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Theorem nzrrng 16013
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
nzrrng  |-  ( R  e. NzRing  ->  R  e.  Ring )

Proof of Theorem nzrrng
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
2 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2isnzr 16011 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  ( 0g
`  R ) ) )
43simplbi 446 1  |-  ( R  e. NzRing  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    =/= wne 2446   ` cfv 5255   0gc0g 13400   Ringcrg 15337   1rcur 15339  NzRingcnzr 16009
This theorem is referenced by:  opprnzr  16016  nzrunit  16018  domnrng  16037  domnchr  16486  nminvr  18180  deg1pw  19506  ply1nz  19507  ply1remlem  19548  ply1rem  19549  facth1  19550  fta1glem1  19551  fta1glem2  19552  uvcf1  27241  lindfind2  27288  mon1pid  27524  mon1psubm  27525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-nzr 16010
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