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Theorem lringring 13266
Description: A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
Assertion
Ref Expression
lringring (𝑅 ∈ LRing → 𝑅 ∈ Ring)

Proof of Theorem lringring
StepHypRef Expression
1 lringnzr 13265 . 2 (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2 nzrring 13258 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 14 1 (𝑅 ∈ LRing → 𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Ringcrg 13110  NzRingcnzr 13254  LRingclring 13262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3135  df-ss 3142  df-nzr 13255  df-lring 13263
This theorem is referenced by:  lringuplu  13268  aprcotr  13274  aprap  13275
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