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Theorem nzrring 19962
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
nzrring (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Proof of Theorem nzrring
StepHypRef Expression
1 eqid 2818 . . 3 (1r𝑅) = (1r𝑅)
2 eqid 2818 . . 3 (0g𝑅) = (0g𝑅)
31, 2isnzr 19960 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r𝑅) ≠ (0g𝑅)))
43simplbi 498 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wne 3013  cfv 6348  0gc0g 16701  1rcur 19180  Ringcrg 19226  NzRingcnzr 19958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-nzr 19959
This theorem is referenced by:  opprnzr  19966  nzrunit  19968  domnring  19997  domnchr  20607  uvcf1  20864  lindfind2  20890  frlmisfrlm  20920  nminvr  23205  deg1pw  24641  ply1nz  24642  ply1remlem  24683  ply1rem  24684  facth1  24685  fta1glem1  24686  fta1glem2  24687  zrhnm  31109  uvcn0  39029  0prjspnlem  39146  mon1pid  39683  mon1psubm  39684  nzrneg1ne0  44068  islindeps2  44466
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