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Theorem lringnz 13398
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 = (1r𝑅)
lringnz.2 0 = (0g𝑅)
Assertion
Ref Expression
lringnz (𝑅 ∈ LRing → 10 )

Proof of Theorem lringnz
StepHypRef Expression
1 lringnzr 13396 . 2 (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2 lringnz.1 . . 3 1 = (1r𝑅)
3 lringnz.2 . . 3 0 = (0g𝑅)
42, 3nzrnz 13388 . 2 (𝑅 ∈ NzRing → 10 )
51, 4syl 14 1 (𝑅 ∈ LRing → 10 )
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  wne 2357  cfv 5228  0gc0g 12722  1rcur 13196  NzRingcnzr 13385  LRingclring 13393
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-nzr 13386  df-lring 13394
This theorem is referenced by:  aprap  13438
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