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Theorem lringnz 13675
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 13673 . 2  |-  ( R  e. LRing  ->  R  e. NzRing )
2 lringnz.1 . . 3  |-  .1.  =  ( 1r `  R )
3 lringnz.2 . . 3  |-  .0.  =  ( 0g `  R )
42, 3nzrnz 13662 . 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 1364    e. wcel 2164    =/= wne 2364   ` cfv 5246   0gc0g 12857   1rcur 13439  NzRingcnzr 13659  LRingclring 13670
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5207  df-fv 5254  df-nzr 13660  df-lring 13671
This theorem is referenced by:  aprap  13766
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