Theorem lringring 20570

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 20569	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		nzrring 20544	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2108  Ringcrg 20262  NzRingcnzr 20540  LRingclring 20566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-ss 3993  df-nzr 20541  df-lring 20567

This theorem is referenced by:  lringuplu  20572