Theorem lringring 13927

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 13926	. 2 |- ( R e. LRing -> R e. NzRing )
2		nzrring 13916	. 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 2175   Ringcrg 13729  NzRingcnzr 13912  LRingclring 13923

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-in 3171  df-ss 3178  df-nzr 13913  df-lring 13924

This theorem is referenced by:  lringuplu  13929  aprcotr  14018  aprap  14019