MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nzrring Structured version   Visualization version   GIF version

Theorem nzrring 19658
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 2778 . . 3 (1r𝑅) = (1r𝑅)
2 eqid 2778 . . 3 (0g𝑅) = (0g𝑅)
31, 2isnzr 19656 . 2 (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r𝑅) ≠ (0g𝑅)))
43simplbi 493 1 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 2969  cfv 6135  0gc0g 16486  1rcur 18888  Ringcrg 18934  NzRingcnzr 19654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-nzr 19655
This theorem is referenced by:  opprnzr  19662  nzrunit  19664  domnring  19693  domnchr  20276  uvcf1  20535  lindfind2  20561  frlmisfrlm  20591  nminvr  22881  deg1pw  24317  ply1nz  24318  ply1remlem  24359  ply1rem  24360  facth1  24361  fta1glem1  24362  fta1glem2  24363  zrhnm  30611  mon1pid  38742  mon1psubm  38743  nzrneg1ne0  42884  islindeps2  43287
  Copyright terms: Public domain W3C validator