Theorem lringring 20627

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

Step	Hyp	Ref	Expression
1		lringnzr 20626	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		nzrring 20599	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2149  Ringcrg 20315  NzRingcnzr 20595  LRingclring 20623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-ss 3930  df-nzr 20596  df-lring 20624

This theorem is referenced by:  lringuplu  20629  dflring2  33728