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Theorem loclly 21230
 Description: If 𝐴 is a local property, then both Locally 𝐴 and 𝑛-Locally 𝐴 simplify to 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
loclly (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)

Proof of Theorem loclly
Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 793 . . . . . . 7 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗𝐴)
2 simpl 473 . . . . . . 7 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → Locally 𝐴 = 𝐴)
31, 2eleqtrrd 2701 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ Locally 𝐴)
4 simprr 795 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
5 llyrest 21228 . . . . . 6 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
63, 4, 5syl2anc 692 . . . . 5 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
76, 2eleqtrd 2700 . . . 4 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
87restnlly 21225 . . 3 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9 id 22 . . 3 (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
108, 9eqtrd 2655 . 2 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11 simprl 793 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗𝐴)
12 simpl 473 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
1311, 12eleqtrrd 2701 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14 simprr 795 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
15 nllyrest 21229 . . . . . 6 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1613, 14, 15syl2anc 692 . . . . 5 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1716, 12eleqtrd 2700 . . . 4 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
1817restnlly 21225 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
19 id 22 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
2018, 19eqtr3d 2657 . 2 (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
2110, 20impbii 199 1 (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  (class class class)co 6615   ↾t crest 16021  Locally clly 21207  𝑛-Locally cnlly 21208 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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-er 7702  df-en 7916  df-fin 7919  df-fi 8277  df-rest 16023  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-nei 20842  df-lly 21209  df-nlly 21210 This theorem is referenced by:  topnlly  21234
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