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Theorem lringring 20442
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 20441 . 2 (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2 nzrring 20418 . 2 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
31, 2syl 17 1 (𝑅 ∈ LRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Ringcrg 20138  NzRingcnzr 20414  LRingclring 20438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-nzr 20415  df-lring 20439
This theorem is referenced by:  lringuplu  20444
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