Theorem lringring 14207

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 e. LRing -> R e. Ring )

Proof of Theorem lringring

Step	Hyp	Ref	Expression
1		lringnzr 14206	. 2 |- ( R e. LRing -> R e. NzRing )
2		nzrring 14196	. 2 |- ( R e. NzRing -> R e. Ring )
3	1, 2	syl 14	1 |- ( R e. LRing -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2202   Ringcrg 14008  NzRingcnzr 14192  LRingclring 14203

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-in 3206  df-ss 3213  df-nzr 14193  df-lring 14204

This theorem is referenced by:  lringuplu  14209  aprcotr  14298  aprap  14299