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Theorem lringnz 13267
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
StepHypRef Expression
1 lringnzr 13265 . 2  |-  ( R  e. LRing  ->  R  e. NzRing )
2 lringnz.1 . . 3  |-  .1.  =  ( 1r `  R )
3 lringnz.2 . . 3  |-  .0.  =  ( 0g `  R )
42, 3nzrnz 13257 . 2  |-  ( R  e. NzRing  ->  .1.  =/=  .0.  )
51, 4syl 14 1  |-  ( R  e. LRing  ->  .1.  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148    =/= wne 2347   ` cfv 5215   0gc0g 12693   1rcur 13073  NzRingcnzr 13254  LRingclring 13262
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-nzr 13255  df-lring 13263
This theorem is referenced by:  aprap  13275
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