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Theorem nzrring 20747
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 2733 . . 3 (1r𝑅) = (1r𝑅)
2 eqid 2733 . . 3 (0g𝑅) = (0g𝑅)
31, 2isnzr 20745 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r𝑅) ≠ (0g𝑅)))
43simplbi 499 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2940  cfv 6497  0gc0g 17326  1rcur 19918  Ringcrg 19969  NzRingcnzr 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-nzr 20744
This theorem is referenced by:  opprnzr  20751  nzrunit  20753  domnring  20782  domnchr  20951  uvcf1  21214  lindfind2  21240  frlmisfrlm  21270  nminvr  24049  deg1pw  25501  ply1nz  25502  ply1remlem  25543  ply1rem  25544  facth1  25545  fta1glem1  25546  fta1glem2  25547  prmidl0  32271  krull  32288  ply1moneq  32335  zrhnm  32607  isdomn4  40670  uvcn0  40773  0prjspnlem  41004  mon1pid  41575  mon1psubm  41576  nzrneg1ne0  46253  islindeps2  46650
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