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

Proof of Theorem lringnz
StepHypRef Expression
1 lringnzr 13673 . 2 (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
2 lringnz.1 . . 3 1 = (1r𝑅)
3 lringnz.2 . . 3 0 = (0g𝑅)
42, 3nzrnz 13662 . 2 (𝑅 ∈ NzRing → 10 )
51, 4syl 14 1 (𝑅 ∈ LRing → 10 )
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  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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