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Theorem domnnzr 16036
Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnnzr  |-  ( R  e. Domn  ->  R  e. NzRing )

Proof of Theorem domnnzr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2283 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdomn 16035 . 2  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  ( 0g `  R
)  ->  ( x  =  ( 0g `  R )  \/  y  =  ( 0g `  R ) ) ) ) )
54simplbi 446 1  |-  ( R  e. Domn  ->  R  e. NzRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400  NzRingcnzr 16009  Domncdomn 16021
This theorem is referenced by:  domnrng  16037  opprdomn  16042  abvn0b  16043  fidomndrng  16048  domnchr  16486  znidomb  16515  nrgdomn  18182  ply1domn  19509  fta1glem1  19551  fta1glem2  19552  fta1b  19555  lgsqrlem4  20583  idomrootle  27511  deg1mhm  27526
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  ax-nul 4149
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-ov 5861  df-domn 16025
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