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Theorem nzrring 19180
 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 2621 . . 3 (1r𝑅) = (1r𝑅)
2 eqid 2621 . . 3 (0g𝑅) = (0g𝑅)
31, 2isnzr 19178 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r𝑅) ≠ (0g𝑅)))
43simplbi 476 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987   ≠ wne 2790  ‘cfv 5847  0gc0g 16021  1rcur 18422  Ringcrg 18468  NzRingcnzr 19176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-nzr 19177 This theorem is referenced by:  opprnzr  19184  nzrunit  19186  domnring  19215  domnchr  19799  uvcf1  20050  lindfind2  20076  frlmisfrlm  20106  nminvr  22383  deg1pw  23784  ply1nz  23785  ply1remlem  23826  ply1rem  23827  facth1  23828  fta1glem1  23829  fta1glem2  23830  zrhnm  29792  mon1pid  37261  mon1psubm  37262  nzrneg1ne0  41154  islindeps2  41557
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