Theorem lringnz 14208

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 14206	. 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 14195	. 2 |- ( R e. NzRing -> .1. =/= .0. )
5	1, 4	syl 14	1 |- ( R e. LRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   =/= wne 2402   ` cfv 5326   0gc0g 13338   1rcur 13971  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-in1 619  ax-in2 620  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-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-nzr 14193  df-lring 14204

This theorem is referenced by:  aprap  14299