Theorem lringring 14152

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 14151	. 2 |- ( R e. LRing -> R e. NzRing )
2		nzrring 14141	. 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 2200   Ringcrg 13954  NzRingcnzr 14137  LRingclring 14148

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-in 3203  df-ss 3210  df-nzr 14138  df-lring 14149

This theorem is referenced by:  lringuplu  14154  aprcotr  14243  aprap  14244