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Theorem lringring 13674
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 13673 . 2 (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2 nzrring 13663 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 14 1 (𝑅 ∈ LRing → 𝑅 ∈ Ring)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2164  Ringcrg 13476  NzRingcnzr 13659  LRingclring 13670
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-in 3159  df-ss 3166  df-nzr 13660  df-lring 13671
This theorem is referenced by:  lringuplu  13676  aprcotr  13765  aprap  13766
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