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Theorem lringnzr 13265
Description: A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
Assertion
Ref Expression
lringnzr (𝑅 ∈ LRing β†’ 𝑅 ∈ NzRing)

Proof of Theorem lringnzr
Dummy variables π‘Ÿ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lring 13263 . . 3 LRing = {π‘Ÿ ∈ NzRing ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(+gβ€˜π‘Ÿ)𝑦) = (1rβ€˜π‘Ÿ) β†’ (π‘₯ ∈ (Unitβ€˜π‘Ÿ) ∨ 𝑦 ∈ (Unitβ€˜π‘Ÿ)))}
21ssrab3 3241 . 2 LRing βŠ† NzRing
32sseli 3151 1 (𝑅 ∈ LRing β†’ 𝑅 ∈ NzRing)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∨ wo 708   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  β€˜cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  1rcur 13073  Unitcui 13187  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-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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3135  df-ss 3142  df-lring 13263
This theorem is referenced by:  lringring  13266  lringnz  13267
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