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Theorem domnnzr 16052
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 2296 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2296 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdomn 16051 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416  NzRingcnzr 16025  Domncdomn 16037
This theorem is referenced by:  domnrng  16053  opprdomn  16058  abvn0b  16059  fidomndrng  16064  domnchr  16502  znidomb  16531  nrgdomn  18198  ply1domn  19525  fta1glem1  19567  fta1glem2  19568  fta1b  19571  lgsqrlem4  20599  idomrootle  27614  deg1mhm  27629
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  ax-nul 4165
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-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-ov 5877  df-domn 16041
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