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Theorem domnnzr 17755
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 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2467 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdomn 17754 . 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 460 1  |-  ( R  e. Domn  ->  R  e. NzRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   0gc0g 14698  NzRingcnzr 17716  Domncdomn 17739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-domn 17743
This theorem is referenced by:  domnrng  17756  opprdomn  17761  abvn0b  17762  fidomndrng  17767  domnchr  18376  znidomb  18407  nrgdomn  21007  ply1domn  22351  fta1glem1  22393  fta1glem2  22394  fta1b  22397  lgsqrlem4  23444  idomrootle  30984  deg1mhm  30999
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