Theorem lringring 20794

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 20793	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		nzrring 20766	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Ringcrg 20459  NzRingcnzr 20762  LRingclring 20790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-ss 3916  df-nzr 20763  df-lring 20791

This theorem is used by:  lringuplu  20796  dflring2  34025