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Theorem nzrrng 16029
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 2296 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
2 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2isnzr 16027 . 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 1696    =/= wne 2459   ` cfv 5271   0gc0g 13416   Ringcrg 15353   1rcur 15355  NzRingcnzr 16025
This theorem is referenced by:  opprnzr  16032  nzrunit  16034  domnrng  16053  domnchr  16502  nminvr  18196  deg1pw  19522  ply1nz  19523  ply1remlem  19564  ply1rem  19565  facth1  19566  fta1glem1  19567  fta1glem2  19568  uvcf1  27344  lindfind2  27391  mon1pid  27627  mon1psubm  27628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-nzr 16026
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