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Theorem nzrring 20034
 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 2824 . . 3 (1r𝑅) = (1r𝑅)
2 eqid 2824 . . 3 (0g𝑅) = (0g𝑅)
31, 2isnzr 20032 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r𝑅) ≠ (0g𝑅)))
43simplbi 501 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115   ≠ wne 3014  ‘cfv 6343  0gc0g 16713  1rcur 19251  Ringcrg 19297  NzRingcnzr 20030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-nzr 20031 This theorem is referenced by:  opprnzr  20038  nzrunit  20040  domnring  20069  domnchr  20231  uvcf1  20488  lindfind2  20514  frlmisfrlm  20544  nminvr  23282  deg1pw  24728  ply1nz  24729  ply1remlem  24770  ply1rem  24771  facth1  24772  fta1glem1  24773  fta1glem2  24774  krull  31025  zrhnm  31271  uvcn0  39393  0prjspnlem  39533  mon1pid  40070  mon1psubm  40071  nzrneg1ne0  44424  islindeps2  44823
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