MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domnnzr Structured version   Unicode version

Theorem domnnzr 18460
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 2420 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2420 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2420 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
41, 2, 3isdomn 18459 . 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 461 1  |-  ( R  e. Domn  ->  R  e. NzRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    = wceq 1437    e. wcel 1867   A.wral 2773   ` cfv 5592  (class class class)co 6296   Basecbs 15081   .rcmulr 15151   0gc0g 15298  NzRingcnzr 18422  Domncdomn 18445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-domn 18449
This theorem is referenced by:  domnring  18461  opprdomn  18466  abvn0b  18467  fidomndrng  18472  domnchr  19040  znidomb  19069  nrgdomn  21611  ply1domn  22979  fta1glem1  23023  fta1glem2  23024  fta1b  23027  lgsqrlem4  24174  idomrootle  35816  deg1mhm  35831
  Copyright terms: Public domain W3C validator