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Theorem llyidm 23336
Description: Idempotence of the "locally" predicate, i.e. being "locally 𝐴 " is a local property. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llyidm Locally Locally 𝐴 = Locally 𝐴

Proof of Theorem llyidm
Dummy variables 𝑗 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 23320 . . . 4 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Top)
2 llyi 23322 . . . . . . 7 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑢𝑗 (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))
3 simprr3 1220 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑗t 𝑢) ∈ Locally 𝐴)
4 simprl 768 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑗)
5 ssidd 3998 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑢)
613ad2ant1 1130 . . . . . . . . . . . 12 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → 𝑗 ∈ Top)
76adantr 480 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑗 ∈ Top)
8 restopn2 23025 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
97, 4, 8syl2anc 583 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑢 ∈ (𝑗t 𝑢) ↔ (𝑢𝑗𝑢𝑢)))
104, 5, 9mpbir2and 710 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢 ∈ (𝑗t 𝑢))
11 simprr2 1219 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑦𝑢)
12 llyi 23322 . . . . . . . . 9 (((𝑗t 𝑢) ∈ Locally 𝐴𝑢 ∈ (𝑗t 𝑢) ∧ 𝑦𝑢) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
133, 10, 11, 12syl3anc 1368 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))
14 restopn2 23025 . . . . . . . . . . . 12 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
157, 4, 14syl2anc 583 . . . . . . . . . . 11 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) ↔ (𝑣𝑗𝑣𝑢)))
16 simpl 482 . . . . . . . . . . 11 ((𝑣𝑗𝑣𝑢) → 𝑣𝑗)
1715, 16syl6bi 253 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (𝑣 ∈ (𝑗t 𝑢) → 𝑣𝑗))
18 simprl 768 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑗)
19 simprr1 1218 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑢)
20 simprr1 1218 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → 𝑢𝑥)
2120adantr 480 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑥)
2219, 21sstrd 3985 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣𝑥)
23 velpw 4600 . . . . . . . . . . . . . 14 (𝑣 ∈ 𝒫 𝑥𝑣𝑥)
2422, 23sylibr 233 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
2518, 24elind 4187 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝑗 ∩ 𝒫 𝑥))
26 simprr2 1219 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑦𝑣)
277adantr 480 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑗 ∈ Top)
28 simplrl 774 . . . . . . . . . . . . . 14 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → 𝑢𝑗)
29 restabs 23013 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑣𝑢𝑢𝑗) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
3027, 19, 28, 29syl3anc 1368 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) = (𝑗t 𝑣))
31 simprr3 1220 . . . . . . . . . . . . 13 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)
3230, 31eqeltrrd 2826 . . . . . . . . . . . 12 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑗t 𝑣) ∈ 𝐴)
3325, 26, 32jca32 515 . . . . . . . . . . 11 ((((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) ∧ (𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
3433ex 412 . . . . . . . . . 10 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ((𝑣𝑗 ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))))
3517, 34syland 602 . . . . . . . . 9 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ((𝑣 ∈ (𝑗t 𝑢) ∧ (𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))))
3635reximdv2 3156 . . . . . . . 8 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → (∃𝑣 ∈ (𝑗t 𝑢)(𝑣𝑢𝑦𝑣 ∧ ((𝑗t 𝑢) ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
3713, 36mpd 15 . . . . . . 7 (((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) ∧ (𝑢𝑗 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝑗t 𝑢) ∈ Locally 𝐴))) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
382, 37rexlimddv 3153 . . . . . 6 ((𝑗 ∈ Locally Locally 𝐴𝑥𝑗𝑦𝑥) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
39383expb 1117 . . . . 5 ((𝑗 ∈ Locally Locally 𝐴 ∧ (𝑥𝑗𝑦𝑥)) → ∃𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
4039ralrimivva 3192 . . . 4 (𝑗 ∈ Locally Locally 𝐴 → ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴))
41 islly 23316 . . . 4 (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥𝑗𝑦𝑥𝑣 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑣 ∧ (𝑗t 𝑣) ∈ 𝐴)))
421, 40, 41sylanbrc 582 . . 3 (𝑗 ∈ Locally Locally 𝐴𝑗 ∈ Locally 𝐴)
4342ssriv 3979 . 2 Locally Locally 𝐴 ⊆ Locally 𝐴
44 llyrest 23333 . . . . 5 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
4544adantl 481 . . . 4 ((⊤ ∧ (𝑗 ∈ Locally 𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
46 llytop 23320 . . . . . 6 (𝑗 ∈ Locally 𝐴𝑗 ∈ Top)
4746ssriv 3979 . . . . 5 Locally 𝐴 ⊆ Top
4847a1i 11 . . . 4 (⊤ → Locally 𝐴 ⊆ Top)
4945, 48restlly 23331 . . 3 (⊤ → Locally 𝐴 ⊆ Locally Locally 𝐴)
5049mptru 1540 . 2 Locally 𝐴 ⊆ Locally Locally 𝐴
5143, 50eqssi 3991 1 Locally Locally 𝐴 = Locally 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wtru 1534  wcel 2098  wral 3053  wrex 3062  cin 3940  wss 3941  𝒫 cpw 4595  (class class class)co 7402  t crest 17371  Topctop 22739  Locally clly 23312
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17373  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-lly 23314
This theorem is referenced by: (None)
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