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Theorem loclly 23384
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 770 . . . . . . 7 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗𝐴)
2 simpl 482 . . . . . . 7 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → Locally 𝐴 = 𝐴)
31, 2eleqtrrd 2831 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ Locally 𝐴)
4 simprr 772 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
5 llyrest 23382 . . . . . 6 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
63, 4, 5syl2anc 583 . . . . 5 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
76, 2eleqtrd 2830 . . . 4 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
87restnlly 23379 . . 3 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9 id 22 . . 3 (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
108, 9eqtrd 2767 . 2 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11 simprl 770 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗𝐴)
12 simpl 482 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
1311, 12eleqtrrd 2831 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14 simprr 772 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
15 nllyrest 23383 . . . . . 6 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1613, 14, 15syl2anc 583 . . . . 5 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1716, 12eleqtrd 2830 . . . 4 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
1817restnlly 23379 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
19 id 22 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
2018, 19eqtr3d 2769 . 2 (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
2110, 20impbii 208 1 (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534  wcel 2099  (class class class)co 7414  t crest 17395  Locally clly 23361  𝑛-Locally cnlly 23362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-en 8958  df-fin 8961  df-fi 9428  df-rest 17397  df-topgen 17418  df-top 22789  df-topon 22806  df-bases 22842  df-nei 22995  df-lly 23363  df-nlly 23364
This theorem is referenced by:  topnlly  23388
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