 𝑗)) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
3117, 21, 22, 28, 30syl22anc 837 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((nei‘𝑗)‘{𝑦}))
32 simprr1 1218 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → 𝑢𝑥)
3332adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
3424, 33sstrd 3904 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
35 velpw 4502 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
3634, 35sylibr 237 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
3731, 36elind 4101 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥))
38 restabs 21879 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3917, 24, 25, 38syl3anc 1368 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
40 simpr33 1262 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)
4139, 40eqeltrrd 2853 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
4237, 41jca 515 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ (𝑣 ∈ 𝒫 𝑢 ∧ (𝑧𝑗𝑧𝑢) ∧ (𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))
43423exp2 1351 . . . . . . . . . . . . 13 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (𝑣 ∈ 𝒫 𝑢 → ((𝑧𝑗𝑧𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))))
4443imp 410 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → ((𝑧𝑗𝑧𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))))
4516, 44sylbid 243 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (𝑧 ∈ (𝑗t 𝑢) → ((𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴))))
4645rexlimdv 3207 . . . . . . . . . 10 ((((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) ∧ 𝑣 ∈ 𝒫 𝑢) → (∃𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))
4746expimpd 457 . . . . . . . . 9 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ((𝑣 ∈ 𝒫 𝑢 ∧ ∃𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝑗t 𝑣) ∈ 𝐴)))
4847reximdv2 3195 . . . . . . . 8 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → (∃𝑣 ∈ 𝒫 𝑢𝑧 ∈ (𝑗t 𝑢)(𝑦𝑧𝑧𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴))
4913, 48mpd 15 . . . . . . 7 (((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝑛-Locally 𝐴))) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
502, 49rexlimddv 3215 . . . . . 6 ((𝑗 ∈ Locally 𝑛-Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
51503expb 1117 . . . . 5 ((𝑗 ∈ Locally 𝑛-Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
5251ralrimivva 3120 . . . 4 (𝑗 ∈ Locally 𝑛-Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴)
53 isnlly 22183 . . . 4 (𝑗 ∈ 𝑛-Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑣) ∈ 𝐴))
541, 52, 53sylanbrc 586 . . 3 (𝑗 ∈ Locally 𝑛-Locally 𝐴𝑗 ∈ 𝑛-Locally 𝐴)
5554ssriv 3898 . 2 Locally 𝑛-Locally 𝐴 ⊆ 𝑛-Locally 𝐴
56 nllyrest 22200 . . . . 5 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
5756adantl 485 . . . 4 ((⊤ ∧ (𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
58 nllytop 22187 . . . . . 6 (𝑗 ∈ 𝑛-Locally 𝐴𝑗 ∈ Top)
5958ssriv 3898 . . . . 5 𝑛-Locally 𝐴 ⊆ Top
6059a1i 11 . . . 4 (⊤ → 𝑛-Locally 𝐴 ⊆ Top)
6157, 60restlly 22197 . . 3 (⊤ → 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴)
6261mptru 1545 . 2 𝑛-Locally 𝐴 ⊆ Locally 𝑛-Locally 𝐴
6355, 62eqssi 3910 1 Locally 𝑛-Locally 𝐴 = 𝑛-Locally 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ∩ cin 3859   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801  ‘cfv 6340  (class class class)co 7156   ↾t crest 16766  Topctop 21607  neicnei 21811  Locally clly 22178  𝑛-Locally cnlly 22179 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-en 8541  df-fin 8544  df-fi 8921  df-rest 16768  df-topgen 16789  df-top 21608  df-topon 21625  df-bases 21660  df-nei 21812  df-lly 22180  df-nlly 22181 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator