Theorem lringnz 14032

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 14030	. 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 14019	. 2 |- ( R e. NzRing -> .1. =/= .0. )
5	1, 4	syl 14	1 |- ( R e. LRing -> .1. =/= .0. )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   =/= wne 2377   ` cfv 5280   0gc0g 13163   1rcur 13796  NzRingcnzr 14016  LRingclring 14027

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 615  ax-in2 616  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-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-nzr 14017  df-lring 14028

This theorem is referenced by:  aprap  14123