Theorem lringnz 14273

Description: A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
lringnz.1	|- .1. = ( 1r ` R )
lringnz.2	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
lringnz	|- ( R e. LRing -> .1. =/= .0. )

Proof of Theorem lringnz

Step	Hyp	Ref	Expression
1		lringnzr 14271	. 2 |- ( R e. LRing -> R e. NzRing )
2		lringnz.1	. . 3 |- .1. = ( 1r ` R )
3		lringnz.2	. . 3 |- .0. = ( 0g ` R )
4	2, 3	nzrnz 14260	. 2 |- ( R e. NzRing -> .1. =/= .0. )
5	1, 4	syl 14	1 |- ( R e. LRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   ` cfv 5333   0gc0g 13402   1rcur 14036  NzRingcnzr 14257  LRingclring 14268

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-nzr 14258  df-lring 14269

This theorem is referenced by:  aprap  14365