Theorem lringring 14000

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 13999	. 2 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2		nzrring 13989	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 14	1 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2177  Ringcrg 13802  NzRingcnzr 13985  LRingclring 13996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-in 3173  df-ss 3180  df-nzr 13986  df-lring 13997

This theorem is referenced by:  lringuplu  14002  aprcotr  14091  aprap  14092