Theorem lringring 20650

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 20649	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		nzrring 20622	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2143  Ringcrg 20319  NzRingcnzr 20618  LRingclring 20646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-ss 3922  df-nzr 20619  df-lring 20647

This theorem is used by:  lringuplu  20652  dflring2  33792