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Theorem loclly 22095
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 486 . . . . . . 7 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → Locally 𝐴 = 𝐴)
31, 2eleqtrrd 2896 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ Locally 𝐴)
4 simprr 772 . . . . . 6 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
5 llyrest 22093 . . . . . 6 ((𝑗 ∈ Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ Locally 𝐴)
63, 4, 5syl2anc 587 . . . . 5 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ Locally 𝐴)
76, 2eleqtrd 2895 . . . 4 ((Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
87restnlly 22090 . . 3 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
9 id 22 . . 3 (Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
108, 9eqtrd 2836 . 2 (Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
11 simprl 770 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗𝐴)
12 simpl 486 . . . . . . 7 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑛-Locally 𝐴 = 𝐴)
1311, 12eleqtrrd 2896 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑗 ∈ 𝑛-Locally 𝐴)
14 simprr 772 . . . . . 6 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → 𝑥𝑗)
15 nllyrest 22094 . . . . . 6 ((𝑗 ∈ 𝑛-Locally 𝐴𝑥𝑗) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1613, 14, 15syl2anc 587 . . . . 5 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝑛-Locally 𝐴)
1716, 12eleqtrd 2895 . . . 4 ((𝑛-Locally 𝐴 = 𝐴 ∧ (𝑗𝐴𝑥𝑗)) → (𝑗t 𝑥) ∈ 𝐴)
1817restnlly 22090 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = Locally 𝐴)
19 id 22 . . 3 (𝑛-Locally 𝐴 = 𝐴 → 𝑛-Locally 𝐴 = 𝐴)
2018, 19eqtr3d 2838 . 2 (𝑛-Locally 𝐴 = 𝐴 → Locally 𝐴 = 𝐴)
2110, 20impbii 212 1 (Locally 𝐴 = 𝐴 ↔ 𝑛-Locally 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  (class class class)co 7139  t crest 16689  Locally clly 22072  𝑛-Locally cnlly 22073
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863  df-rest 16691  df-topgen 16712  df-top 21502  df-topon 21519  df-bases 21554  df-nei 21706  df-lly 22074  df-nlly 22075
This theorem is referenced by:  topnlly  22099
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