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Theorem domnnzr 16357
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 2438 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2438 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2438 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdomn 16356 . 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 448 1  |-  ( R  e. Domn  ->  R  e. NzRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532   0gc0g 13725  NzRingcnzr 16330  Domncdomn 16342
This theorem is referenced by:  domnrng  16358  opprdomn  16363  abvn0b  16364  fidomndrng  16369  domnchr  16815  znidomb  16844  nrgdomn  18709  ply1domn  20048  fta1glem1  20090  fta1glem2  20091  fta1b  20094  lgsqrlem4  21130  idomrootle  27490  deg1mhm  27505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-domn 16346
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