Theorem lringring 14041

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 14040	. 2 |- ( R e. LRing -> R e. NzRing )
2		nzrring 14030	. 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 2177   Ringcrg 13843  NzRingcnzr 14026  LRingclring 14037

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 3176  df-ss 3183  df-nzr 14027  df-lring 14038

This theorem is referenced by:  lringuplu  14043  aprcotr  14132  aprap  14133